Digital Object Identifier (DOI) 101007s00220-009-0901-6Commun Math Phys 292 871ndash912 (2009) Communications in

MathematicalPhysics

Cardy Algebras and Sewing Constraints I

Liang Kong12 Ingo Runkel3

1 Max-Planck-Institut fuumlr Mathematik Vivatsgasse 7 53111 Bonn GermanyE-mail kongmpim-bonnmpgde

2 Hausdorff Research Institute for Mathematics Poppelsdorfer Allee 45 53115 Bonn Germany3 Department of Mathematics Kingrsquos College London Strand London WC2R 2LS United Kingdom

E-mail ingorunkelkclacuk

Received 28 March 2009 Accepted 7 June 2009Published online 13 August 2009 ndash copy Springer-Verlag 2009

Abstract This is part one of a two-part work that relates two different approaches totwo-dimensional open-closed rational conformal field theory In part one we review thedefinition of a Cardy algebra which captures the necessary consistency conditions of thetheory at genus 0 and 1 We investigate the properties of these algebras and prove unique-ness and existence theorems One implication is that under certain natural assumptionsevery rational closed CFT is extendable to an open-closed CFT The relation of Cardyalgebras to the solutions of the sewing constraints is the topic of part two

Contents

1 Introduction and Summary 8712 Preliminaries on Tensor Categories 875

21 Tensor categories and (co)lax tensor functors 87522 Algebras in tensor categories 87823 Modular tensor categories 88424 The functors T and R 885

3 Cardy Algebras 89131 Modular invariance 89132 Two definitions 89533 Uniqueness and existence theorems 899

A Appendix 905A1 Proof of Lemma 27 905A2 Proof of Lemma 320 906

1 Introduction and Summary

This is part I of a two-part work which relates two different approaches to two-dimen-sional open-closed rational conformal field theory (CFT)

872 L Kong I Runkel

The first approach uses a three-dimensional topological field theory to express cor-relators of the open-closed CFT [FeFRSFj] Here one starts from a modular tensorcategory which defines a three-dimensional topological field theory [RTT] and froma special symmetric Frobenius algebra in this modular tensor category To each open-closed world sheet X one assigns a 3-bordism MX with embedded ribbon graph con-structed from this Frobenius algebra To the boundary of MX the topological field theoryassigns a vector space B(X) and to MX itself a vector CX isin B(X) One proves thatthis collection of vectors CX provides a so-called solution to the sewing constraints [Fj]If the modular tensor category is the category of representations of a suitable vertexoperator algebra the spaces B(X) are spaces of conformal blocks and the CX are thecorrelators of an open-closed CFT In this approach one thus makes an ansatz for thecorrelators on all world sheets simultaneously and then proves that they obey the neces-sary consistency conditions The relation to CFT rests on convergence and factorisationproperties of higher genus conformal blocks and the precise list of conditions the vertexoperator algebra has to fulfill for these properties to hold is not known However froma physical perspective one expects that interesting classes of models [WFK] will haveall the necessary properties

The second approach uses the theory of vertex operator algebras to construct directlythe correlators of the genus 0 and genus 1 open-closed CFT [HK1HK2K3] More pre-cisely in this approach one uses a notion of CFT defined in [K3 Sect 1] (and called partialCFT1 there) where one glues Riemann surfaces around punctures with local coordinatesas in [VH1] instead of gluing around parametrised circles as in [Se] This approach isbased on the precise relation between genus-0 CFT and vertex operator algebras [H1]and on the fact that the category of modules over a rational vertex operator algebra isa modular tensor category [HLH2] Let us call a vertex operator algebra rational if itsatisfies the conditions in [H2 Sect 1] If one analyses the consistency conditions of agenus-01 open-closed CFT one arrives at a structure called Cardy CV |CV otimesV - algebrain [K3] It is formulated in purely categorical terms in the categories CV and CV otimesV ofmodules over the rational vertex operator algebras V and V otimes V respectively Cardyalgebras (Definition 37) are the central objects in part I of this work and we will describetheir relation to CFT in slightly more detail below The data in a Cardy algebra amountsto an open-closed CFT on a generating set of world sheets from which the entire CFTcan be obtained by repeated gluing The conditions on this data are necessary for thisprocedure to give a consistent genus-01 open-closed CFT

The two approaches just outlined start at opposite ends of the same problem Inboth cases the difficulty to obtain a complete answer lies in the lack of control over theproperties of higher genus conformal blocks Nonetheless both approaches give rise tonotions formulated in entirely categorical terms and we can compare the structures onefinds In part II we will come to the satisfying conclusion that giving a solution to thesewing constraints is essentially equivalent in a sense made precise in part II to givinga Cardy algebra

To motivate the notion of a Cardy algebra and our interest in it we would like to outlinehow it emerges when formulating closed CFT and open-closed CFT at genus-01 in thelanguage of vertex operator algebras The next one and a half pages together with a few

1 The qualifier lsquopartialrsquo refers to the fact that the gluing of punctures is only defined if the coordinates ζ1ζ2 around two punctures can be analytically extended to a large enough region containing no other puncturesso that the identification ζ1 sim 1ζ2 is well-defined That is if ζ1 can be extended to a disc of radius r then ζ2must be defined on a disc of radius greater than 1r Both discs must not contain further punctures

Cardy Algebras and Sewing Constraints I 873

remarks in the main text are the only places where we make reference to vertex operatoralgebras The reader who is not familiar with this structure is invited to skip ahead

All types of field algebras occurring below are called self-dual if they are endowedwith non-degenerate invariant bilinear forms

A genus-0 closed CFT is equivalent to an algebra over a partial dioperad consisting ofspheres with arbitrary in-coming and out-going punctures The dioperad structure allowsto compose one in-going and one out-going puncture of distinct spheres so that the resultis again a sphere Such an algebra with additional natural properties is canonically equiv-alent to a so-called self-dual conformal full field algebra [HK2K1] A conformal fullfield algebra contains chiral and anti-chiral parts the easiest nontrivial example is givenby V otimesV where V is a vertex operator algebra A conformal full field algebra containingV otimes V as a subalgebra is called a conformal full field algebra over V otimes V When V isrational the category of self-dual conformal full field algebras over V otimesV is isomorphicto the category of commutative symmetric Frobenius algebras in CV otimesV [K1 Thm 415]

Similarly a genus-0 open CFT is an algebra over a partial dioperad consisting ofdisks with an arbitrary number of in-coming and out-going boundary punctures Suchan algebra with additional natural properties is canonically equivalent to a self-dualopen-string vertex operator algebra as defined in [HK1] A vertex operator algebra V isnaturally an open-string vertex operator algebra An open-string vertex operator algebracontaining V as a subalgebra in its meromorphic centre is called an open-string vertexoperator algebra over V When V is rational the category of self-dual open-string vertexoperator algebras over V is isomorphic to the category of symmetric Frobenius algebrasin CV see [HK1 Thm 43] and [K3 Thm 610]

Finally a genus-0 open-closed CFT is an algebra over the Swiss-cheese partial dio-perad which consists of disks with both interior punctures and boundary puncturesand is equipped with an action of the partial spheres dioperad Such an algebra can beconstructed from a so-called self-dual open-closed field algebra [K2] It consists of aself-dual conformal full field algebra Acl a self-dual open-string vertex operator algebraAop and interactions between Acl and Aop satisfying certain compatibility conditionsNamely if Acl is defined over V otimes V and Aop over V one requires that the boundarycondition on a disc is V -invariant in the sense that both the chiral copy V otimes 1 and theanti-chiral copy 1 otimes V of V in Acl give the copy of V in Aop in the limit of the insertionpoint approaching a point on the boundary of the disc [K2 Def 125] An open-closedfield algebra with V -invariant boundary condition is called an open-closed field algebraover V When V is rational the category of self-dual open-closed field algebras overV is isomorphic to the category of triples (Aop|Acl ιcl-op) where Acl is a commuta-tive symmetric Frobenius CV otimesV -algebra Aop a symmetric Frobenius CV -algebra andιcl-op an algebra homomorphism T (Acl) rarr Aop satisfying a centre condition (given in(320) below) see [K2 Thm 314] and [K3 Sect 62] Here T CV otimesV rarr CV is theHuang-Lepowsky tensor product functor [HL]

The genus-1 theory does not provide new data as it is determined by taking tracesof genus-0 correlators but it does provide two additional consistency conditions themodular invariance condition for one-point correlators on the torus [So] and the Cardycondition for boundary-two-point correlators on the annulus [C2Lw] Their categor-ical formulations have been worked out in [HK3K3] Adding them to the axioms ofa self-dual open-closed field algebra over V finally results in the notion of a CardyCV |CV otimesV -algebra One can prove that the category of self-dual open-closed field alge-bras over a rational vertex operator algebra V satisfying the two genus-1 consistencyconditions is isomorphic to the category of Cardy CV |CV otimesV -algebras [K3 Thm 615]

874 L Kong I Runkel

If V is rational then so is V otimes V [DMZHK2] Thus both CV and CV otimesV aremodular tensor categories In fact CV otimesV sim= CV (CV )minus (see [FHL Thm 474] and[DMZ Thm 27]) where the minus sign relates to the particular braiding used for CV otimesV Namely for a given modular tensor category D we denote by Dminus the modular tensorcategory obtained from D by inverting braiding and twist We will also sometimes writeD+ for D The product amounts to taking direct sums of pairs of objects and tensorproducts of morphisms spaces The definition of a Cardy algebra can be stated in a waythat no longer makes reference to the vertex operator algebra V and therefore makessense in an arbitrary modular tensor category C Abbreviating C2plusmn equiv C+ Cminus this leadsto the definition of a Cardy C|C2plusmn-algebra

The relation to genus-01 open-closed CFT outlined above is the main motivation forour interest in Cardy C|C2plusmn-algebras In part I of this work we investigate how much onecan learn about Cardy algebras in the categorical setting and without the assumptionthat the modular tensor category C is given by CV for some V We briefly summariseour approach and results below

In Sect 21ndash23 we recall some basic notions we will need such as (co)lax tensorfunctors Frobenius functors and modular tensor categories In Sect 24 we study thefunctor T C2plusmn rarr C which is defined by the tensor product on C via T (oplusi Ai times Bi ) =oplusi Ai otimes Bi for Ai Bi isin C Using the braiding of C one can turn T into a tensor functorA tensor functor is automatically also a Frobenius functor and so takes a Frobeniusalgebra A in its domain category to a Frobenius algebra F(A) in its target category

An important object in this work is the functor R C rarr C2plusmn also defined in Sect 24We show that R is left and right adjoint to T As a consequence R is automatically alax and colax tensor functor but it is in general not a tensor functor However we willshow that it is still a Frobenius functor and so takes Frobenius algebras in C to Frobeniusalgebras in C2plusmn In fact it also preserves the properties simple special and symmetricof a Frobenius algebra In the case C = CV the functor R CV rarr CV otimesV was firstconstructed in [Li1Li2] using techniques from vertex operator algebras This motivatedthe present construction and notation The functor R was also considered in a slightlydifferent context in [ENO2]

The above results imply that R and T form an ambidextrous adjunction and we willuse this adjunction to transport algebraic structures between C and C2plusmn For examplethe algebra homomorphism ιcl-op T (Acl) rarr Aop in C is transported to an algebrahomomorphism ιcl-op Acl rarr R(Aop) in C2plusmn This gives rise to an alternative definitionof a Cardy C|C2plusmn-algebra as a triple (Aop|Acl ιcl-op)

To prepare the definition of a Cardy algebra in Sect 31 we discuss the so-calledmodular invariance condition for algebras in C2plusmn (Definition 31 below) We show thatwhen Acl is simple the modular invariance condition can be replaced by an easier con-dition on the quantum dimension of Acl (namely the dimension of Acl has to be that ofthe modular tensor category C) see Theorem 34

In Sect 32 we give the two definitions of a Cardy algebra and prove their equiva-lence Section 33 contains our main results We first show that for each special symmet-ric Frobenius algebra A in C (see Sect 22 for the definition of special) one obtains aCardy algebra (A|Z(A) e) where Z(A) is the full centre of A (Theorem 318) Thefull centre [Fj Def 49] is a subobject of R(A) and e Z(A) rarr R(A) is the canonicalembedding Next we prove a uniqueness theorem (Theorem 321) which states that if(Aop|Acl ιcl-op) is a Cardy algebra such that dim Aop = 0 and Acl is simple then Aopis special and (Aop|Acl ιcl-op) is isomorphic to (Aop|Z(Aop) e) When combined withpart II of this work this result amounts to [Fj Thm 426] and provides an alternative

Cardy Algebras and Sewing Constraints I 875

(and shorter) proof Finally we show that for every simple modular invariant commu-tative symmetric Frobenius algebra Acl in C2plusmn there exists a simple special symmetricFrobenius algebra Aop and an algebra homomorphism ιcl-op Acl rarr R(Aop) such that(Aop Acl ιcl-op) is a Cardy algebra (Theorem 322) This theorem is closely related toa result announced in [Muuml2] and provides an independent proof in the framework ofCardy algebras

In physical terms these two theorems mean that a rational open-closed CFT with aunique closed vacuum state can be uniquely reconstructed from its correlators involvingonly discs with boundary punctures and that every closed CFT with unique vacuum andleftright rational chiral algebra V otimes V occurs as part of such an open-closed CFT

2 Preliminaries on Tensor Categories

In this section we review some basic facts of tensor categories and fix our conventionsand notations along the way

21 Tensor categories and (co)lax tensor functors In a tensor (or monoidal) categoryC with tensor product bifunctor otimes and unit object 1 for U VW isin C we denote the

associator U otimes (V otimes W )sim=minusrarr (U otimes V ) otimes W by αUVW the left unit isomorphism

1 otimes Usim=minusrarr U by lU and the right unit isomorphism U otimes 1

sim=minusrarr U by rU If C is braidedfor U V isin C we write the braiding isomorphism as cUV U otimes V rarr V otimes U

Let C1 and C2 be two tensor categories with units 11 and 12 respectively For simplic-ity we will often write otimes α l r for the data of both C1 and C2 Lax and colax tensorfunctors are defined as follows see eg [Y Ch I3] or [Ln Ch I12]

Definition 21 A lax tensor functor G C1 rarr C2 is a functor equipped with a mor-phism φG

0 12 rarr G(11) in C2 and a natural transformation φG2 otimes(G times G) rarr G otimes

such that the following three diagrams commute

G(A)otimes (G(B)otimes G(C))α

idG(A)otimesφG2

(G(A)otimes G(B))otimes G(C)

φG2 otimesidG(C)

G(A)otimes G(B otimes C)

φG2

G(A otimes B)otimes G(C)

φG2

G(A otimes (B otimes C))

G(α) G((A otimes B)otimes C)

(21)

12 otimes G(A)

φG0 otimesidG(A)

lG(A) G(A)

G(lminus1A )

G(11)otimes G(A)

φG2 G(11 otimes A)

G(A)otimes 12

idG(A)otimesφG0

rG(A) G(A)

G(rminus1A )

G(A)otimes G(11)

φG2 G(A otimes 11)

(22)

Definition 22 A colax tensor functor is a functor F C1 rarr C2 equipped with a mor-phismψ F

0 F(11) rarr 12 in C2 and a natural transformationψ F2 Fotimes rarr otimes(F times F)

such that the following three diagrams commute

876 L Kong I Runkel

F(A)otimes (F(B)otimes F(C))α (F(A)otimes F(B))otimes F(C)

F(A)otimes F(B otimes C)

idF(A)otimesψ F2

F(A otimes B)otimes F(C)

ψF2 otimesidF(C)

F(A otimes (B otimes C))F(α)

ψF2

F((A otimes B)otimes C)

ψF2

(23)

12 otimes F(A) F(A)lminus1F(A)

F(11)otimes F(A)

ψF0 otimesidF(A)

F(11 otimes A)ψF

2

F(lA)

F(A)otimes 12 F(A)rminus1

F(A)

F(A)otimes F(11)

idF(A)otimesψ F0

F(A otimes 11)

F(rA)

ψF2

(24)

We denote a lax tensor functor by (G φG2 φ

G0 ) or just G and a colax tensor functor

by (F ψ F2 ψ

F0 ) or F

Definition 23 A tensor functor T C1 rarr C2 is a lax tensor functor (T φT2 φ

T0 ) such

that φT0 φ

T2 are both isomorphisms

A tensor functor (T φT2 φ

T0 ) is automatically a colax tensor functor (T ψT

2 ψT0 )

with ψT0 = (φT

0 )minus1 and ψT

2 = (φT2 )

minus1In the next section we will discuss algebras in tensor categories The defining proper-

ties (21) and (22) of a lax tensor functor are analogues of the associativity the left-unitand the right-unit properties of an algebra Indeed a lax tensor functor G C1 rarr C2maps a C1-algebra to a C2-algebra Similarly (23) and (24) are analogues of the coasso-ciativity the left-counit and the right-counit properties of a coalgebra and a colax tensorfunctor F C1 rarr C2 maps a C1-coalgebra to a C2-coalgebra We will later make useof functors that take Frobenius algebras to Frobenius algebras This requires a strongercondition than being lax and colax and leads to the notion of a lsquofunctor with Frobeniusstructurersquo or lsquoFrobenius monoidal functorrsquo [SzDPP] which we will simply refer to asFrobenius functor

Definition 24 A Frobenius functor F C1 rarr C2 is a tuple F equiv (F φF2 φ

F0 ψ

F2 ψ

F0 )

such that (F φF2 φ

F0 ) is a lax tensor functor (F ψ F

2 ψF0 ) is a colax tensor functor

and such that the following two diagrams commute

F(A)otimes (F(B)otimes F(C))α (F(A)otimes F(B))otimes F(C)

φF2 otimesidF(C)

F(A)otimes F(B otimes C)

idF(A)otimesψ F2

φF2

F(A otimes B)otimes F(C)

F(A otimes (B otimes C))F(α) F((A otimes B)otimes C)

ψF2

(25)

Cardy Algebras and Sewing Constraints I 877

F(A)otimes (F(B)otimes F(C))

idF(A)otimesφF2

(F(A)otimes F(B))otimes F(C)αminus1

F(A)otimes F(B otimes C) F(A otimes B)otimes F(C)

ψF2 otimesidF(C)

φF2

F(A otimes (B otimes C))

ψF2

F((A otimes B)otimes C)F(αminus1)

(26)

Proposition 25 If (F φF2 φ

F0 ) is a tensor functor then F is a Frobenius functor with

ψ F0 = (φF

0 )minus1 and ψ F

2 = (φF2 )

minus1

Proof Since F is a tensor functor it is lax and colax If we replace ψ F2 by (φF

2 )minus1 in

(25) and (26) both commuting diagrams are equivalent to (21) which holds becauseF is lax Thus F is a Frobenius functor

The converse statement does not hold For example the functor R which we definein Sect 24 is Frobenius but not tensor

Let us recall the notion of adjunctions and adjoint functors [Ma Ch IV1]

Definition 26 An adjunction from C1 to C2 is a triple 〈FG χ〉 where F and G arefunctors

F C1 rarr C2 G C2 rarr C1

and χ is a natural isomorphism which assigns to each pair of objects A1 isin C1 A2 isin C2a bijective map

χA1A2 HomC2(F(A1) A2)sim=minusminusrarr HomC1(A1G(A2))

which is natural in both A1 and A2 F is called a left-adjoint of G and G is called aright-adjoint of F

For simplicity we will often abbreviate χA1A2 as χ Associated to each adjunction

〈FG χ〉 there are two natural transformations idC1

δminusrarr G F and FGρminusrarr idC2 where

idC1 and idC2 are identity functors given by

δA1 = χ(idF(A1)) ρA2 = χminus1(idG(A2)) (27)

for Ai isin Ci i = 1 2 They satisfy the following two identities

GδGminusrarr G FG

Gρminusrarr G = GidGminusminusrarr G F

Fδminusrarr FG FρFminusrarr F = F

idFminusminusrarr F (28)

We have for g F(A1) rarr A2 and f A1 rarr G(A2)

χ(g) = G(g) δA1 χminus1( f ) = ρA2 F( f ) (29)

For simplicity δA1 and ρA2 are often abbreviated as δ and ρ respectivelyLet 〈FG χ〉 be an adjunction from a tensor category C1 to a tensor category C2

and (F ψ F2 ψ

F0 ) a colax tensor functor from C1 to C2 We can define a morphism

878 L Kong I Runkel

φG0 11 rarr G(12) and a natural transformation φG

2 otimes (G times G) rarr G otimes by forA B isin C2

φG0 = χ(ψ F

0 ) = 11δ11minusrarr G F(11)

G(ψ F0 )minusminusminusminusrarr G(12)

φG2 = χ((ρA otimes ρB) ψ F

2 ) = G(A)otimes G(B)δminusrarr G F (G(A)otimes G(B))

G(ψ F2 )minusminusminusminusrarr G (FG(A)otimes FG(B))

G(ρAotimesρB )minusminusminusminusminusminusrarr G(A otimes B) (210)

where we have used the first identity in (29) Notice that φG2 is natural because it is a

composition of natural transformations One can easily show that ψ F0 and ψ F

2 can bere-obtained from φG

0 and φG2 as follows

ψ F0 = χminus1(φG

0 ) = F(11)FφG

0minusminusrarr FG(12)ρminusrarr 12

ψ F2 = χminus1(φG

2 (δ otimes δ)) = F(U otimes V )F(δotimesδ)minusminusminusminusrarr F (G F(U )otimes G F(V ))

FφG2minusminusrarr FG (F(U )otimes F(V ))

ρminusrarr F(U )otimes F(V ) (211)

for U V isin C1 The following result is standard for the sake of completeness we givea proof in Appendix A1

Lemma 27 (F ψ F2 ψ

F0 ) is a colax tensor functor iff (G φG

2 φG0 ) is a lax tensor

functor

22 Algebras in tensor categories An algebra in a tensor category C or a C-algebra isa triple A = (Am η) where A is an object of C m (the multiplication) is a morphismA otimes A rarr A such that m (m otimes idA) αAAA = m (idA otimes m) and η (the unit) is amorphism 1 rarr A such that m (idA otimes η) = idA rA and m (ηotimes idA) = idA lAIf C is braided and m cAA = m then A is called commutative

A left A-module is a pair (MmM ) where M isin C and m M is a morphism AotimesM rarr Msuch that mM(idAotimesm M ) = m M(m AotimesidM )αAAM and m M(ηAotimesidM ) = idMlM Right A-modules and A-bimodules are defined similarly

Definition 28 Let C be a tensor category and let A be an algebra in C

(i) A is called simple iff it is simple as a bimodule over itselfLet C be in addition k-linear for k a field

(ii) A is called absolutely simple iff the space of A-bimodule maps from A to itself isone-dimensional dimk HomA|A(A A) = 1

(iii) A is called haploid iff dimk Hom(1 A) = 1 [FS Def 43]

In the following we will assume that all tensor categories are strict to avoid spelling outassociators and unit constraints

A C-coalgebra A = (A ε) is defined analogously to a C-algebra ie A rarrA otimes A and ε A rarr 1 obey coassociativity and counit conditions

If C is braided and if A and B are C-algebras there are two in general non-isomorphicalgebra structures on A otimes B We choose A otimes B to be the C-algebra with multiplicationm AotimesB = (m Aotimesm B)(idAotimescminus1

ABotimesidB) and unitηAotimesB = ηAotimesηB Similarly if A and Bare C-coalgebras then A otimes B becomes a C-coalgebra if we choose the comultiplicationAotimesB = (idA otimes cAB otimes idB) (A otimesB) and the counit εAotimesB = εA otimes εB

Cardy Algebras and Sewing Constraints I 879

Definition 29 A Frobenius algebra A = (Am η ε) is an algebra and a coalge-bra such that the coproduct is an intertwiner of A-bimodules

(idA otimes m) (otimes idA) = otimes m = (m otimes idA) (idA otimes)

We will use the following graphical representation for the morphisms of a Frobeniusalgebra

m =A A

A

η =A

=A A

A

ε =A

(212)

A Frobenius algebra A in a k-linear tensor category for k a field is called special iffm = ζ idA and ε η = ξ id1 for nonzero constants ζ ξ isin k If ζ = 1 we call A nor-malised-special A Frobenius algebra homomorphism between two Frobenius algebrasis both an algebra homomorphism and a coalgebra homomorphism

A (strictly) sovereign tensor category is a tensor category equipped with a left and aright duality which agree on objects and morphisms (see eg [BiFS] for more details)We will write the dualities as

Uor U

= dU Uor otimes U rarr 1U Uor

= dU U otimes Uor rarr 1

U Uor

= bU 1 rarr U otimes UorUor U

= bU 1 rarr Uor otimes U

(213)

In terms of these we define the left and right dimension of an object U as

diml U = dU bU dimr U = dU bU (214)

both of which are elements of Hom(1 1)Let now C be a sovereign tensor category For a Frobenius algebra A in C we define

two morphisms

A =

A

Aor

primeA =

A

Aor

(215)

Definition 210 A Frobenius algebra A is symmetric iff A = primeA

The following lemma shows that under certain conditions we do not need to distin-guish the various notions of simplicity in Definition 28

Lemma 211 Let A be a commutative symmetric Frobenius algebra in a C-linear semi-simple sovereign braided tensor category C and suppose that diml A = 0 Then thefollowing are equivalent

880 L Kong I Runkel

(i) A is simple(ii) A is absolutely simple

(iii) A is haploid

Proof (ii)hArr(iii) A is haploid iff it is absolutely simple as a left module over itself[FS Eq (417)] Furthermore for a commutative algebra we have HomA(A A) =HomA|A(A A) and so A is haploid iff it is absolutely simple(i)rArr(ii) If A is simple then every nonzero element of HomA|A(A A) is invertibleHence this space forms a division algebra over C and is therefore isomorphic to C(iii)rArr(i) Since C is semi-simple and A is haploid also Hom(A 1) is one-dimensionalThe counit ε is a nonzero element in this space and so gives a basis This implies firstlythat ε η = 0 and secondly that there is a constant β isin C such that

β middot ε = dA (idAor otimes m) (bA otimes idA) (216)

Composing with η from the right yields β εη = diml A The right-hand side is nonzeroand so β = 0 By [FRS Lem 311] A is special We have already proved (ii)hArr(iii)and so A is absolutely simple A special Frobenius algebra in a semi-simple categoryhas a semi-simple category of bimodules (apply [FS Prop 524] to the algebra tensoredwith its opposite algebra) For semi-simple C-linear categories simple and absolutelysimple are equivalent2 Thus A is simple Remark 212 For a Frobenius algebra A the morphisms (215) are invertible and henceA sim= Aor In this case one has diml A = dimr A [FS Rem 363] and so we could havestated the above lemma equivalently with the condition dimr A = 0

Let F C1 rarr C2 be a lax tensor functor between two tensor categories C1 C2 and

let (Am A ηA) be an algebra in C1 Define morphisms F(A)otimes F(A)m F(A)minusminusminusrarr F(A) and

12ηF(A)minusminusminusrarr F(A) as

m F(A) = F(m A) φF2 ηF(A) = F(ηA) φF

0 (217)

Then (F(A)m F(A) ηF(A)) is an algebra in C2 [JS Prop 55] If f A rarr B is analgebra homomorphism between two algebras A B isin C1 then F( f ) F(A) rarr F(B)is also an algebra homomorphism If (MmM ) is a left (or right) A-module in C1 then(F(M) F(mM )φF

2 ) is a left (or right) F(A)-module if M has a A-bimodule structurethen F(M) naturally has a F(A)-bimodule structure

Similarly if (AA εA) is a coalgebra in C1 and F C1 rarr C2 is a colax tensor func-

tor then F(A) with coproduct F(A)F(A)minusminusminusrarr F(A) otimes F(A) and counit F(A)

εF(A)minusminusminusrarr 12given by

F(A) = ψ F2 F(A) εF(A) = ψ F

0 F(εA) (218)

is a coalgebra in C2 If f A rarr B is a coalgebra homomorphism between two coalgebrasA B isin C1 then F( f ) F(A) rarr F(B) is also a coalgebra homomorphism

2 To see this note that if U is simple then the C-vector space Hom(UU ) is a division algebra and henceHom(UU ) = C idU Conversely if U is not simple then U = U1 oplus U2 and Hom(UU ) contains at leasttwo linearly independent elements namely idU1 and idU2

Cardy Algebras and Sewing Constraints I 881

Proposition 2133 If F C1 rarr C2 is a Frobenius functor and (Am A ηAA εA) aFrobenius algebra in C1 then (F(A)m F(A) ηF(A)F(A) εF(A)) is a Frobenius alge-bra in C2

Proof One Frobenius property (m F(A)otimes idF(A)) (idF(A)otimesF(A)) = F(A) m F(A)follows from the commutativity of the following diagram (we spell out the associativityisomorphisms)

F(A)otimes F(A)idF(A)otimesF(A)

φF2

F(A)otimes F(A otimes A)idF(A)otimesψF

2

φF2

F(A)otimes (F(A)otimes F(A))

αF(A)F(A)F(A)

F(A otimes A)

F(idAotimesA)

F(m A)

F(A otimes (A otimes A))

F(αAAA)

(F(A)otimes F(A))otimes F(A)

φF2 otimesidF(A)

F((A otimes A)otimes A)

F(m AotimesidA)

ψF2 F(A otimes A)otimes F(A)

F(m A)otimesidF(A)

F(A)

F(A) F(A otimes A)ψF

2 F(A)otimes F(A)

(219)

The commutativity of the upper-left subdiagram follows from the naturalness of φF2 that

of the upper-right subdiagram follows from (25) that of the lower-left subdiagram fol-lows from the Frobenius properties of A and that of the lower-right subdiagram followsfrom the naturalness of ψ F

2 The proof of the other Frobenius property is similar Proposition 214 If F C1 rarr C2 is a tensor functor and A a Frobenius algebra in C1then

(i) F(A) has a natural structure of Frobenius algebra as given in Proposition 213(ii) If A is (normalised-)special so is F(A)

Proof Part (i) follows from Propositions 25 and 213 Part (ii) is a straightforwardverification of the definition using ψ F

2 = (φF2 )

minus1 and ψ F0 = (φF

0 )minus1

Let C1 C2 be sovereign tensor categories and F C1 rarr C2 a Frobenius functor Wedefine two morphisms IF(Aor) I prime

F(Aor) F(Aor) rarr F(A)or for a Frobenius algebra A inC1 as follows

IF(Aor) = ((ψ F0 F(dA) φF

2 )otimes idF(A)or) (idF(Aor) otimes bF(A))

I primeF(Aor) = (idF(A)or otimes (ψ F

0 F(dA) φF2 )) (bF(A) otimes idF(Aor))

(220)

It is easy to see that these are isomorphisms

Lemma 215 If F C1 rarr C2 is a Frobenius functor and A a Frobenius algebra in C1then

F(A) = IF(Aor) F(A) primeF(A) = I prime

F(Aor) F(primeA) (221)

3 After the preprint of the present paper appeared we noticed that this proposition is also proved in[DP Cor 5]

882 L Kong I Runkel

Proof We only prove the first equality the second one can be seen in the same way Bydefinition we have

IF(Aor) F(A)

= ((ψ F0 F(dA) φF

2 )otimes idF(A)or) (idF(Aor) otimes bF(A)) F(A)

=[(ψ F

0 F(dA) φF2 ) (F(A)otimes idF(A))

]otimes idF(A)or

(idF(A) otimes bF(A))

For the term inside the square brackets we find

ψ F0 F(dA) φF

2 (F(A)otimes idF(A)) = ψ F0 F(dA) F(A otimes idA) φF

2

= ψ F0 F(dA (A otimes idA)) φF

2 = ψ F0 F(εA m A) φF

2 (222)

On the other hand by definitionF(A) = [((ψ F0 F(εA) (F(m A)φF

2 ))otimes idF(A)or] (idF(A) otimes bF(A)) This demonstrates the first equality in (221) Proposition 216 Let F C1 rarr C2 be a tensor functor G C2 rarr C1 a functor〈FG χ〉 an adjunction A a C1-algebra and B a C2-algebra Then f A rarr G(B)is an algebra homomorphism if and only if f = χminus1( f ) F(A) rarr B is an algebrahomomorphism

Proof We need to show that

mG(B) ( f otimes f ) = f m A and f ηA = ηG(B) (223)

is equivalent to

m B ( f otimes f ) = f m F(A) and f ηF(A) = ηBs (224)

We first prove that the first identity in (223) is equivalent to the first identity in (224)For the left-hand side of the first identity in (223) we have the following equalities

mG(B) ( f otimes f )(1)= G(m B) φG

2 ( f otimes f )(2)= G(m B) G(ρ otimes ρ) G(ψ F

2 ) δ ( f otimes f )(3)= G(m B) G(ρ otimes ρ) G(ψ F

2 ) G F( f otimes f ) δ(4)= G(m B) G(ρ otimes ρ) G(F( f )otimes F( f )) G(ψ F

2 ) δ(5)= G

(m B (ρ otimes ρ) (F( f )otimes F( f )) ψ F

2

) δ

(6)= χ(

m B (ρ otimes ρ) (F( f )otimes F( f )) ψ F2

) (225)

where (1) is the definition of mG(B) in (217) (2) is the second identity in (210) (3)and (4) are naturality of δ and ψ F

2 respectively step (5) is functoriality of G and finallystep (6) is (29) For the right hand side of the first identity in (223) we get

f m A(1)= Gρ δG ( f m A)

(2)= Gρ G F( f m A) δ(3)= G(ρ F( f m A)) δ(4)= χ(ρ F( f ) F(m A)) (226)

Cardy Algebras and Sewing Constraints I 883

where (1) is the adjunction property (28) (2) is naturality of δ (3) functoriality of Gand (4) amounts to (29) and functoriality of F

On the other hand we see that the first equality in (224) is equivalent to

m B (ρ otimes ρ) (F( f )otimes F( f )) = ρ F( f ) F(m A) φF2 (227)

Using that φF2 is invertible with inverse (φF

2 )minus1 = ψ F

2 and that χ is an isomorphismit follows that the statement that (225) is equal to (226) is equivalent to the identity(227)

Now we prove that the second identity in (223) is equivalent to the second identity in(224) Using (217) and (210) we can write ηG(B) = G(ηB)φF

0 = G(ηB)G(ψ F0 )δ1

Together with (29) this shows that the second identity in (223) is equivalent to

f ηA = χ(ηB ψ F0 ) (228)

On the other hand the second identity in (224) is equivalent to

ρ F( f ) F(ηA) φF0 = ηB (229)

which by φF0 = (ψ F

0 )minus1 and (29) is further equivalent to (228)

Definition 217 Let (Am A ηAA εA) and (Bm B ηBB εB) be two Frobeniusalgebras in a tensor category C For f A rarr B we define f lowast B rarr A by

f lowast = ((εB m B)otimes idA) (idB otimes f otimes idA) (idB otimes (A ηA)) (230)

The following lemma is immediate from the definition of (middot)lowast and the properties ofFrobenius algebras We omit the proof

Lemma 218 Let C be a tensor category let A BC be Frobenius algebras in C andlet f A rarr B and g B rarr C be morphisms

(i) (g f )lowast = f lowast glowast(ii) f is a monomorphism iff f lowast is an epimorphism

(iii) f is an algebra map iff f lowast is a coalgebra map(iv) If f is a homomorphism of Frobenius algebras then f lowast f = idA and f f lowast =

idB(v) If C is sovereign and if A and B are symmetric then f lowastlowast = f

Let C and D be tensor categories and let F C rarr D be a Frobenius functor GivenFrobenius algebras A B in C and a morphism f A rarr B the next lemma shows how(middot)lowast behaves under F

Lemma 219 F( f lowast) = F( f )lowast

Proof The definition of the structure morphisms of the Frobenius algebra F(A) is givenin (217) and (218) Substituting these definitions gives

F( f )lowast =[(ψ F

0 F(εB) F(m B) φF2

)otimes idF(A)

] [

idF(B) otimes F( f )otimes idF(A)]

[idF(B) otimes

(ψ F

2 F(A) F(ηA) φF0

)]

= (ψ F0 otimes idF(A)) [

F (εB m B (idB otimes f ))otimes idF(A)]

(φF2 otimes idF(A)) (idF(B) otimes ψ F

2 )

[idF(B) otimes F(A ηA)

] (idF(B) otimes φF0 ) (231)

884 L Kong I Runkel

In the middle line of the last expression we can use the defining property (25) of F namely we substitute (φF

2 otimes idF(A)) (idF(B) otimes ψ F2 ) = ψ F

2 φF2 Then ψ F

2 can bemoved to the left and φF

2 to the right until they can be omitted against ψ F0 and φF

0 respectively using (22) and (24) This results in

F( f )lowast = F ((εB m B (idB otimes f ))otimes idA) F (idB otimes (A ηA)) (232)

which is nothing but F( f lowast)

23 Modular tensor categories Let C be a modular tensor category [TBK] ie anabelian semi-simple finite C-linear ribbon category with simple tensor unit 1 and a non-degeneracy condition on the braiding (to be stated in a moment) We denote the set ofequivalence classes of simple objects in C by I elements in I by i j k isin I and theirrepresentatives by Ui U j Uk We also set U0 = 1 and for an index k isin I we define kby Uk

sim= Uork

Since the tensor unit is simple we shall for modular tensor categories identifyHom(1 1) sim= C (cf footnote 2) Define numbers si j isin C by4

si j = UiU j (233)

They obey si j = s ji and s0i = dim Ui see eg [BK Sect 31] (In a ribbon cate-gory the left and right dimension (214) of Ui coincide and are denoted by dim Ui )The non-degeneracy condition on the braiding of a modular tensor category is that the|I|times|I|-matrix s should be invertible In fact [BK Thm 317]

sumkisinI

sik sk j = Dim C δij (234)

where Dim C = sumiisinI(dim Ui )

2 One can show (even in the weaker context of fusioncategories over C) that Dim C ge 1 [ENO1 Thm 23] In particular Dim C = 0 We fixonce and for all a square root

radicDim C of Dim C

Let us fix a basis λα(i j)kN k

i jα=1 in HomC(Ui otimesU j Uk) and the dual basis ϒ(i j)k

α N ki j

α=1

in HomC(UkUi otimesU j ) The duality of the bases means that λα(i j)k ϒ(i j)kβ = δαβ idUk

We also fix λ(0i)i = λ(i0)i = idUi We denote the basis vectors graphically as follows

λα(i j)k = α

Uk

Ui U j

ϒ(i j)kα = α

Uk

Ui U j

(235)

4 In the graphical notation used below we have given an orientation to the ribbons indicated by the arrowsFor example it is understood that this orientation determines which of the duality morphisms in (213) to use

Cardy Algebras and Sewing Constraints I 885

For V isin C we also choose a basis b(iα)V of HomC(VUi ) and the dual basis bV(iβ)

of HomC(Ui V ) for i isin I such that b(iα)V bV(iβ) = δαβ idUi We use the graphical

notation

b(iα)V = α

Ui

V

bV(iα) = α

Ui

V

(236)

Given two modular tensor categories C and D by C D we mean the tensor productof additive categories over C [BK Def 1115] ie the category whose objects are directsums of pairs V times W of objects V isin C and W isin D and whose morphism spaces are

HomCD(V times W V prime times W prime) = HomC(V V prime)otimesC HomC(WW prime) (237)

for pairs and direct sums of these if the objects are direct sums of pairsIf we replace the braiding and the twist in C by the antibraiding cminus1 and the antitwist

θminus1 respectively we obtain another ribbon category structure on C In order to distin-guish these two distinct structures we denote (C c θ) and (C cminus1 θminus1) by C+ and Cminusrespectively As in the Introduction we will abbreviate

C2plusmn = C+ Cminus (238)

Note that a set of representatives of the simple objects in C2plusmn is given by Ui times U j fori j isin I

For the remainder of Sect 2 we fix a modular tensor category C

24 The functors T and R The tensor product bifunctor otimes can be naturally extendedto a functor T C2plusmn rarr C Namely T (oplusN

i=1Vi times Wi ) = oplusNi=1Vi otimes Wi for all Vi Wi isin C

and N isin N The functor T becomes a tensor functor as follows For φT0 1 rarr T (1 times 1)

take φT0 = id1 (or lminus1

1 in the non-strict case) Next notice that for U VW X isin C

T (U times V )otimes T (W times X)= (U otimes V )otimes (W otimes X)

T ((U times V )otimes (W times X))= (U otimes W )otimes (V otimes X)(239)

We define φT2 T (U times V )otimes T (W times X) rarr T ((U times V )otimes (W times X)) by

φT2 = idU otimes cminus1

W V otimes idX (240)

(In the non-strict case the appropriate associators have to be added) The above definitionof φT

2 can be naturally extended to a morphism φT2 T (M1 otimes M2) rarr T (M1)otimes T (M2)

for any pair of objects M1M2 in C2plusmn The following result can be checked by directcalculation [JS Prop 52]

Lemma 220 The triple (T φT2 φ

T0 ) gives a tensor functor

886 L Kong I Runkel

In particular (T φT2 φ

T0 ψ

T2 ψ

T0 ) where ψT

2 = (φT2 )

minus1 and ψT0 = (φT

0 )minus1 gives

a Frobenius functorDefine the functor R C rarr C2plusmn as follows for A isin C and f isin HomC(A B)

R(A) =oplusiisinI

(A otimes Uori )times Ui R( f ) =

oplusiisinI

( f otimes idUori)times idUi (241)

This functor was also considered in a slightly different context in [ENO2 Prop 23]The family of isomorphisms γ R

A = oplusiisinI DimCdim Ui

id(AotimesUori )timesUi

isin Aut(R(A)) defines a

natural isomorphism γ R R rarr ROur next aim is to show that R is left and right adjoint to T in other words R

and T form an ambidextrous adjunction (see eg [Ld] for a discussion of ambidextrousadjunctions) To this end we introduce two linear isomorphisms for A isin C and M isin C2plusmn

χ HomC(T (M) A) minusrarr HomC2plusmn(M R(A))

χ HomC(A T (M)) minusrarr HomC2plusmn(R(A)M)(242)

If we decompose M as M = oplusNn=1 Ml

n times Mrn then χ and χ are given by

χ Noplus

n=1

fn

Mln Mr

n

A

rarrNoplus

n=1

oplusiisinI

sumα

fn

Mln

Mrn

A

α

Uori

times α

Mrn

Ui

(243)

and

χ Noplus

n=1 gn

Mln Mr

n

A

rarrNoplus

n=1

oplusiisinI

sumα gn

Mln

Mrn

A

α

Uori

times α

Ui

Mrn

DimCdim Ui

(244)

Notice that χ and χ are independent of the choice of basis

Theorem 221 〈T R χ〉 and 〈R T χminus1〉 are adjunctions ie R is both left and rightadjoint of T

Proof Write M as M = oplusNn=1 Ml

n times Mrn The isomorphism χ amounts to the following

composition of natural isomorphisms

HomC(T (M) A) = oplusnHomC(Mln otimes Mr

n A)sim= oplusni HomC(Ml

n otimes Ui A)otimes HomC(Mrn Ui )

sim= oplusni HomC(Mln A otimes Uor

i )otimes HomC(Mrn Ui ) = HomC2plusmn(M R(A)) (245)

Cardy Algebras and Sewing Constraints I 887

Thus χ is natural Let (γ RA )

lowast HomC2plusmn(R(A)M) rarr HomC2plusmn(R(A)M) denote the

pull-back of γ RA The isomorphism χ is equal to the composition of (γ R

A )lowast and the

following sequence of natural isomorphisms

HomC(A T (M)) = oplusn HomC(AMln otimes Mr

n)

sim= oplusni HomC(AMln otimes Ui )otimes HomC(Ui Mr

n)

sim= oplusni HomC(A otimes Uori Ml

n)otimes HomC(Ui Mrn) = HomC2plusmn(R(A)M) (246)

We have proved that both χ and χ are natural isomorphisms There are four natural transformations associated to χ and χ namely

idC2plusmnδminusrarr RT

ρminusrarr idC2plusmn and idCδminusrarr T R

ρminusrarr idC (247)

defined by for A isin C M isin C2plusmn

δM = χ(idT (M)) ρA = χminus1(idR(A))

ρM = χ (idT (M)) δA = χminus1(idR(A))(248)

They can be expressed graphically as follows with M = oplusNn=1 Ml

n times Mrn

δM =oplusni

sumα

Mln

Mln Mr

n Uori

αtimes

Ui

Mrn

α

ρA =oplusiisinI

A

A

Uori Ui

(249)

ρM =oplusni

sumα

Mln

Mln

Mrn Uor

i

α times

Ui

Mrn

α DimCdim Ui

δA =oplusiisinI

A

A Uori Ui

dim UiDimC

Note that

ρM δM = Dim C middot idM and ρA δA = idA (250)

Lemma 222 The functors T and R as maps on the sets of morphisms have left inversesand thus are injective

Proof Let f A rarr B be a morphism in C We define a map Q R HomC2plusmn(R(A)

R(B)) rarr HomC(A B) by f prime rarr ρB T ( f prime) δA Then we have

Q R R( f ) = ρB T R( f ) δA = ρB δB f = f (251)

888 L Kong I Runkel

where we used naturality of δ and (250) in the second and third equalities respectivelySo Q R is a left inverse of R on morphisms Thus R is injective on morphisms Similarlylet g M rarr N be a morphism in C2plusmn We define a map QT HomC(T (M) T (N )) rarrHomC2plusmn(M N ) by gprime rarr (Dim C)minus1 middot ρN R(gprime) δM Then we have

QT T (g) = (Dim C)minus1 middot ρN RT (g) δM = (Dim C)minus1 middot ρN δN g = g (252)

So QT is a left inverse of T on morphisms Thus T is injective on morphisms Using (29) and (250) one can express the two inverse maps χminus1 χminus1 as follows

for f isin HomC2plusmn(M R(A)) and g isin HomC2plusmn(R(A)M)

χminus1( f ) = ρ T ( f ) χminus1(g) = T (g) δ (253)

By Proposition 25 and Lemma 27 R is both a lax and colax tensor functor In particularφR

0 1 times 1 rarr R(1) is given by

φR0 = χ (ψT

0 ) = R(ψT0 ) δ1 times 1 = id1times1 (254)

and φR2 R(A)otimes R(B) rarr R(A otimes B) by φR

2 = R(ρA otimes ρB) R(ψT2 ) δ which can

be expressed graphically as

φR2 =

oplusi jkisinI

sumα

A

A

Uori B

B

Uorj

Uork

α

times

Ui U j

α

Uk

(255)

Similarly ψ R0 R(1) rarr 1 times 1 is given by

ψ R0 = ρ1 R(φT

0 ) = Dim C id1times1 (256)

and ψ R2 R(A otimes B) rarr R(A) otimes R(B) by ψ R

2 = ρ R(φT2 ) R(δA otimes δB) which in

graphical notation reads

ψ R2 =

oplusi jkisinI

sumα

A

A Uori

B

B Uorj

Uork

α

times

Ui U j

α

Uk

dim Ui dim U j

dim Uk Dim C

(257)

If C has more than one simple object then R does not take the tensor unit of C to thetensor unit of C2plusmn and so is clearly not a tensor functor However we will show that R isstill a Frobenius functor This will imply that if A is a Frobenius algebra in C then

R(A) = (R(A)m R(A) ηR(A)R(A) εR(A)) (258)

Cardy Algebras and Sewing Constraints I 889

is a Frobenius algebra in C2plusmn where the structure morphisms were given in (217) and(218) In the case A = 1 it was proved in [Muuml1 Prop 41] (see also [Fr Lem 619]and [K1 Thm 52]) that (258) is a commutative simple symmetric normalised-specialFrobenius algebra in C2plusmn In fact given a Frobenius algebra A in C it is straightforwardto verify that the structure morphisms in (258) are precisely those of (A times 1)otimes R(1)cf Sect 22

Proposition 223 (R φR2 φ

R0 ψ

R2 ψ

R0 ) is a Frobenius functor

Proof Using the explicit graphical expression of φR2 φ

R0 ψ

R2 ψ

R0 it is easy to see that

the commutativity of the diagrams (25) and (26) are equivalent to the statement thatR(1) with structure morphisms as in (258) is a Frobenius algebra in C2plusmn The latterstatement is true by [Muuml1 Prop 41]

From Lemma 220 and Proposition 223 we see that T and R take Frobenius alge-bras to Frobenius algebras The following two propositions show how the properties ofFrobenius algebras are transported

Proposition 224 Let A be a Frobenius algebra in C2plusmn Then T (A) is a Frobenius algebrain C and

(i) A is symmetric iff T (A) is symmetric(ii) A is (normalised-)special iff T (A) is (normalised-)special

Proof For part (i) write A as a direct sum oplusNn=1 Al

n times Arn Then the maps IT (A) I prime

T (A) T (Aor) rarr T (A)or defined in (220) are given by

IT (A) = I primeT (A) = oplusN

n=1c(Aln)

or(Arn)

or (259)

Therefore by (221) T (A) = primeT (A) is equivalent to T (A) = T (prime

A) Since byLemma 222 T is injective on morphisms this proves part (i) Part (ii) can be checkedin the same way for example the condition mT (A) T (A) = ζ idT (A) is easily checkedto be equivalent to T (m A A) = ζ T (idA) Proposition 225 Let A be a Frobenius algebra A in C Then R(A) is a Frobeniusalgebra in C2plusmn and

(i) A is symmetric iff R(A) is symmetric(ii) A is (normalised-)special iff R(A) is (normalised-)special

Proof Recall that the structure morphisms of the Frobenius algebra R(A) are equal tothose of (A times 1) otimes R(1) Using this equality part (i) and (ii) follow because R(1) issymmetric and normalised-special For example

m R(A) R(A) = [(m A A)times id1] otimes idR(1) = R(m A A) (260)

so that m R(A) R(A) = ζ idR(A) is equivalent to R(m A A) = ζ R(idA) which byLemma 222 is equivalent to m A A = ζ idA

The functor R has one additional property not shared by T namely R takes absolutelysimple algebras to absolutely simple algebras We will see explicitly in Sect 33 thatthis is not true for T

890 L Kong I Runkel

Lemma 226 For a C-algebra A the map

R HomA|A(A A) rarr HomR(A)|R(A)(R(A) R(A)) (261)

given by f rarr R( f ) is well-defined and an isomorphism

Proof Since R is a lax tensor functor R(A) is naturally a R(A)-bimodule It is easy tosee that R in (261) is a well-defined map R(A) is also naturally a R(1)-bimodule whichcan be identified with the induced R(1)-bimodule structure on (A times 1)otimes R(1) wherethe left R(1) action on (Atimes1)otimes R(1) is given by (idAtimes1 otimesm R(1))(cminus1

Atimes1R(1)otimes idR(1))We have the following natural isomorphisms

HomR(1)|R(1)(R(A) R(A))sim=minusrarr HomC2plusmn(A times 1 R(A))

χminus1

minusminusrarr HomC(A A) (262)

which by (253) are given by for f isin HomR(1)|R(1)(R(A) R(A))

f rarr f prime = f (idAtimes1 otimes ηR(1)) rarr f primeprime = ρ T ( f prime) (263)

and its inverse is given by for g isin HomC(A A)

g rarr gprime = R(g) δ rarr gprimeprime = (idAtimes1 otimes m R(1)) (gprime otimes idR(1)) (264)

where gprimeprime is indeed a R(1)-bimodule map due to the commutativity of R(1) It is easy tocheck that gprimeprime = R(g) in (264) Therefore R gives an isomorphism from HomC(A A)to HomR(1)|R(1)(R(A) R(A)) Moreover one verifies that R(g) is an R(A)-bimodulemap iff g is an A-bimodule map In other words R g rarr R(g) gives an isomorphism

HomA|A(A A)sim=minusrarr HomR(A)|R(A)(R(A) R(A))

Corollary 227 Let A be a C-algebra

(i) A is absolutely simple iff R(A) is absolutely simple(ii) Let A be in addition Frobenius Then A is simple and special iff R(A) is simple and

special

Proof Part (i) immediately follows from Lemma 226 The statement of part (ii) withoutthe qualifier lsquosimplersquo is proved in Proposition 225 But as in the proof of (iii)rArr(i) inLemma 211 a special Frobenius algebra in a semi-simple category has a semi-simplecategory of bimodules and for a semi-simple C-linear category simple and absolutelysimple are equivalent Part (ii) then follows from part (i)

The following lemma will be needed in Sect 32 below to discuss the properties ofCardy algebras

Lemma 228 Let A be a Frobenius algebra in C Then (δA)lowast = ρA

Proof Recall from (250) that δA is a morphism A rarr T R(A) Since T and R are bothFrobenius functors T R(A) is a Frobenius algebra in C Substituting the definitions aftera short calculation one finds

Cardy Algebras and Sewing Constraints I 891

εT R(A) mT R(A) = Dim(C)oplusiisinI

A Uori Ui A U or

ı Uı

(265)

Substituting this in the definition of (δA)lowast gives again after a short calculation the mor-

phism ρA At an intermediate step one uses that the part of the morphism (δA)lowast which

is made up of Ui and Uı ribbons their duals and the basis morphisms λ(iı)0 andϒ(iı)0can be replaced by 1

dim Uimiddot dUi

3 Cardy Algebras

In this section we start by investigating the properties of Frobenius algebras which sat-isfy the so-called modular invariance condition We then give two definitions of a Cardyalgebra and prove their equivalence Finally in Sect 33 we study the properties of thesealgebras and state our main results

We fix a modular tensor category C Recall that C2plusmn is an abbreviation for C+ Cminus

31 Modular invariance In C2plusmn we define the object K and the morphism ω K rarr Kas

K =oplus

i jisinIUi times U j ω =

sumi jisinI

dim Ui dim U j

Dim C idUi timesU j (31)

They have the property (see eg [BK Cor 3111])

ω

K

(Uk times Ul )or

(Uk times Ul )or

Ui times U j

Ui times U j

= δik δ jlDim C

dim Ui dim U jbUi timesU j dUi timesU j (32)

892 L Kong I Runkel

Definition 31

(i) Let A B be objects of C2plusmn A morphism f A otimes B rarr B is called S-invariant iff

A

B

W

W

f =

A

B

W

W

f

K

ω(33)

holds for all for W isin C2plusmn(ii) A C2plusmn-algebra (Aclmcl ηcl) is called modular invariant iff θAcl = idAcl and mcl

is S-invariant

Lemma 32 The morphism f A otimes B rarr B is S-invariant if and only if

A

B

Ui times U j

Ui times U j

f = Dim Cdim Ui dim U j

sumα

A

B

B

f

Ui times U j

Ui times U j

α

α

(34)

holds for all i j isin I

Proof Condition (33) holds for all W iff it holds for all W = Ui times U j i j isin I so itis enough to show that the right-hand side of (33) with W = Ui times U j is equal to theright-hand side of (34) Recall the notation for basis morphisms in (236) Starting from(33) write

idBor otimes idUi timesU j =⎛⎝sum

klα

(b(ktimeslα)

B

)or (

bB(ktimeslα)

)or⎞⎠ otimes idUi timesU j (35)

and then apply (32) The graphical representation of the resulting morphism can bedeformed to give (34) Remark 33 As shown in [K3 Sect 61] the modular invariance condition of a C2plusmn-alge-bra exactly coincides with the modular invariance condition for torus 1-point correlationfunctions of a genus-01 closed CFT In particular the condition θAcl = idAcl is equiv-alent to invariance under the modular transformation T τ rarr τ + 1 and the condition(34) with f = mcl is equivalent to invariance under S τ rarr minus 1

τ Combining the

modular invariance condition with the genus zero properties of a genus-01 closed CFTresults in a modular invariant commutative symmetric Frobenius algebra in C2plusmn

Cardy Algebras and Sewing Constraints I 893

Let Acl be a modular invariant C2plusmn-algebra Evaluating (34) for f = mcl composingit with ηcl otimes idUi timesU j and taking the trace implies the following identity

Zi j = 1

DimCAclUi timesU j where

(36)

Zi j = dimC HomC2plusmn(UitimesU j Acl)

Decomposing Acl into simple objects this gives

Zi j =sum

klisinISik Zkl Sminus1

l j where Si j = si jradic

DimC (37)

which in CFT terms is of course nothing but the invariance of the torus partition functionunder the modular S-transformation

The following theorem gives a simple criterion for modular invariance

Theorem 34 Let Acl be a haploid commutative symmetric Frobenius algebra in C2plusmn

(i) If Acl is modular invariant then dim Acl = Dim C(ii) If dim Acl = Dim C then Acl is special and modular invariant

Proof Part (i) Since Acl is haploid for i = j = 0 Equation (36) reduces to 1 =dim AclDimCPart (ii) By the same reasoning as in the proof of (iii)rArr(i) in Lemma 211 one showsthat Acl is special Thus mcl cl = ζclidAcl for some ζcl = 0

By [KO Thm 45] the category (C2plusmn)locAcl

of local Acl-modules is again a mod-

ular tensor category and Dim (C2plusmn)locAcl

= (Dim C dim Acl)2 (see [Fr Prop 321 amp

Rem 323] for the same statement in the notation used here) Thus by assumption wehave Dim(C2plusmn)loc

Acl= 1 It then follows from [ENO1 Thm 23] that up to isomorphism

(C2plusmn)locAcl

has a unique simple object (namely the tensor unit) In other words every simplelocal Acl-module is isomorphic to Acl (seen as a left-module over itself)

We have the following isomorphisms between morphism spaces [FS Prop 47 amp 411]

HomAcl(Acl otimes (Ui times U j ) Acl) sim= HomC2plusmn(Ui times U j Acl)

HomAcl(Acl Acl otimes (Ui times U j )) sim= HomC2plusmn(AclUi times U j )

Using these to transport the bases (236) from the right to the left we obtain bases bα(i j)αof HomAcl(Acl otimes (Ui times U j ) Acl) and b(i j)

β β of HomAcl(Acl Acl otimes (Ui times U j )) Thesecan be expressed graphically as

894 L Kong I Runkel

bα(i j) =

Acl

Acl

Acl

Ui times U j

α

b(i j)β = DimC

dim Ui dim U j

1

ζcl

Acl

Ui times U jAcl

Acl

β

(38)

where the nonzero factor in b(i j)β is included for convenience Notice that bα(i j) b(i j)

β isa left Acl-module map Since Acl is simple as a left module over itself we have

bα(i j) b(i j)β = λαβ idAcl (39)

for some λαβ isin C By computing tr(bα(i j) b(i j)β ) it is easy to verify that λαβ = δαβ

We will now prove the following identity

1

ζcl

Acl

Acl

Ui times U j

Ui times U j

Acl

=sumα

DimCdim Ui dim U j

1

ζcl

Acl

Ui times U j

Ui times U j

Acl

Acl

Acl

α

α

(310)

One checks that the left-hand side of this equation is an idempotent which we denote byζminus1

cl PlAcl(Ui timesU j ) cf [Fr Sect 31] By [Fr Prop 41] the image Im(ζminus1

cl PlAcl(Ui timesU j ))

is a local Acl-module and hence isomorphic to AoplusNcl for some N isin Zge0

All left-module morphisms from Acl to Aclotimes(Ui timesU j ) are linear combinations of the

b(i j)β Furthermore one verifies that ζminus1

cl PlAcl(Ui timesU j )b(i j)

β = b(i j)β Therefore the b(i j)

β

describe precisely the image of the idempotent ie ζminus1cl Pl

Acl(Ui timesU j ) = sum

α b(i j)α bα(i j)

which is nothing but (310) Composing (310) with ζcl middot εcl otimes idUi timesU j from the left (iefrom the top) produces (34)

In addition since Acl is commutative symmetric Frobenius it satisfies θAcl = idAcl

[Fr Prop 225] Altogether this shows that Acl is modular invariant Remark 35 As we were writing this paper we heard that the results in Theorem 34were obtained independently by Kitaev and Muumlger [Ki]

Remark 36 Setting i = j = 0 in (36) gives the identity dim Acl = Z00 Dim C [KRProp 23] Combining this with Theorem 34 (ii) one may wonder if a general modularinvariant commutative symmetric Frobenius algebra Acl in C2plusmn is isomorphic to a directsum of simple such algebras However this is not so For example one can take the com-mutative symmetric Frobenius algebra Acl = C[x]x2 in the category of vector spacesequipped with the non-degenerate trace ε(ax + b) = a In this case the modular invari-ance condition holds automatically but Acl is clearly not a direct sum of two algebras

Cardy Algebras and Sewing Constraints I 895

For a general modular tensor category C the algebra C[x]x2 R(1) understood as analgebra in C2plusmn via the braided monoidal isomorphism Vect f (C) C2plusmn rarr C2plusmn providesanother counter-example

32 Two definitions Define a morphism PlA A rarr A for a Frobenius algebra A in C

or C2plusmn as follows [Fr Sect 24]

PlA =

A

A

A

m

(311)

If A is also commutative and obeys m A A = ζA idA we have PlA = ζA idA In partic-

ular this holds if A is commutative and special Using the fact that the Frobenius algebraR(1) is commutative and normalised-special one can check that Pl

R(A) R(A) rarr R(A)takes the following form

PlR(A) =

oplusiisinI

A

A

Uori

Uori

A

m

times

Ui

Ui

(312)

With these ingredients we can now give the first definition of a Cardy C|C2plusmn-algebrawhich was introduced in [K3 Def 514] cf Remark 315 below

Definition 37 (CardyC|C2plusmn-algebra I) A CardyC|C2plusmn-algebra is a triple (Aop|Acl ιcl-op)where (Aclmcl ηclcl εcl) is a modular invariant commutative symmetric Frobe-nius C2plusmn-algebra (Aopmop ηopop εop) is a symmetric Frobenius C-algebra andιcl-op Acl rarr R(Aop) an algebra homomorphism such that the following conditionsare satisfied

(i) Centre condition

Acl R(Aop)

R(Aop)

R(Aop)

ιcl-op

m R(Aop)

=

Acl

ιcl-op

R(Aop)

R(Aop)

R(Aop)

m R(Aop)

(313)

896 L Kong I Runkel

(ii) Cardy condition

ιcl-op ιlowastcl-op =

R(Aop)

R(Aop)

R(Aop)

(314)

Remark 38

(i) The name ldquoCardy C|C2plusmn-algebrardquo in Definition 37 was chosen because many ofthe important ingredients were first studied by Cardy the modular invariance ofthe closed theory [C1] the consistency of the annulus amplitude [C2] and thebulk-boundary OPE [CL] On the other hand the boundary-boundary OPE andthe OPE analogue of the centre condition were first considered in [Lw]

(ii) One can easily see that in the special case that C is the category Vect f (C) of finite-dimensional C-vector spaces a Cardy C|C2plusmn-algebra gives exactly the algebraicformulation of two-dimensional open-closed topological field theory over C (cfRemark 614 in [K3]) see [Lz Sect 48] [Mo Thm 11] [AN Thm 45] [LPCor 43] [MS Sect 22] When passing to a general modular tensor category Cthere are two important differences to the two-dimensional topological field the-ory Firstly the algebras Acl and Aop now live in different categories which inparticular affects the formulation of the centre condition and the Cardy conditionSecondly the modular invariance condition has to be imposed on Acl In the caseC = Vect f (C) modular invariance holds automatically

Definition 39 A homomorphism of Cardy C|C2plusmn-algebras (A(1)op |A(1)cl ι(1)cl-op) rarr (A(2)op

|A(2)cl ι(2)cl-op) is a pair ( fop fcl) of Frobenius algebra homomorphisms fop A(1)op rarr A(2)op

and fcl A(1)cl rarr A(2)cl such that the diagram

A(1)cl

fcl

ι(1)cl-op

A(2)cl

ι(2)cl-op

R(A(1)op )

R( fop) R(A(2)op )

(315)

commutes

Remark 310 Since a homomorphism of Frobenius algebras is invertible (cf Lemma218 (iv)) a homomorphism of Cardy algebras is always an isomorphism

For a homomorphism ( fop fcl) of Cardy C|C2plusmn-algebras using the commutativity of(315) and the fact that fcl and fop are both algebra and coalgebra homomorphisms itis easy to show that (315) commutes iff

R(A(1)op )R( fop)

(ι(1)cl-op)

lowast

R(A(2)op )

(ι(2)cl-op)

lowast

A(1)cl

fcl A(2)cl

(316)

Cardy Algebras and Sewing Constraints I 897

commutesLet (Aop|Acl ιcl-op) be a Cardy C|C2plusmn-algebra Define the morphism

ιcl-op = χminus1(ιcl-op) T (Acl) minusrarr Aop (317)

Decompose Acl as Acl = oplusNn=1Cl

n times Crn such that Cl

1 times Cr1 = ηcl(1 times 1) We use ι(n)cl-op

to denote the restriction of ιcl-op to Cln times Cr

n and ι(n)cl-op to denote the restriction of ιcl-op

to Cln otimes Cr

n We introduce the following graphical notation

ι(n)cl-op =

Cln Cr

n

Aop

(318)

By (243) ιcl-op can be expressed in terms of ιcl-op as follows

ιcl-op =Noplus

n=1

oplusiisinI

sumα

Cln

Crn

Aop

α

Uori

times α

Crn

Ui

(319)

Lemma 311 The centre condition (313) is equivalent to the following condition in C

Cln Cr

n

Aop

Aop

Aop

=

Cln Cr

n

Aop

Aop

Aop

for n = 1 N (320)

Proof First insert (319) and the definition (217) of m R(Aop) into (313) Then apply thecommutativity of R(1) to the left hand side of (313) The equivalence between (313)and (320) follows immediately Remark 312 The centre condition (320) is very natural from the open-closed confor-mal field theory point of view Correlators on the upper half plane are expressed in termsof conformal blocks on the full complex plane The objects Cl

n and Crn are associated to

the field insertion at a point z in the upper half plane and at the complex conjugate pointz in the lower half plane respectively The object Aop corresponds to a field inserted ata point r on the real axis The centre condition (320) simply says that the correlationfunctions in the disjoint domains |z| gt r gt 0 and r gt |z| gt 0 are analytic continuationsof each other see [K2 Prop 118]

898 L Kong I Runkel

Recall that we define ιlowastcl-op Aop rarr T (Acl) as in (230) We introduce the graphicalnotation

ιlowastcl-op =Noplus

n=1

Cln Cr

n

Aop

=Noplus

n=1

Cln Cr

n

Aop

Aop T (minus1Acl) (321)

where the second equality follows from (215) (221) and (259)

Lemma 313 The Cardy condition (314) is equivalent to the following identity in C

Noplusn=1

DimCdim Ui

sumα

Cln

Crn

Crn

α

α

Aop

Aop Uori

Uori

=Aop

Aop

Aop

Uori

Uori

op

mop for all i isin I (322)

Proof By (244) (312) and (319) it is easy to see that (322) is equivalent to thefollowing identity

ιcl-op χ(ιlowastcl-op) = PlR(Aop)

(323)

Therefore it is enough to show that

χ(ιlowastcl-op) = ιlowastcl-op (324)

We have

χminus1(ιlowastcl-op)(1)= T (ιlowastcl-op) δ (2)= T (ιcl-op)

lowast δlowastlowast (3)= (ρ T (ιcl-op)

)lowast (4)= (ιcl-op

)lowast (325)

In step (1) we use the expression (253) for χminus1 step (2) follows from Lemma 218 (v)and Lemma 219 Step (3) is Lemma 218 (i) and Lemma 228 and finally step (4)amounts to substituting (253) and (317) Acting with χ on both sides of the aboveequality produces (324)

Combining Lemmas 311 and 313 and Proposition 216 we obtain the followingequivalent definition of Cardy C|C2plusmn-algebra (recall the graphical notation (318) forιcl-op and (321) for ιlowastcl-op)

Cardy Algebras and Sewing Constraints I 899

Definition 314 (CardyC|C2plusmn-algebra II) A CardyC-algebra is a triple (Aop|Acl ιcl-op)where Acl is a commutative symmetric Frobenius C2plusmn-algebra satisfying property (34)with f = mcl Aop is a symmetric Frobenius C-algebra and ιcl-op T (Acl) rarr Aop isan algebra homomorphism satisfying the conditions (320) and (322)

Remark 315 Up to a choice of normalisation Definition 314 is the same as the orig-inal one in [K3 Def 613] The difference between the two definitions is the factorDim C dim Ui on the left hand side of (322) which in [K3 Def 613] is given byradic

Dim C dim Ui The two definitions are related by rescaling the coproduct cl andcounit εcl of Acl by 1

radicDim C and

radicDim C respectively We chose the convention in

(322) to remove all dimension factors from the expression (314) for the Cardy condition

33 Uniqueness and existence theorems In this subsection we investigate the structureof Cardy algebras We start with the following proposition which when combined withthe results of part II provides an alternative proof of [Fj Prop 422]

Proposition 316 Let (Aop|Acl ιcl-op) be a Cardy C|C2plusmn-algebra If Acl is simple anddim Aop = 0 then Aop is simple and special

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis therefore haploid Restricting the Cardy condition (322) to the case Ui = 1 andcomposing both sides with εop from the left we see that εop kills all terms associated toU j times 1 isin Acl in the sum except for a single 1 times 1 term Thus we obtain the followingidentity

β εop = dAop (mop otimes idAorop) (idAop otimes bAop) (326)

where β isin C Composing with ηop from the right in turn implies that βεop ηop =dim Aop which is nonzero by assumption Thus also β = 0 and εop is a nonzero multipleof the morphism on the right hand side of (326) By [FRS Lem 311] Aop is special

Since Aop is a special Frobenius algebra Aop is semi-simple as an Aop-bimodule(apply [FS Prop 524] to Aop tensored with its opposite algebra) Suppose Aop is not

simple so that we can write Aop = A(1)op oplus A(2)op for nonzero Aop-bimodules A(1)op and A(2)op We denote the canonical embeddings and projections associated to this decompositionas ι12 and π12 We have the identities

mop (ι1 otimes ι2) = 0 εop ηop =2sum

i=1

εop ιi πi ηop (327)

The first identity follows since π1 mop (ι1 otimes ι2) = 0 (as mop gives the left action of

Aop on Aop and hence it preserves A(2)op ) and similarly π2 mop (ι1 otimes ι2) = 0 Thesecond identity is just the completeness of ι12 π12

Since εop ηop = 0 without losing generality we can assume εop ι1 π1 ηop = 0Using that π2 is a bimodule map we compute

π2 [LHS of (322)]Ui =1 ι1 π1 ηop = π2 [RHS of (322)]Ui =1 ι1 π1 ηop

= PlA(2)op

π2 ι1 π1 ηop = 0 (328)

900 L Kong I Runkel

On the other hand using that Acl is haploid that ιcl-op is an algebra map and thatι lowastcl-op is a coalgebra map one can check that the left-hand side of (328) is equal toλ(εop ι1 π1 ηop) π2 ηop for some λ = 0 This implies that π2 ηop = 0 Thusηop = sum

i ιi πi ηop = ι1 π1 ηop Hence we have

0 = π2 ι2 = π2 mop (ηop otimes ι2) = π2 mop ((ι1 π1 ηop)otimes ι2) (329)

However the right-hand side is zero by (327) This is a contradiction and hence Aopmust be simple

To formulate the next theorem we need the notion of the full centre of an algebra [FjDef 49] Recall that an algebra A in a braided tensor category has a left centre and aright centre [VZO] both of which are sub-algebras of A Of these two we will onlyneed the left centre The following definition is [Fr Def 231] which in our setting isequivalent to that of [VZO]

Definition 317 Let A be a symmetric special Frobenius algebra such that m A A =ζA idA

(i) The left centre Cl(A) of A is the image of the idempotent ζminus1A Pl

A(ii) The full centre Z(A) is Cl(R(A))

That ζminus1A Pl

A is an idempotent follows from [FRS Lem 52] when keeping track ofthe factors ζA ([FRS] assumes normalised-special ie ζA = 1) Note that Cl(A) is againan object of C while Z(A) is an object of C2plusmn Let el

A Cl(A) rarr A be the embeddingof Cl(A) into A The left centre is in fact the maximal subobject of A such that

m A cAA (elA otimes idA) = m A (330)

see [Fr Lem 232] This observation explains the name left centre and also makes theconnection to [O Def 15]

The full centre is by definition the image of the idempotent ζminus1A Pl

R(A) R(A) rarrR(A) Since C2plusmn is abelian the idempotent splits and we obtain the embedding andrestriction morphisms

e Z(A) rarr R(A) and r R(A) Z(A) (331)

which obey r e = idZ(A) and e r = ζminus1A Pl

R(A) It follows from Proposition 225 and

[Fr Prop 237] that Z(A) is a commutative symmetric Frobenius algebra in C2plusmn withstructure morphisms5

m Z(A) = r m R(A) (e otimes e) ηZ(A) = r ηR(A)

Z(A) = ζA middot (r otimes r) R(A) e εZ(A) = ζminus1A middot εR(A) e

(332)

Moreover if A is simple then Z(A) is simple and if A is simple and dim A = 0 thenZ(A) is simple and special The normalisation of the counit is such that

εZ(A) ηZ(A) = ζminus2A dim A DimC (333)

5 The normalisation of product and unit is the standard one The factors in the coproduct and counit haveto be included in order for (A|Z(A) e) to be a Cardy algebra see Theorem 318 below The normalisation ofthe counit enters the Cardy condition (314) through the definition of ( middot )lowast

Cardy Algebras and Sewing Constraints I 901

Theorem 318 Let A be a special symmetric Frobenius C-algebra Then (A|Z(A) e)is a Cardy C|C2plusmn-algebra

The proof of this theorem makes use the following two lemmas

Lemma 319 e Z(A) rarr R(A) is an algebra map and elowast = ζA middot r

Proof It follows from [Fr Lem 229] (or by direct calculation using in particularm R(A) R(A) = ζAidR(A)) that

m R(A) (e otimes e) = ζminus1A middot Pl

R(A) m R(A) (e otimes e) (334)

Substituting e r = ζminus1A Pl

R(A) shows that e is compatible with multiplication For theunit one finds

e ηZ(A) = e r ηR(A) = ζminus1A Pl

R(A) ηR(A) = ηR(A) (335)

Thus e is an algebra map For the second statement one computes

elowast (1)= ζA

R(A)

r

R(A)

R(A)

Z(A)

R(A)

Z(A)

r

e

e

(2)=

R(A)

r

R(A)

R(A)

R(A)

Z(A)

(3)=

R(A)

r

R(A)

Z(A)

(4)= ζA middot r (336)

where in (1) the definitions (230) and (332) have been substituted step (2) is e r = ζminus1

A PlR(A) step (3) uses that R(A) is symmetric Frobenius and step (4) is again

e r = ζminus1A Pl

R(A) Lemma 320 Let A be a symmetric Frobenius algebra in C The morphism

PlR(A) m R(A)

(Pl

R(A) otimes PlR(A)

) R(A)otimes R(A) minusrarr R(A) (337)

is S-invariant

The proof of this lemma is a slightly lengthy explicit calculation and has been deferredto Appendix A2

902 L Kong I Runkel

Proof of Theorem 318 That e is an algebra map was proved in Lemma 319 The centrecondition (313) holds by property (330) of the left centre The Cardy condition (314)also is an immediate consequence of Lemma 319

ιcl-op ιlowastcl-op = e (ζA r) = PlR(A) (338)

The full centre Z(A) is a commutative symmetric Frobenius algebra It remains to provemodular invariance That θZ(A) = idZ(A) is implied by commutativity and symmetryof Z(A) [Fr Prop 225] The S-invariance condition (33) follows from Lemma 320In (337) substitute Pl

R(A) = ζA e r and then put the resulting morphism into (33)Compose the resulting equation with e otimes idW from the right (ie from the bottom) andsubstitute the definition (332) of m Z(A) This results in the statement that m Z(A) isS-invariant

The following theorem is analogous to [LR Prop 29] and [Fj Thm 426] whichroughly speaking answer the question under which circumstances the restriction of atwo-dimensional conformal field theory to the boundary already determines the entireconformal field theory The first work is set in Minkowski space and uses operator alge-bras and subfactors while the second work is set in Euclidean space and uses modulartensor categories

Theorem 321 Let (A|Acl ιcl-op) be a Cardy C|C2plusmn-algebra such that dim A = 0 andAcl is simple Then A is special and (A|Acl ιcl-op) sim= (A|Z(A) e) as Cardy algebras

Proof By Proposition 316 A is simple and special Since Acl is simple the algebramap ιcl-op Acl rarr R(A) is either zero or a monomorphism But ιcl-op ηcl = ηR(A)and so ιcl-op = 0 Thus ιcl-op is monic By Lemma 218 (ii) ζminus1

A ιlowastcl-op is epi The Cardycondition (314) implies

ιcl-op ζminus1A ιlowastcl-op = ζminus1

A PlR(A) = e r (339)

Composing this with e r from the left yields e r ιcl-op ζminus1A ιlowastcl-op = e r =

ιcl-op ζminus1A ιlowastcl-op Since ζminus1

A ιlowastcl-op is epi we have

e r ιcl-op = ιcl-op (340)

Actually (340) also follows from (313) and specialness of R(A) We will prove that

( fop fcl) (A|Acl ιcl-op) minusrarr (A|Z(A) e) where fop = idA fcl = r ιcl-op

(341)

is an isomorphism of Cardy algebrasfcl is an algebra map Compatibility with the units follows since ιcl-op is an algebramap

fcl ηcl = r ιcl-op ηcl = r ηR(A) = ηZ(A) (342)

Compatibility with the multiplication also follows since ιcl-op is an algebra map

m Z(A) ( fcl otimes fcl) = r m R(A) (e otimes e) (r otimes r) (ιcl-op otimes ιcl-op)

= r m R(A) (ιcl-op otimes ιcl-op) = r ιcl-op mcl = fcl mcl (343)

where in the second step we used (340)

Cardy Algebras and Sewing Constraints I 903

fcl is an isomorphism As above since fcl is an algebra map and since Acl is simple fcl

has to be monic By Lemma 319 rlowast = ζminus1A e Thus f lowast

cl = ιlowastcl-op rlowast = ζminus1A ιlowastcl-op e and

fcl f lowastcl = r ιcl-op ζminus1

A ιlowastcl-op e = r e r e = idZ(A) (344)

and so fcl is also epi and hence isofcl is a coalgebra map Since fcl is an algebra map so is f minus1

cl By (344) f minus1cl = f lowast

cland by Lemma 218 (iii) this implies that fcl is a also coalgebra mapThe diagram (315) commutes Commutativity of (315) is equivalent to e fcl = ιcl-opwhich holds by (340)

Let A be a special symmetric Frobenius algebra So far we have seen that (A|Z(A) e)is a Cardy algebra and that all Cardy algebras with Aop = A and simple Acl are of thisform It is now natural to ask if every simple Acl does occur as part of a Cardy algebraThe following theorem provides an affirmative answer Recall that for an A-left moduleM the object Mor otimesA M is an algebra (see eg [KR Lem 42])

Theorem 322 If Acl is a simple modular invariant commutative symmetric FrobeniusC2plusmn-algebra then there exist a simple special symmetric Frobenius C-algebra A and amorphism ιcl-op Acl rarr R(A) such that

(i) Acl sim= Z(A) as Frobenius algebras(ii) (A|Acl ιcl-op) is a Cardy C|C2plusmn-algebra

(iii) T (Acl) sim= oplusκisinJ Morκ otimesA Mκ as algebras where MκκisinJ is a set of representa-

tives of the isomorphism classes of simple A-left modules

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis haploid It then follows from Theorem 34 that Acl is special By Proposition 224T (Acl) is a special symmetric Frobenius algebra in C Thus T (Acl) = oplusi Ai where theAi are simple symmetric Frobenius algebras We will show that at least one of the Aiis special Since T (Acl) is special we have mT (Acl) T (Acl) = ζ idT (Acl) for someζ isin C

times Restricting this to the summand Ai shows mi i = ζ idAi FurthermoreεT (Acl) ηT (Acl) = ξ id1 for some ξ isin C

times But εT (Acl) ηT (Acl) = sumi εi ηi and so

at least one of the εi ηi has to be nonzero Therefore at least one of the Ai is speciallet A equiv Ai be this summand We denote the embedding A rarr T (Acl) by e0 and therestriction T (Acl) A by r0 Notice that r0 is an algebra homomorphism Define

ιcl-op = χ (r0) Acl minusrarr R(A) (345)

By Proposition 216 ιcl-op is an algebra homomorphism Next we verify the centre con-dition (313) or rather its equivalent form (320) By substituting the definitions onecan convince oneself that the commutativity mcl cAclAcl = mcl of Acl in C2plusmn implies thecondition mT (Acl) = mT (Acl) in C see [K2 Prop 36] Here T (Acl)otimes T (Acl) rarrT (Acl)otimes T (Acl) is given by

=oplusmn

(idCl

notimes cCl

m Crnotimes idCr

m

)

(cCl

m Clnotimes cminus1

Crn C

rm

)

(idCl

motimes cminus1

Cln C

rm

otimes idCrn

)

(346)

and we decomposed Acl as Acl = oplusnCln times Cr

n As a consequence we obtain the identity

r0 mT (Acl) (idAcl otimes e0) = r0 mT (Acl) (idAcl otimes e0) (347)

904 L Kong I Runkel

Using that r0 is an algebra map and that by definition ιcl-op = r0 we obtain (320)In order to show that (A|Acl ιcl-op) is a Cardy algebra it remains to show that the

Cardy condition (314) is satisfied We will demonstrate this via a detour by first provingthat Acl sim= Z(A) as Frobenius algebras

Recall the notations e and r given in (331) Using the centre condition (320) onecan check that Pl

R(A) ιcl-op = m R(A) R(A) ιcl-op By specialness of A we have

m A A = ζA idA and so together with e r = ζminus1A Pl

R(A) we get

e r ιcl-op = ιcl-op (348)

Next consider the morphism

fcl = r ιcl-op Acl minusrarr Z(A) (349)

By the same derivation as in (342) and (343) one sees that fcl is an algebra map Inparticular fcl ηcl = ηZ(A) = 0 and so fcl = 0 Since Acl is simple fcl has to be amonomorphism

By the same argument as used in the proof of Theorem 34 (ii) up to isomorphismAcl is the unique simple local Acl-(left-)module The algebra monomorphism fcl turnsZ(A) into an Acl-module Since Z(A) is commutative it is local as an Acl-module andso Z(A) sim= AoplusN

cl for some N ge 1 By construction A is a simple special symmetricFrobenius algebra Proposition 225 and Corollary 227 show that R(A) inherits all theseproperties and thus Z(A) is simple (see the comment below Eq (332)) By Theorem318 Z(A) is modular invariant and then by Theorem 34 (i) dim Z(A) = Dim C Thisimplies that N = 1 in Z(A) sim= AoplusN

cl and so fcl is in fact an isomorphismSince Acl and Z(A) are both haploid we have εZ(A) fcl = ξ εcl for some ξ isin C

timesThe counit uniquely determines the Frobenius structure on Acl and Z(A) (see eg [FRSLemma 37]) so that fcl is a coalgebra isomorphism iff ξ = 1 To compute ξ we composethe above identity with ηcl from the right Defining ζcl via εcl ηcl = ζminus1

cl Dim C middot id1 andusing (333) gives ξ = dim A ζclζ

2A By rescaling the comultiplication and the counit

of A and consequently changing ζA we can always achieve ξ = 1 This proves part (i)of the theorem

Equation (348) implies that ιcl-op = e fcl Since fcl is an isomorphism of Frobeniusalgebras by Lemmas 218 and 319 we have

ιcl-op ιlowastcl-op = e fcl f lowastcl elowast = ζA e r = Pl

R(A) (350)

Thus (A|Acl ιcl-op) is a Cardy algebra This proves part (ii) of the theoremPart (iii) can be seen as follows By [KR Prop 43] T Z(A) sim= oplusκisinJ Mor

κ otimesA Mκ asalgebras Together with the observation that T ( fcl) T (Acl) rarr T Z(A) is an isomor-phism of algebras this proves part (iii) Remark 323 Part (i) of Theorem 322 was announced by Muumlger [Muuml2] We provide anindependent proof in the setting of Cardy algebras

The above theorem together with Lemma 211 and Theorem 34 shows that a sim-ple commutative symmetric Frobenius C2plusmn-algebra Acl with dim Acl = Dim C is alwayspart of a Cardy algebra (Aop|Acl ιcl-op) for some simple special symmetric Frobeniusalgebra Aop in C However the above proof also illustrates that Aop is not unique Thisraises the question how two Cardy algebras with a given Acl can differ This question isanswered by [KR Thm 11] which in the present framework can be restated as follows

Cardy Algebras and Sewing Constraints I 905

Theorem 324 If (A(i)op |A(i)cl ι(i)cl-op) i = 1 2 are two Cardy C|C2plusmn-algebras such that

A(i)cl is simple and dim A(i)op = 0 for i = 1 2 then A(1)clsim= A(2)cl as algebras if and only

if A(1)op and A(2)op are Morita equivalent

Proof Theorem 11 in [KR] is stated for A(i)op being non-degenerate algebras and A(i)cl =Z(A(i)op ) for i = 1 2 By Proposition 316 A(i)op are simple and special for i = 1 2

Then by [KR Lem 21] A(i)op are non-degenerate algebras By Theorem 321 we have

A(i)clsim= Z(A(i)op ) as Frobenius algebras Finally by [KR Thm 11] Z(A(1)op ) sim= Z(A(2)op )

as algebras iff A(1)op and A(2)op are Morita equivalent Let Cmax(C2plusmn) be the set of equivalence classes [B] of simple modular invariant

commutative symmetric Frobenius algebras B in C2plusmn Two such algebras B and B prime areequivalent if B and B prime are isomorphic as algebras (but not necessarily as Frobenius alge-bras) Let Msimp(C) be the set of Morita classes of simple special symmetric Frobeniusalgebras in C Define the map z Msimp(C) rarr Cmax(C2plusmn) by z A rarr [Z(A)] whereA denotes the Morita class of A From Theorem 322 (i) and [KR Thm 11] we learn

Corollary 325 The map z Msimp(C) rarr Cmax(C2plusmn) is a bijection

A Appendix

A1 Proof of Lemma 27 We will show that if (F ψ F2 ψ

F0 ) is a colax tensor functor

from C1 to C2 then (G φG2 φ

G0 ) is a lax tensor functor from C2 to C1 Applying this

result to the opposed categories then gives the converse statementWe need to show that φG

0 and φG2 make the diagrams (21) and (22) commute We

first prove the commutativity of (21) Consider the following diagram

G(A)otimes (G(B)otimes G(C))

GψF2 δ

idG(A)otimesφG2 G(A)otimes G(B otimes C)

GψF2 δ

G(FG(A)otimes F(G(B)otimes G(C)))

G(F(idG(A))otimesF(φG2 ))

G(idFG(A)otimesψ F2 )

G(FG(A)otimes FG(B otimes C))

G(ρAotimesρBotimesC )

G(FG(A)otimes (FG(B)otimes FG(C)))

G(ρAotimes(ρBotimesρC )) G(A otimes (B otimes C))

(A1)

The top subdiagram is commutative because of the naturality of Gψ F2 δ The commu-

tativity of the bottom subdiagram follows from the following identities

(ρB otimes ρC ) ψ F2 = (ρB otimes ρC ) ψ F

2 ρF Fδ

= ρBotimesC FG(ρB otimes ρC ) FG(ψ F2 ) Fδ

= ρBotimesC F(φG2 ) (A2)

as a map F(G(B) otimes G(C)) rarr B otimes C The commutativity of (A1) implies that thecomposition of maps in the left column in (21) can be replaced by

G(ρA otimes (ρB otimes ρC )) G((idFG(A) otimes ψ F

2 ) ψ F2

) δ (A3)

906 L Kong I Runkel

Similarly we can show that the composition of maps in the right column in (21) can bereplaced by

G((ρA otimes ρB)otimes ρC ) G((ψ F

2 otimes idFG(C)) ψ F2

) δ (A4)

Using the commutativity of (23) it is easy to see that (21) with the left and right col-umns of (21) replaced by (A3) and (A4) respectively is commutative Hence (21) iscommutative

Now we prove the commutativity of the first diagram in (22)

φG2 (φG

0 otimes idG(A))

(1)= G(ρ12 otimes ρA) Gψ F2 δ

[(Gψ F

0 δ11)otimes idG(A)

]

(2)= G(ρ12 otimes ρA) Gψ F2 G F(Gψ0 otimes idG(A)) G F(δ11 otimes idG(A)) δ

(3)= G(ρ12 otimes ρA) G(FG(ψ F0 )otimes idFG(A)) Gψ F

2 G F(δ11 otimes idG(A)) δ(4)= G(id12 otimes ρA) G

([ψ F

0 ρF(11) (Fδ)11] otimes idFG(A)

) Gψ F

2 δ(5)= G(id12 otimes ρA) G(ψ F

0 otimes idFG(A)) Gψ F2 δ

(6)= G(id12 otimes ρA) G(lminus1FG(A)) G F(lG(A)) δ

(7)= G(lminus1A ) GρA δG lG(A)

(8)= G(lminus1A ) lG(A) (A5)

where in step (1) we substituted the definition of φG0 φ

G2 given in (210) in step (2)

we used the naturality of δ in step (3) we used the naturality of Gψ F2 in step (4) we

switched the position between Gρ12 and G FG(ψ F0 ) and the position between Gψ2 and

G F(δ12 otimes idG(A)) using the naturality of ρ and FψG2 respectively in step (5) we applied

the second identity in (28) in step (6) we used (24) in step (7) we used the naturalityof lminus1 and δ in step (8) we used the first identity in (28)

The proof of the commutativity of the second diagram in (22) is similar Thus wehave shown that G is a lax tensor functor

A2 Proof of Lemma 320 To prepare the proof recall that for a given object B isin Cthe modular group P SL(2Z) acts on the space oplusi HomC(B otimes Ui Ui ) see eg [BKSect 31] and [K3 Eq (455)] We will only need the action of S and Sminus1 Let f isinoplusi HomC(B otimes Ui Ui ) Then

S oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A6)

Cardy Algebras and Sewing Constraints I 907

Sminus1 oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A7)

By Lemma 32 to establish that (337) is S-invariant it is enough to prove the identity(34) when f is given by (337) Using (A6) and (A7) we can see that Eq (34) simplysays that oplusi j [RHS of (34)] is invariant under the action of S times S Consider the elementg of oplus jkisinIHomC2plusmn(R(A)otimes (Uor

j times Uk)Uorj times Uk) given by

g =oplusjkisinI

sumα

R(A)

R(A)

R(A)

R(A)

Uorj times Uk

Uorj times Uk

α

α

(A8)

By the above arguments proving S-invariance of (337) is equivalent to proving invari-ance of g under the action of S times S

For i isin I we denote by gi the component of g in(oplus jisinIHomC(A otimes Uor

i otimes Uorj U

orj )

)otimes (opluskisinIHomC(Ui otimes UkUk))

We view the second Hom-space in the above tensor product as a Hom-space in C+ insteadof Cminus It is enough to show that gi is invariant under the action of S times Sminus1 Note thatthe action of Sminus1 in C+ is equivalent to that of S in Cminus

The morphism gi can be canonically identified with a bilinear pairing

( middot middot )i ⎛⎝oplus

jisinIHomC(Uor

j A otimes Uori otimes Uor

j )

⎞⎠

times(oplus

kisinIHomC(UkUi otimes Uk)

)minusrarr C (A9)

as follows For h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) we set

(h1 h2)i = (dim U j dim Uk)minus1 trUor

j timesUk[gi (h1 times h2)] (A10)

908 L Kong I Runkel

When substituting the explicit form of the product m R(A) of R(A) = (A times 1) otimes R(1)after a short calculation one finds

(h1 h2)i =sumα

1

dim U j

A

A

A

A

A A

A α

α

h1

h2

U jUk

Uorj

Ui

Uk (A11)

Here the top morphism PlR(A) has been simplified with the help of the identity

PR(A) m R(A) (PR(A) otimes PR(A))

=(oplus

iisinI((m A A)otimes idUor

i)times idUi

) m R(A) (PR(A) otimes PR(A)) (A12)

which can be checked by direct calculation along the same lines as in the proof of [FrLem 310]

The action of the modular transformation S on oplusiisinI(B otimes Ui Ui ) for B isin C nat-urally induces an action on oplusiisinI(Ui B otimes Ui ) [K3 Prop 514] which we denote bySlowast In the present case we get an action of Slowast on oplus jisinIHomC(Uor

j A otimes Uori otimes Uor

j ) and

opluskisinIHomC(UkUi otimes Uk) Then to show gi is invariant under the action of S times Sminus1

amounts to showing that

(h1 h2)i = ((Sminus1)lowast h1 Slowast h2)i (A13)

Cardy Algebras and Sewing Constraints I 909

for all h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) We have

((Sminus1)lowast h1 Slowast h2)i =sum

mnα

dim Un

DimC

A

A

A

A

A A

A

Uk

α

α

h1

h2

Ui

U j

UmUn

(A14)

Now drag the upper vertex indexed by α in the above graph along its Uorm -leg until it

meets the lower vertex also indexed by α then sum over α and m This gives

((Sminus1)lowast h1 Slowast h2)i =sum

n

dim Un

DimC

A

A

A

A

A A

A

Uk

Un

h1

h2

Ui

U j

(A15)

910 L Kong I Runkel

If we just look at the neighbourhood of the Un-loop in the above graph we see thefollowing subgraphs

sumn

dim Un

DimC

A Aor

Uorj U j

Uk Uork

Un=

sumα

1

dim U j

A Aor Uk Uork

Uorj U j

αα

(A16)

where we have applied [BK Cor 3111] Substituting this subgraph back to the originalgraph in (A15) we obtain

((Sminus1)lowast h1 Slowast h2)i =sumα

1

dim U j

A

A

A

A

A A

A

A

A

Uorj

α

α

h1

h2m A

Ui

U jUk

Uk

(A17)

The graph in (A17) is equal to that in (A11) In order to see this we first drag theldquobubblerdquo (m A A) along A lines and through the m A vertex (because m A A is abimodule map) until it reaches the lower-left leg of the upper vertex indexed by α Thendrag the m A vertex along the (red) dotted line in the above graph Finally we applythe associativity of A (A12) and [Fr Lem 311] Then we see that the graph in (A17)exactly matches with the one in (A11)

Acknowledgements We would like to thank the organisers of the Oberwolfach Arbeitsgemeinschaft ldquoAlge-braic structures in conformal field theoriesrdquo (April 2007) where this work was started for an inspiring meetingWe would further like to thank the Hausdorff Institute for Mathematics in Bonn and the organisers of the stim-ulating meeting ldquoGeometry and Physicsrdquo (May 2008) We are indebted to Alexei Davydov Jens FjelstadJuumlrgen Fuchs Yi-Zhi Huang Alexei Kitaev Urs Schreiber Christoph Schweigert Stephan Stolz and PeterTeichner for helpful discussions andor comments on a draft of this paper The research of IR was partiallysupported by the EPSRC First Grant EPE0050471 the PPARC rolling grant PPC5071451 and the MarieCurie network lsquoSuperstring Theoryrsquo (MRTN-CT-2004-512194)

Cardy Algebras and Sewing Constraints I 911

References

[AN] Alexeevski A Natanzon SM Noncommutative two-dimensional topological field theories andhurwitz numbers for real algebraic curves Sel Math New Ser 12 307ndash377 (2006)

[Bi] Bichon J Cosovereign hopf algebras J Pure Appl Alg 157 121ndash133 (2001)[BK] Bakalov B Kirillov AA Lectures on Tensor Categories and Modular Functors Providence

RI Amer Math Soc 2001[C1] Cardy JL Operator content of two-dimensional conformal invariant theories Nucl Phys

B 270 186ndash204 (1986)[C2] Cardy JL Boundary conditions fusion rules and the verlinde formula Nucl Phys B 324

581ndash596 (1989)[CL] Cardy JL Lewellen DC Bulk and boundary operators in conformal field theory Phys Lett

B 259 274ndash278 (1991)[DMZ] Dong C-Y Mason G Zhu Y-C Discrete series of the Virasoro algebra and the moonshine

module In Algebraic Groups and Their Generalizations Quantum and infinite-dimensional Meth-ods Proc Symp Pure Math 56 Part 2 Providence RI Amer MathSoc 1994 pp 295ndash316

[DP] Day B Pastro C Note on frobenius monoidal functors New York J Math 14 733ndash742 (2008)[ENO1] Etingof PI Nikshych D Ostrik V On fusion categories Ann Math 162 581ndash642 (2005)[ENO2] Etingof PI Nikshych D Ostrik V An analogue of radfordrsquos s4 formula for finite tensor

categories Int Math Research Notices 54 2915ndash2933 (2004)[Fe] Felder G Froumlhlich J Fuchs J Schweigert C Correlation functions and boundary conditions in

rational conformal field theory and three-dimensional topology Comp Math 131 189ndash237 (2002)[FHL] Frenkel IB Huang Y-Z Lepowsky J On axiomatic approaches to vertex operator algebras

and modules Mem Amer Math Soc 104 (1993)[Fj] Fjelstad J Fuchs J Runkel I Schweigert C Uniqueness of openclosed rational cft with

given algebra of open states Adv Theor Math Phys 12 1283ndash1375 (2008)[FK] Froumlhlich J King C The chern-simons theory and knot polynomials Commun Math

Phys 126 167ndash199 (1989)[Fr] Froumlhlich J Fuchs J Runkel I Schweigert C Correspondences of ribbon categories Adv

Math 199 192ndash329 (2006)[FRS] Fuchs J Runkel I Schweigert C Tft construction of rcft correlators I partition functions Nucl

Phys B 646 353ndash497 (2002)[FS] Fuchs J Schweigert C Category theory for conformal boundary conditions Fields Inst Com-

mun 39 25ndash71 (2003)[H1] Huang Y-Z Two-dimensional conformal geometry and vertex operator algebras Progress in

Mathematics Vol 148 Boston Birkhaumluser 1997[H2] Huang Y-Z Rigidity and modularity of vertex tensor categories Comm Contemp Math 10

871ndash911 (2008)[HK1] Huang Y-Z Kong L Open-string vertex algebra category and operads Commun Math

Phys 250 433ndash471 (2004)[HK2] Huang Y-Z Kong L Full field algebras Commun Math Phys 272 345ndash396 (2007)[HK3] Huang Y-Z Kong L Modular invariance for conformal full field algebras httparxivorgabs

math0609570v2[mathQA] 2006[HL] Huang Y-Z Lepowsky J Tensor products of modules for a vertex operator algebra and vertex

tensor categories In Lie Theory and Geometry in honor of Bertram Kostant ed R BrylinskiJ-L Brylinski V Guillemin V Kac Boston Birkhaumluser 1994 pp 349ndash383

[JS] Joyal A Street R Braided tensor categories Adv Math 102 20ndash78 (1993)[K1] Kong L Full field algebras operads and tensor categories Adv Math 213 271ndash340 (2007)[K2] Kong L Open-closed field algebras Commun Math Phys 280 207ndash261 (2008)[K3] Kong L Cardy condition for open-closed field algebras Commun Math Phys 283 25ndash92 (2008)[Ki] Kitaev A Private communication[KO] Kirillov AA Ostrik V On q-analog of mckay correspondence and ade classification of sl(2)

conformal field theories Adv Math 171 183ndash227 (2002)[KR] Kong L Runkel I Morita classes of algebras in modular tensor categories Adv Math 219

1548ndash1576 (2008)[Ld] Lauda AD Frobenius algebras and ambidextrous adjunctions Theo Appl Cat 16 84ndash

122 (2006)[Li1] Li H-S Regular representations of vertex operator algebras Commun Contemp Math 4

639ndash683 (2002)[Li2] Li H-S Regular representations and huang-lepowsky tensor functors for vertex operator

algebras J Alge 255 423ndash462 (2002)

872 L Kong I Runkel

The first approach uses a three-dimensional topological field theory to express cor-relators of the open-closed CFT [FeFRSFj] Here one starts from a modular tensorcategory which defines a three-dimensional topological field theory [RTT] and froma special symmetric Frobenius algebra in this modular tensor category To each open-closed world sheet X one assigns a 3-bordism MX with embedded ribbon graph con-structed from this Frobenius algebra To the boundary of MX the topological field theoryassigns a vector space B(X) and to MX itself a vector CX isin B(X) One proves thatthis collection of vectors CX provides a so-called solution to the sewing constraints [Fj]If the modular tensor category is the category of representations of a suitable vertexoperator algebra the spaces B(X) are spaces of conformal blocks and the CX are thecorrelators of an open-closed CFT In this approach one thus makes an ansatz for thecorrelators on all world sheets simultaneously and then proves that they obey the neces-sary consistency conditions The relation to CFT rests on convergence and factorisationproperties of higher genus conformal blocks and the precise list of conditions the vertexoperator algebra has to fulfill for these properties to hold is not known However froma physical perspective one expects that interesting classes of models [WFK] will haveall the necessary properties

The second approach uses the theory of vertex operator algebras to construct directlythe correlators of the genus 0 and genus 1 open-closed CFT [HK1HK2K3] More pre-cisely in this approach one uses a notion of CFT defined in [K3 Sect 1] (and called partialCFT1 there) where one glues Riemann surfaces around punctures with local coordinatesas in [VH1] instead of gluing around parametrised circles as in [Se] This approach isbased on the precise relation between genus-0 CFT and vertex operator algebras [H1]and on the fact that the category of modules over a rational vertex operator algebra isa modular tensor category [HLH2] Let us call a vertex operator algebra rational if itsatisfies the conditions in [H2 Sect 1] If one analyses the consistency conditions of agenus-01 open-closed CFT one arrives at a structure called Cardy CV |CV otimesV - algebrain [K3] It is formulated in purely categorical terms in the categories CV and CV otimesV ofmodules over the rational vertex operator algebras V and V otimes V respectively Cardyalgebras (Definition 37) are the central objects in part I of this work and we will describetheir relation to CFT in slightly more detail below The data in a Cardy algebra amountsto an open-closed CFT on a generating set of world sheets from which the entire CFTcan be obtained by repeated gluing The conditions on this data are necessary for thisprocedure to give a consistent genus-01 open-closed CFT

The two approaches just outlined start at opposite ends of the same problem Inboth cases the difficulty to obtain a complete answer lies in the lack of control over theproperties of higher genus conformal blocks Nonetheless both approaches give rise tonotions formulated in entirely categorical terms and we can compare the structures onefinds In part II we will come to the satisfying conclusion that giving a solution to thesewing constraints is essentially equivalent in a sense made precise in part II to givinga Cardy algebra

To motivate the notion of a Cardy algebra and our interest in it we would like to outlinehow it emerges when formulating closed CFT and open-closed CFT at genus-01 in thelanguage of vertex operator algebras The next one and a half pages together with a few

1 The qualifier lsquopartialrsquo refers to the fact that the gluing of punctures is only defined if the coordinates ζ1ζ2 around two punctures can be analytically extended to a large enough region containing no other puncturesso that the identification ζ1 sim 1ζ2 is well-defined That is if ζ1 can be extended to a disc of radius r then ζ2must be defined on a disc of radius greater than 1r Both discs must not contain further punctures

Cardy Algebras and Sewing Constraints I 873

remarks in the main text are the only places where we make reference to vertex operatoralgebras The reader who is not familiar with this structure is invited to skip ahead

All types of field algebras occurring below are called self-dual if they are endowedwith non-degenerate invariant bilinear forms

A genus-0 closed CFT is equivalent to an algebra over a partial dioperad consisting ofspheres with arbitrary in-coming and out-going punctures The dioperad structure allowsto compose one in-going and one out-going puncture of distinct spheres so that the resultis again a sphere Such an algebra with additional natural properties is canonically equiv-alent to a so-called self-dual conformal full field algebra [HK2K1] A conformal fullfield algebra contains chiral and anti-chiral parts the easiest nontrivial example is givenby V otimesV where V is a vertex operator algebra A conformal full field algebra containingV otimes V as a subalgebra is called a conformal full field algebra over V otimes V When V isrational the category of self-dual conformal full field algebras over V otimesV is isomorphicto the category of commutative symmetric Frobenius algebras in CV otimesV [K1 Thm 415]

Similarly a genus-0 open CFT is an algebra over a partial dioperad consisting ofdisks with an arbitrary number of in-coming and out-going boundary punctures Suchan algebra with additional natural properties is canonically equivalent to a self-dualopen-string vertex operator algebra as defined in [HK1] A vertex operator algebra V isnaturally an open-string vertex operator algebra An open-string vertex operator algebracontaining V as a subalgebra in its meromorphic centre is called an open-string vertexoperator algebra over V When V is rational the category of self-dual open-string vertexoperator algebras over V is isomorphic to the category of symmetric Frobenius algebrasin CV see [HK1 Thm 43] and [K3 Thm 610]

Finally a genus-0 open-closed CFT is an algebra over the Swiss-cheese partial dio-perad which consists of disks with both interior punctures and boundary puncturesand is equipped with an action of the partial spheres dioperad Such an algebra can beconstructed from a so-called self-dual open-closed field algebra [K2] It consists of aself-dual conformal full field algebra Acl a self-dual open-string vertex operator algebraAop and interactions between Acl and Aop satisfying certain compatibility conditionsNamely if Acl is defined over V otimes V and Aop over V one requires that the boundarycondition on a disc is V -invariant in the sense that both the chiral copy V otimes 1 and theanti-chiral copy 1 otimes V of V in Acl give the copy of V in Aop in the limit of the insertionpoint approaching a point on the boundary of the disc [K2 Def 125] An open-closedfield algebra with V -invariant boundary condition is called an open-closed field algebraover V When V is rational the category of self-dual open-closed field algebras overV is isomorphic to the category of triples (Aop|Acl ιcl-op) where Acl is a commuta-tive symmetric Frobenius CV otimesV -algebra Aop a symmetric Frobenius CV -algebra andιcl-op an algebra homomorphism T (Acl) rarr Aop satisfying a centre condition (given in(320) below) see [K2 Thm 314] and [K3 Sect 62] Here T CV otimesV rarr CV is theHuang-Lepowsky tensor product functor [HL]

The genus-1 theory does not provide new data as it is determined by taking tracesof genus-0 correlators but it does provide two additional consistency conditions themodular invariance condition for one-point correlators on the torus [So] and the Cardycondition for boundary-two-point correlators on the annulus [C2Lw] Their categor-ical formulations have been worked out in [HK3K3] Adding them to the axioms ofa self-dual open-closed field algebra over V finally results in the notion of a CardyCV |CV otimesV -algebra One can prove that the category of self-dual open-closed field alge-bras over a rational vertex operator algebra V satisfying the two genus-1 consistencyconditions is isomorphic to the category of Cardy CV |CV otimesV -algebras [K3 Thm 615]

874 L Kong I Runkel

If V is rational then so is V otimes V [DMZHK2] Thus both CV and CV otimesV aremodular tensor categories In fact CV otimesV sim= CV (CV )minus (see [FHL Thm 474] and[DMZ Thm 27]) where the minus sign relates to the particular braiding used for CV otimesV Namely for a given modular tensor category D we denote by Dminus the modular tensorcategory obtained from D by inverting braiding and twist We will also sometimes writeD+ for D The product amounts to taking direct sums of pairs of objects and tensorproducts of morphisms spaces The definition of a Cardy algebra can be stated in a waythat no longer makes reference to the vertex operator algebra V and therefore makessense in an arbitrary modular tensor category C Abbreviating C2plusmn equiv C+ Cminus this leadsto the definition of a Cardy C|C2plusmn-algebra

The relation to genus-01 open-closed CFT outlined above is the main motivation forour interest in Cardy C|C2plusmn-algebras In part I of this work we investigate how much onecan learn about Cardy algebras in the categorical setting and without the assumptionthat the modular tensor category C is given by CV for some V We briefly summariseour approach and results below

In Sect 21ndash23 we recall some basic notions we will need such as (co)lax tensorfunctors Frobenius functors and modular tensor categories In Sect 24 we study thefunctor T C2plusmn rarr C which is defined by the tensor product on C via T (oplusi Ai times Bi ) =oplusi Ai otimes Bi for Ai Bi isin C Using the braiding of C one can turn T into a tensor functorA tensor functor is automatically also a Frobenius functor and so takes a Frobeniusalgebra A in its domain category to a Frobenius algebra F(A) in its target category

An important object in this work is the functor R C rarr C2plusmn also defined in Sect 24We show that R is left and right adjoint to T As a consequence R is automatically alax and colax tensor functor but it is in general not a tensor functor However we willshow that it is still a Frobenius functor and so takes Frobenius algebras in C to Frobeniusalgebras in C2plusmn In fact it also preserves the properties simple special and symmetricof a Frobenius algebra In the case C = CV the functor R CV rarr CV otimesV was firstconstructed in [Li1Li2] using techniques from vertex operator algebras This motivatedthe present construction and notation The functor R was also considered in a slightlydifferent context in [ENO2]

The above results imply that R and T form an ambidextrous adjunction and we willuse this adjunction to transport algebraic structures between C and C2plusmn For examplethe algebra homomorphism ιcl-op T (Acl) rarr Aop in C is transported to an algebrahomomorphism ιcl-op Acl rarr R(Aop) in C2plusmn This gives rise to an alternative definitionof a Cardy C|C2plusmn-algebra as a triple (Aop|Acl ιcl-op)

To prepare the definition of a Cardy algebra in Sect 31 we discuss the so-calledmodular invariance condition for algebras in C2plusmn (Definition 31 below) We show thatwhen Acl is simple the modular invariance condition can be replaced by an easier con-dition on the quantum dimension of Acl (namely the dimension of Acl has to be that ofthe modular tensor category C) see Theorem 34

In Sect 32 we give the two definitions of a Cardy algebra and prove their equiva-lence Section 33 contains our main results We first show that for each special symmet-ric Frobenius algebra A in C (see Sect 22 for the definition of special) one obtains aCardy algebra (A|Z(A) e) where Z(A) is the full centre of A (Theorem 318) Thefull centre [Fj Def 49] is a subobject of R(A) and e Z(A) rarr R(A) is the canonicalembedding Next we prove a uniqueness theorem (Theorem 321) which states that if(Aop|Acl ιcl-op) is a Cardy algebra such that dim Aop = 0 and Acl is simple then Aopis special and (Aop|Acl ιcl-op) is isomorphic to (Aop|Z(Aop) e) When combined withpart II of this work this result amounts to [Fj Thm 426] and provides an alternative

Cardy Algebras and Sewing Constraints I 875

(and shorter) proof Finally we show that for every simple modular invariant commu-tative symmetric Frobenius algebra Acl in C2plusmn there exists a simple special symmetricFrobenius algebra Aop and an algebra homomorphism ιcl-op Acl rarr R(Aop) such that(Aop Acl ιcl-op) is a Cardy algebra (Theorem 322) This theorem is closely related toa result announced in [Muuml2] and provides an independent proof in the framework ofCardy algebras

In physical terms these two theorems mean that a rational open-closed CFT with aunique closed vacuum state can be uniquely reconstructed from its correlators involvingonly discs with boundary punctures and that every closed CFT with unique vacuum andleftright rational chiral algebra V otimes V occurs as part of such an open-closed CFT

2 Preliminaries on Tensor Categories

In this section we review some basic facts of tensor categories and fix our conventionsand notations along the way

21 Tensor categories and (co)lax tensor functors In a tensor (or monoidal) categoryC with tensor product bifunctor otimes and unit object 1 for U VW isin C we denote the

associator U otimes (V otimes W )sim=minusrarr (U otimes V ) otimes W by αUVW the left unit isomorphism

1 otimes Usim=minusrarr U by lU and the right unit isomorphism U otimes 1

sim=minusrarr U by rU If C is braidedfor U V isin C we write the braiding isomorphism as cUV U otimes V rarr V otimes U

Let C1 and C2 be two tensor categories with units 11 and 12 respectively For simplic-ity we will often write otimes α l r for the data of both C1 and C2 Lax and colax tensorfunctors are defined as follows see eg [Y Ch I3] or [Ln Ch I12]

Definition 21 A lax tensor functor G C1 rarr C2 is a functor equipped with a mor-phism φG

0 12 rarr G(11) in C2 and a natural transformation φG2 otimes(G times G) rarr G otimes

such that the following three diagrams commute

G(A)otimes (G(B)otimes G(C))α

idG(A)otimesφG2

(G(A)otimes G(B))otimes G(C)

φG2 otimesidG(C)

G(A)otimes G(B otimes C)

φG2

G(A otimes B)otimes G(C)

φG2

G(A otimes (B otimes C))

G(α) G((A otimes B)otimes C)

(21)

12 otimes G(A)

φG0 otimesidG(A)

lG(A) G(A)

G(lminus1A )

G(11)otimes G(A)

φG2 G(11 otimes A)

G(A)otimes 12

idG(A)otimesφG0

rG(A) G(A)

G(rminus1A )

G(A)otimes G(11)

φG2 G(A otimes 11)

(22)

Definition 22 A colax tensor functor is a functor F C1 rarr C2 equipped with a mor-phismψ F

0 F(11) rarr 12 in C2 and a natural transformationψ F2 Fotimes rarr otimes(F times F)

such that the following three diagrams commute

876 L Kong I Runkel

F(A)otimes (F(B)otimes F(C))α (F(A)otimes F(B))otimes F(C)

F(A)otimes F(B otimes C)

idF(A)otimesψ F2

F(A otimes B)otimes F(C)

ψF2 otimesidF(C)

F(A otimes (B otimes C))F(α)

ψF2

F((A otimes B)otimes C)

ψF2

(23)

12 otimes F(A) F(A)lminus1F(A)

F(11)otimes F(A)

ψF0 otimesidF(A)

F(11 otimes A)ψF

2

F(lA)

F(A)otimes 12 F(A)rminus1

F(A)

F(A)otimes F(11)

idF(A)otimesψ F0

F(A otimes 11)

F(rA)

ψF2

(24)

We denote a lax tensor functor by (G φG2 φ

G0 ) or just G and a colax tensor functor

by (F ψ F2 ψ

F0 ) or F

Definition 23 A tensor functor T C1 rarr C2 is a lax tensor functor (T φT2 φ

T0 ) such

that φT0 φ

T2 are both isomorphisms

A tensor functor (T φT2 φ

T0 ) is automatically a colax tensor functor (T ψT

2 ψT0 )

with ψT0 = (φT

0 )minus1 and ψT

2 = (φT2 )

minus1In the next section we will discuss algebras in tensor categories The defining proper-

ties (21) and (22) of a lax tensor functor are analogues of the associativity the left-unitand the right-unit properties of an algebra Indeed a lax tensor functor G C1 rarr C2maps a C1-algebra to a C2-algebra Similarly (23) and (24) are analogues of the coasso-ciativity the left-counit and the right-counit properties of a coalgebra and a colax tensorfunctor F C1 rarr C2 maps a C1-coalgebra to a C2-coalgebra We will later make useof functors that take Frobenius algebras to Frobenius algebras This requires a strongercondition than being lax and colax and leads to the notion of a lsquofunctor with Frobeniusstructurersquo or lsquoFrobenius monoidal functorrsquo [SzDPP] which we will simply refer to asFrobenius functor

Definition 24 A Frobenius functor F C1 rarr C2 is a tuple F equiv (F φF2 φ

F0 ψ

F2 ψ

F0 )

such that (F φF2 φ

F0 ) is a lax tensor functor (F ψ F

2 ψF0 ) is a colax tensor functor

and such that the following two diagrams commute

F(A)otimes (F(B)otimes F(C))α (F(A)otimes F(B))otimes F(C)

φF2 otimesidF(C)

F(A)otimes F(B otimes C)

idF(A)otimesψ F2

φF2

F(A otimes B)otimes F(C)

F(A otimes (B otimes C))F(α) F((A otimes B)otimes C)

ψF2

(25)

Cardy Algebras and Sewing Constraints I 877

F(A)otimes (F(B)otimes F(C))

idF(A)otimesφF2

(F(A)otimes F(B))otimes F(C)αminus1

F(A)otimes F(B otimes C) F(A otimes B)otimes F(C)

ψF2 otimesidF(C)

φF2

F(A otimes (B otimes C))

ψF2

F((A otimes B)otimes C)F(αminus1)

(26)

Proposition 25 If (F φF2 φ

F0 ) is a tensor functor then F is a Frobenius functor with

ψ F0 = (φF

0 )minus1 and ψ F

2 = (φF2 )

minus1

Proof Since F is a tensor functor it is lax and colax If we replace ψ F2 by (φF

2 )minus1 in

(25) and (26) both commuting diagrams are equivalent to (21) which holds becauseF is lax Thus F is a Frobenius functor

The converse statement does not hold For example the functor R which we definein Sect 24 is Frobenius but not tensor

Let us recall the notion of adjunctions and adjoint functors [Ma Ch IV1]

Definition 26 An adjunction from C1 to C2 is a triple 〈FG χ〉 where F and G arefunctors

F C1 rarr C2 G C2 rarr C1

and χ is a natural isomorphism which assigns to each pair of objects A1 isin C1 A2 isin C2a bijective map

χA1A2 HomC2(F(A1) A2)sim=minusminusrarr HomC1(A1G(A2))

which is natural in both A1 and A2 F is called a left-adjoint of G and G is called aright-adjoint of F

For simplicity we will often abbreviate χA1A2 as χ Associated to each adjunction

〈FG χ〉 there are two natural transformations idC1

δminusrarr G F and FGρminusrarr idC2 where

idC1 and idC2 are identity functors given by

δA1 = χ(idF(A1)) ρA2 = χminus1(idG(A2)) (27)

for Ai isin Ci i = 1 2 They satisfy the following two identities

GδGminusrarr G FG

Gρminusrarr G = GidGminusminusrarr G F

Fδminusrarr FG FρFminusrarr F = F

idFminusminusrarr F (28)

We have for g F(A1) rarr A2 and f A1 rarr G(A2)

χ(g) = G(g) δA1 χminus1( f ) = ρA2 F( f ) (29)

For simplicity δA1 and ρA2 are often abbreviated as δ and ρ respectivelyLet 〈FG χ〉 be an adjunction from a tensor category C1 to a tensor category C2

and (F ψ F2 ψ

F0 ) a colax tensor functor from C1 to C2 We can define a morphism

878 L Kong I Runkel

φG0 11 rarr G(12) and a natural transformation φG

2 otimes (G times G) rarr G otimes by forA B isin C2

φG0 = χ(ψ F

0 ) = 11δ11minusrarr G F(11)

G(ψ F0 )minusminusminusminusrarr G(12)

φG2 = χ((ρA otimes ρB) ψ F

2 ) = G(A)otimes G(B)δminusrarr G F (G(A)otimes G(B))

G(ψ F2 )minusminusminusminusrarr G (FG(A)otimes FG(B))

G(ρAotimesρB )minusminusminusminusminusminusrarr G(A otimes B) (210)

where we have used the first identity in (29) Notice that φG2 is natural because it is a

composition of natural transformations One can easily show that ψ F0 and ψ F

2 can bere-obtained from φG

0 and φG2 as follows

ψ F0 = χminus1(φG

0 ) = F(11)FφG

0minusminusrarr FG(12)ρminusrarr 12

ψ F2 = χminus1(φG

2 (δ otimes δ)) = F(U otimes V )F(δotimesδ)minusminusminusminusrarr F (G F(U )otimes G F(V ))

FφG2minusminusrarr FG (F(U )otimes F(V ))

ρminusrarr F(U )otimes F(V ) (211)

for U V isin C1 The following result is standard for the sake of completeness we givea proof in Appendix A1

Lemma 27 (F ψ F2 ψ

F0 ) is a colax tensor functor iff (G φG

2 φG0 ) is a lax tensor

functor

22 Algebras in tensor categories An algebra in a tensor category C or a C-algebra isa triple A = (Am η) where A is an object of C m (the multiplication) is a morphismA otimes A rarr A such that m (m otimes idA) αAAA = m (idA otimes m) and η (the unit) is amorphism 1 rarr A such that m (idA otimes η) = idA rA and m (ηotimes idA) = idA lAIf C is braided and m cAA = m then A is called commutative

A left A-module is a pair (MmM ) where M isin C and m M is a morphism AotimesM rarr Msuch that mM(idAotimesm M ) = m M(m AotimesidM )αAAM and m M(ηAotimesidM ) = idMlM Right A-modules and A-bimodules are defined similarly

Definition 28 Let C be a tensor category and let A be an algebra in C

(i) A is called simple iff it is simple as a bimodule over itselfLet C be in addition k-linear for k a field

(ii) A is called absolutely simple iff the space of A-bimodule maps from A to itself isone-dimensional dimk HomA|A(A A) = 1

(iii) A is called haploid iff dimk Hom(1 A) = 1 [FS Def 43]

In the following we will assume that all tensor categories are strict to avoid spelling outassociators and unit constraints

A C-coalgebra A = (A ε) is defined analogously to a C-algebra ie A rarrA otimes A and ε A rarr 1 obey coassociativity and counit conditions

If C is braided and if A and B are C-algebras there are two in general non-isomorphicalgebra structures on A otimes B We choose A otimes B to be the C-algebra with multiplicationm AotimesB = (m Aotimesm B)(idAotimescminus1

ABotimesidB) and unitηAotimesB = ηAotimesηB Similarly if A and Bare C-coalgebras then A otimes B becomes a C-coalgebra if we choose the comultiplicationAotimesB = (idA otimes cAB otimes idB) (A otimesB) and the counit εAotimesB = εA otimes εB

Cardy Algebras and Sewing Constraints I 879

Definition 29 A Frobenius algebra A = (Am η ε) is an algebra and a coalge-bra such that the coproduct is an intertwiner of A-bimodules

(idA otimes m) (otimes idA) = otimes m = (m otimes idA) (idA otimes)

We will use the following graphical representation for the morphisms of a Frobeniusalgebra

m =A A

A

η =A

=A A

A

ε =A

(212)

A Frobenius algebra A in a k-linear tensor category for k a field is called special iffm = ζ idA and ε η = ξ id1 for nonzero constants ζ ξ isin k If ζ = 1 we call A nor-malised-special A Frobenius algebra homomorphism between two Frobenius algebrasis both an algebra homomorphism and a coalgebra homomorphism

A (strictly) sovereign tensor category is a tensor category equipped with a left and aright duality which agree on objects and morphisms (see eg [BiFS] for more details)We will write the dualities as

Uor U

= dU Uor otimes U rarr 1U Uor

= dU U otimes Uor rarr 1

U Uor

= bU 1 rarr U otimes UorUor U

= bU 1 rarr Uor otimes U

(213)

In terms of these we define the left and right dimension of an object U as

diml U = dU bU dimr U = dU bU (214)

both of which are elements of Hom(1 1)Let now C be a sovereign tensor category For a Frobenius algebra A in C we define

two morphisms

A =

A

Aor

primeA =

A

Aor

(215)

Definition 210 A Frobenius algebra A is symmetric iff A = primeA

The following lemma shows that under certain conditions we do not need to distin-guish the various notions of simplicity in Definition 28

Lemma 211 Let A be a commutative symmetric Frobenius algebra in a C-linear semi-simple sovereign braided tensor category C and suppose that diml A = 0 Then thefollowing are equivalent

880 L Kong I Runkel

(i) A is simple(ii) A is absolutely simple

(iii) A is haploid

Proof (ii)hArr(iii) A is haploid iff it is absolutely simple as a left module over itself[FS Eq (417)] Furthermore for a commutative algebra we have HomA(A A) =HomA|A(A A) and so A is haploid iff it is absolutely simple(i)rArr(ii) If A is simple then every nonzero element of HomA|A(A A) is invertibleHence this space forms a division algebra over C and is therefore isomorphic to C(iii)rArr(i) Since C is semi-simple and A is haploid also Hom(A 1) is one-dimensionalThe counit ε is a nonzero element in this space and so gives a basis This implies firstlythat ε η = 0 and secondly that there is a constant β isin C such that

β middot ε = dA (idAor otimes m) (bA otimes idA) (216)

Composing with η from the right yields β εη = diml A The right-hand side is nonzeroand so β = 0 By [FRS Lem 311] A is special We have already proved (ii)hArr(iii)and so A is absolutely simple A special Frobenius algebra in a semi-simple categoryhas a semi-simple category of bimodules (apply [FS Prop 524] to the algebra tensoredwith its opposite algebra) For semi-simple C-linear categories simple and absolutelysimple are equivalent2 Thus A is simple Remark 212 For a Frobenius algebra A the morphisms (215) are invertible and henceA sim= Aor In this case one has diml A = dimr A [FS Rem 363] and so we could havestated the above lemma equivalently with the condition dimr A = 0

Let F C1 rarr C2 be a lax tensor functor between two tensor categories C1 C2 and

let (Am A ηA) be an algebra in C1 Define morphisms F(A)otimes F(A)m F(A)minusminusminusrarr F(A) and

12ηF(A)minusminusminusrarr F(A) as

m F(A) = F(m A) φF2 ηF(A) = F(ηA) φF

0 (217)

Then (F(A)m F(A) ηF(A)) is an algebra in C2 [JS Prop 55] If f A rarr B is analgebra homomorphism between two algebras A B isin C1 then F( f ) F(A) rarr F(B)is also an algebra homomorphism If (MmM ) is a left (or right) A-module in C1 then(F(M) F(mM )φF

2 ) is a left (or right) F(A)-module if M has a A-bimodule structurethen F(M) naturally has a F(A)-bimodule structure

Similarly if (AA εA) is a coalgebra in C1 and F C1 rarr C2 is a colax tensor func-

tor then F(A) with coproduct F(A)F(A)minusminusminusrarr F(A) otimes F(A) and counit F(A)

εF(A)minusminusminusrarr 12given by

F(A) = ψ F2 F(A) εF(A) = ψ F

0 F(εA) (218)

is a coalgebra in C2 If f A rarr B is a coalgebra homomorphism between two coalgebrasA B isin C1 then F( f ) F(A) rarr F(B) is also a coalgebra homomorphism

2 To see this note that if U is simple then the C-vector space Hom(UU ) is a division algebra and henceHom(UU ) = C idU Conversely if U is not simple then U = U1 oplus U2 and Hom(UU ) contains at leasttwo linearly independent elements namely idU1 and idU2

Cardy Algebras and Sewing Constraints I 881

Proposition 2133 If F C1 rarr C2 is a Frobenius functor and (Am A ηAA εA) aFrobenius algebra in C1 then (F(A)m F(A) ηF(A)F(A) εF(A)) is a Frobenius alge-bra in C2

Proof One Frobenius property (m F(A)otimes idF(A)) (idF(A)otimesF(A)) = F(A) m F(A)follows from the commutativity of the following diagram (we spell out the associativityisomorphisms)

F(A)otimes F(A)idF(A)otimesF(A)

φF2

F(A)otimes F(A otimes A)idF(A)otimesψF

2

φF2

F(A)otimes (F(A)otimes F(A))

αF(A)F(A)F(A)

F(A otimes A)

F(idAotimesA)

F(m A)

F(A otimes (A otimes A))

F(αAAA)

(F(A)otimes F(A))otimes F(A)

φF2 otimesidF(A)

F((A otimes A)otimes A)

F(m AotimesidA)

ψF2 F(A otimes A)otimes F(A)

F(m A)otimesidF(A)

F(A)

F(A) F(A otimes A)ψF

2 F(A)otimes F(A)

(219)

The commutativity of the upper-left subdiagram follows from the naturalness of φF2 that

of the upper-right subdiagram follows from (25) that of the lower-left subdiagram fol-lows from the Frobenius properties of A and that of the lower-right subdiagram followsfrom the naturalness of ψ F

2 The proof of the other Frobenius property is similar Proposition 214 If F C1 rarr C2 is a tensor functor and A a Frobenius algebra in C1then

(i) F(A) has a natural structure of Frobenius algebra as given in Proposition 213(ii) If A is (normalised-)special so is F(A)

Proof Part (i) follows from Propositions 25 and 213 Part (ii) is a straightforwardverification of the definition using ψ F

2 = (φF2 )

minus1 and ψ F0 = (φF

0 )minus1

Let C1 C2 be sovereign tensor categories and F C1 rarr C2 a Frobenius functor Wedefine two morphisms IF(Aor) I prime

F(Aor) F(Aor) rarr F(A)or for a Frobenius algebra A inC1 as follows

IF(Aor) = ((ψ F0 F(dA) φF

2 )otimes idF(A)or) (idF(Aor) otimes bF(A))

I primeF(Aor) = (idF(A)or otimes (ψ F

0 F(dA) φF2 )) (bF(A) otimes idF(Aor))

(220)

It is easy to see that these are isomorphisms

Lemma 215 If F C1 rarr C2 is a Frobenius functor and A a Frobenius algebra in C1then

F(A) = IF(Aor) F(A) primeF(A) = I prime

F(Aor) F(primeA) (221)

3 After the preprint of the present paper appeared we noticed that this proposition is also proved in[DP Cor 5]

882 L Kong I Runkel

Proof We only prove the first equality the second one can be seen in the same way Bydefinition we have

IF(Aor) F(A)

= ((ψ F0 F(dA) φF

2 )otimes idF(A)or) (idF(Aor) otimes bF(A)) F(A)

=[(ψ F

0 F(dA) φF2 ) (F(A)otimes idF(A))

]otimes idF(A)or

(idF(A) otimes bF(A))

For the term inside the square brackets we find

ψ F0 F(dA) φF

2 (F(A)otimes idF(A)) = ψ F0 F(dA) F(A otimes idA) φF

2

= ψ F0 F(dA (A otimes idA)) φF

2 = ψ F0 F(εA m A) φF

2 (222)

On the other hand by definitionF(A) = [((ψ F0 F(εA) (F(m A)φF

2 ))otimes idF(A)or] (idF(A) otimes bF(A)) This demonstrates the first equality in (221) Proposition 216 Let F C1 rarr C2 be a tensor functor G C2 rarr C1 a functor〈FG χ〉 an adjunction A a C1-algebra and B a C2-algebra Then f A rarr G(B)is an algebra homomorphism if and only if f = χminus1( f ) F(A) rarr B is an algebrahomomorphism

Proof We need to show that

mG(B) ( f otimes f ) = f m A and f ηA = ηG(B) (223)

is equivalent to

m B ( f otimes f ) = f m F(A) and f ηF(A) = ηBs (224)

We first prove that the first identity in (223) is equivalent to the first identity in (224)For the left-hand side of the first identity in (223) we have the following equalities

mG(B) ( f otimes f )(1)= G(m B) φG

2 ( f otimes f )(2)= G(m B) G(ρ otimes ρ) G(ψ F

2 ) δ ( f otimes f )(3)= G(m B) G(ρ otimes ρ) G(ψ F

2 ) G F( f otimes f ) δ(4)= G(m B) G(ρ otimes ρ) G(F( f )otimes F( f )) G(ψ F

2 ) δ(5)= G

(m B (ρ otimes ρ) (F( f )otimes F( f )) ψ F

2

) δ

(6)= χ(

m B (ρ otimes ρ) (F( f )otimes F( f )) ψ F2

) (225)

where (1) is the definition of mG(B) in (217) (2) is the second identity in (210) (3)and (4) are naturality of δ and ψ F

2 respectively step (5) is functoriality of G and finallystep (6) is (29) For the right hand side of the first identity in (223) we get

f m A(1)= Gρ δG ( f m A)

(2)= Gρ G F( f m A) δ(3)= G(ρ F( f m A)) δ(4)= χ(ρ F( f ) F(m A)) (226)

Cardy Algebras and Sewing Constraints I 883

where (1) is the adjunction property (28) (2) is naturality of δ (3) functoriality of Gand (4) amounts to (29) and functoriality of F

On the other hand we see that the first equality in (224) is equivalent to

m B (ρ otimes ρ) (F( f )otimes F( f )) = ρ F( f ) F(m A) φF2 (227)

Using that φF2 is invertible with inverse (φF

2 )minus1 = ψ F

2 and that χ is an isomorphismit follows that the statement that (225) is equal to (226) is equivalent to the identity(227)

Now we prove that the second identity in (223) is equivalent to the second identity in(224) Using (217) and (210) we can write ηG(B) = G(ηB)φF

0 = G(ηB)G(ψ F0 )δ1

Together with (29) this shows that the second identity in (223) is equivalent to

f ηA = χ(ηB ψ F0 ) (228)

On the other hand the second identity in (224) is equivalent to

ρ F( f ) F(ηA) φF0 = ηB (229)

which by φF0 = (ψ F

0 )minus1 and (29) is further equivalent to (228)

Definition 217 Let (Am A ηAA εA) and (Bm B ηBB εB) be two Frobeniusalgebras in a tensor category C For f A rarr B we define f lowast B rarr A by

f lowast = ((εB m B)otimes idA) (idB otimes f otimes idA) (idB otimes (A ηA)) (230)

The following lemma is immediate from the definition of (middot)lowast and the properties ofFrobenius algebras We omit the proof

Lemma 218 Let C be a tensor category let A BC be Frobenius algebras in C andlet f A rarr B and g B rarr C be morphisms

(i) (g f )lowast = f lowast glowast(ii) f is a monomorphism iff f lowast is an epimorphism

(iii) f is an algebra map iff f lowast is a coalgebra map(iv) If f is a homomorphism of Frobenius algebras then f lowast f = idA and f f lowast =

idB(v) If C is sovereign and if A and B are symmetric then f lowastlowast = f

Let C and D be tensor categories and let F C rarr D be a Frobenius functor GivenFrobenius algebras A B in C and a morphism f A rarr B the next lemma shows how(middot)lowast behaves under F

Lemma 219 F( f lowast) = F( f )lowast

Proof The definition of the structure morphisms of the Frobenius algebra F(A) is givenin (217) and (218) Substituting these definitions gives

F( f )lowast =[(ψ F

0 F(εB) F(m B) φF2

)otimes idF(A)

] [

idF(B) otimes F( f )otimes idF(A)]

[idF(B) otimes

(ψ F

2 F(A) F(ηA) φF0

)]

= (ψ F0 otimes idF(A)) [

F (εB m B (idB otimes f ))otimes idF(A)]

(φF2 otimes idF(A)) (idF(B) otimes ψ F

2 )

[idF(B) otimes F(A ηA)

] (idF(B) otimes φF0 ) (231)

884 L Kong I Runkel

In the middle line of the last expression we can use the defining property (25) of F namely we substitute (φF

2 otimes idF(A)) (idF(B) otimes ψ F2 ) = ψ F

2 φF2 Then ψ F

2 can bemoved to the left and φF

2 to the right until they can be omitted against ψ F0 and φF

0 respectively using (22) and (24) This results in

F( f )lowast = F ((εB m B (idB otimes f ))otimes idA) F (idB otimes (A ηA)) (232)

which is nothing but F( f lowast)

23 Modular tensor categories Let C be a modular tensor category [TBK] ie anabelian semi-simple finite C-linear ribbon category with simple tensor unit 1 and a non-degeneracy condition on the braiding (to be stated in a moment) We denote the set ofequivalence classes of simple objects in C by I elements in I by i j k isin I and theirrepresentatives by Ui U j Uk We also set U0 = 1 and for an index k isin I we define kby Uk

sim= Uork

Since the tensor unit is simple we shall for modular tensor categories identifyHom(1 1) sim= C (cf footnote 2) Define numbers si j isin C by4

si j = UiU j (233)

They obey si j = s ji and s0i = dim Ui see eg [BK Sect 31] (In a ribbon cate-gory the left and right dimension (214) of Ui coincide and are denoted by dim Ui )The non-degeneracy condition on the braiding of a modular tensor category is that the|I|times|I|-matrix s should be invertible In fact [BK Thm 317]

sumkisinI

sik sk j = Dim C δij (234)

where Dim C = sumiisinI(dim Ui )

2 One can show (even in the weaker context of fusioncategories over C) that Dim C ge 1 [ENO1 Thm 23] In particular Dim C = 0 We fixonce and for all a square root

radicDim C of Dim C

Let us fix a basis λα(i j)kN k

i jα=1 in HomC(Ui otimesU j Uk) and the dual basis ϒ(i j)k

α N ki j

α=1

in HomC(UkUi otimesU j ) The duality of the bases means that λα(i j)k ϒ(i j)kβ = δαβ idUk

We also fix λ(0i)i = λ(i0)i = idUi We denote the basis vectors graphically as follows

λα(i j)k = α

Uk

Ui U j

ϒ(i j)kα = α

Uk

Ui U j

(235)

4 In the graphical notation used below we have given an orientation to the ribbons indicated by the arrowsFor example it is understood that this orientation determines which of the duality morphisms in (213) to use

Cardy Algebras and Sewing Constraints I 885

For V isin C we also choose a basis b(iα)V of HomC(VUi ) and the dual basis bV(iβ)

of HomC(Ui V ) for i isin I such that b(iα)V bV(iβ) = δαβ idUi We use the graphical

notation

b(iα)V = α

Ui

V

bV(iα) = α

Ui

V

(236)

Given two modular tensor categories C and D by C D we mean the tensor productof additive categories over C [BK Def 1115] ie the category whose objects are directsums of pairs V times W of objects V isin C and W isin D and whose morphism spaces are

HomCD(V times W V prime times W prime) = HomC(V V prime)otimesC HomC(WW prime) (237)

for pairs and direct sums of these if the objects are direct sums of pairsIf we replace the braiding and the twist in C by the antibraiding cminus1 and the antitwist

θminus1 respectively we obtain another ribbon category structure on C In order to distin-guish these two distinct structures we denote (C c θ) and (C cminus1 θminus1) by C+ and Cminusrespectively As in the Introduction we will abbreviate

C2plusmn = C+ Cminus (238)

Note that a set of representatives of the simple objects in C2plusmn is given by Ui times U j fori j isin I

For the remainder of Sect 2 we fix a modular tensor category C

24 The functors T and R The tensor product bifunctor otimes can be naturally extendedto a functor T C2plusmn rarr C Namely T (oplusN

i=1Vi times Wi ) = oplusNi=1Vi otimes Wi for all Vi Wi isin C

and N isin N The functor T becomes a tensor functor as follows For φT0 1 rarr T (1 times 1)

take φT0 = id1 (or lminus1

1 in the non-strict case) Next notice that for U VW X isin C

T (U times V )otimes T (W times X)= (U otimes V )otimes (W otimes X)

T ((U times V )otimes (W times X))= (U otimes W )otimes (V otimes X)(239)

We define φT2 T (U times V )otimes T (W times X) rarr T ((U times V )otimes (W times X)) by

φT2 = idU otimes cminus1

W V otimes idX (240)

(In the non-strict case the appropriate associators have to be added) The above definitionof φT

2 can be naturally extended to a morphism φT2 T (M1 otimes M2) rarr T (M1)otimes T (M2)

for any pair of objects M1M2 in C2plusmn The following result can be checked by directcalculation [JS Prop 52]

Lemma 220 The triple (T φT2 φ

T0 ) gives a tensor functor

886 L Kong I Runkel

In particular (T φT2 φ

T0 ψ

T2 ψ

T0 ) where ψT

2 = (φT2 )

minus1 and ψT0 = (φT

0 )minus1 gives

a Frobenius functorDefine the functor R C rarr C2plusmn as follows for A isin C and f isin HomC(A B)

R(A) =oplusiisinI

(A otimes Uori )times Ui R( f ) =

oplusiisinI

( f otimes idUori)times idUi (241)

This functor was also considered in a slightly different context in [ENO2 Prop 23]The family of isomorphisms γ R

A = oplusiisinI DimCdim Ui

id(AotimesUori )timesUi

isin Aut(R(A)) defines a

natural isomorphism γ R R rarr ROur next aim is to show that R is left and right adjoint to T in other words R

and T form an ambidextrous adjunction (see eg [Ld] for a discussion of ambidextrousadjunctions) To this end we introduce two linear isomorphisms for A isin C and M isin C2plusmn

χ HomC(T (M) A) minusrarr HomC2plusmn(M R(A))

χ HomC(A T (M)) minusrarr HomC2plusmn(R(A)M)(242)

If we decompose M as M = oplusNn=1 Ml

n times Mrn then χ and χ are given by

χ Noplus

n=1

fn

Mln Mr

n

A

rarrNoplus

n=1

oplusiisinI

sumα

fn

Mln

Mrn

A

α

Uori

times α

Mrn

Ui

(243)

and

χ Noplus

n=1 gn

Mln Mr

n

A

rarrNoplus

n=1

oplusiisinI

sumα gn

Mln

Mrn

A

α

Uori

times α

Ui

Mrn

DimCdim Ui

(244)

Notice that χ and χ are independent of the choice of basis

Theorem 221 〈T R χ〉 and 〈R T χminus1〉 are adjunctions ie R is both left and rightadjoint of T

Proof Write M as M = oplusNn=1 Ml

n times Mrn The isomorphism χ amounts to the following

composition of natural isomorphisms

HomC(T (M) A) = oplusnHomC(Mln otimes Mr

n A)sim= oplusni HomC(Ml

n otimes Ui A)otimes HomC(Mrn Ui )

sim= oplusni HomC(Mln A otimes Uor

i )otimes HomC(Mrn Ui ) = HomC2plusmn(M R(A)) (245)

Cardy Algebras and Sewing Constraints I 887

Thus χ is natural Let (γ RA )

lowast HomC2plusmn(R(A)M) rarr HomC2plusmn(R(A)M) denote the

pull-back of γ RA The isomorphism χ is equal to the composition of (γ R

A )lowast and the

following sequence of natural isomorphisms

HomC(A T (M)) = oplusn HomC(AMln otimes Mr

n)

sim= oplusni HomC(AMln otimes Ui )otimes HomC(Ui Mr

n)

sim= oplusni HomC(A otimes Uori Ml

n)otimes HomC(Ui Mrn) = HomC2plusmn(R(A)M) (246)

We have proved that both χ and χ are natural isomorphisms There are four natural transformations associated to χ and χ namely

idC2plusmnδminusrarr RT

ρminusrarr idC2plusmn and idCδminusrarr T R

ρminusrarr idC (247)

defined by for A isin C M isin C2plusmn

δM = χ(idT (M)) ρA = χminus1(idR(A))

ρM = χ (idT (M)) δA = χminus1(idR(A))(248)

They can be expressed graphically as follows with M = oplusNn=1 Ml

n times Mrn

δM =oplusni

sumα

Mln

Mln Mr

n Uori

αtimes

Ui

Mrn

α

ρA =oplusiisinI

A

A

Uori Ui

(249)

ρM =oplusni

sumα

Mln

Mln

Mrn Uor

i

α times

Ui

Mrn

α DimCdim Ui

δA =oplusiisinI

A

A Uori Ui

dim UiDimC

Note that

ρM δM = Dim C middot idM and ρA δA = idA (250)

Lemma 222 The functors T and R as maps on the sets of morphisms have left inversesand thus are injective

Proof Let f A rarr B be a morphism in C We define a map Q R HomC2plusmn(R(A)

R(B)) rarr HomC(A B) by f prime rarr ρB T ( f prime) δA Then we have

Q R R( f ) = ρB T R( f ) δA = ρB δB f = f (251)

888 L Kong I Runkel

where we used naturality of δ and (250) in the second and third equalities respectivelySo Q R is a left inverse of R on morphisms Thus R is injective on morphisms Similarlylet g M rarr N be a morphism in C2plusmn We define a map QT HomC(T (M) T (N )) rarrHomC2plusmn(M N ) by gprime rarr (Dim C)minus1 middot ρN R(gprime) δM Then we have

QT T (g) = (Dim C)minus1 middot ρN RT (g) δM = (Dim C)minus1 middot ρN δN g = g (252)

So QT is a left inverse of T on morphisms Thus T is injective on morphisms Using (29) and (250) one can express the two inverse maps χminus1 χminus1 as follows

for f isin HomC2plusmn(M R(A)) and g isin HomC2plusmn(R(A)M)

χminus1( f ) = ρ T ( f ) χminus1(g) = T (g) δ (253)

By Proposition 25 and Lemma 27 R is both a lax and colax tensor functor In particularφR

0 1 times 1 rarr R(1) is given by

φR0 = χ (ψT

0 ) = R(ψT0 ) δ1 times 1 = id1times1 (254)

and φR2 R(A)otimes R(B) rarr R(A otimes B) by φR

2 = R(ρA otimes ρB) R(ψT2 ) δ which can

be expressed graphically as

φR2 =

oplusi jkisinI

sumα

A

A

Uori B

B

Uorj

Uork

α

times

Ui U j

α

Uk

(255)

Similarly ψ R0 R(1) rarr 1 times 1 is given by

ψ R0 = ρ1 R(φT

0 ) = Dim C id1times1 (256)

and ψ R2 R(A otimes B) rarr R(A) otimes R(B) by ψ R

2 = ρ R(φT2 ) R(δA otimes δB) which in

graphical notation reads

ψ R2 =

oplusi jkisinI

sumα

A

A Uori

B

B Uorj

Uork

α

times

Ui U j

α

Uk

dim Ui dim U j

dim Uk Dim C

(257)

If C has more than one simple object then R does not take the tensor unit of C to thetensor unit of C2plusmn and so is clearly not a tensor functor However we will show that R isstill a Frobenius functor This will imply that if A is a Frobenius algebra in C then

R(A) = (R(A)m R(A) ηR(A)R(A) εR(A)) (258)

Cardy Algebras and Sewing Constraints I 889

is a Frobenius algebra in C2plusmn where the structure morphisms were given in (217) and(218) In the case A = 1 it was proved in [Muuml1 Prop 41] (see also [Fr Lem 619]and [K1 Thm 52]) that (258) is a commutative simple symmetric normalised-specialFrobenius algebra in C2plusmn In fact given a Frobenius algebra A in C it is straightforwardto verify that the structure morphisms in (258) are precisely those of (A times 1)otimes R(1)cf Sect 22

Proposition 223 (R φR2 φ

R0 ψ

R2 ψ

R0 ) is a Frobenius functor

Proof Using the explicit graphical expression of φR2 φ

R0 ψ

R2 ψ

R0 it is easy to see that

the commutativity of the diagrams (25) and (26) are equivalent to the statement thatR(1) with structure morphisms as in (258) is a Frobenius algebra in C2plusmn The latterstatement is true by [Muuml1 Prop 41]

From Lemma 220 and Proposition 223 we see that T and R take Frobenius alge-bras to Frobenius algebras The following two propositions show how the properties ofFrobenius algebras are transported

Proposition 224 Let A be a Frobenius algebra in C2plusmn Then T (A) is a Frobenius algebrain C and

(i) A is symmetric iff T (A) is symmetric(ii) A is (normalised-)special iff T (A) is (normalised-)special

Proof For part (i) write A as a direct sum oplusNn=1 Al

n times Arn Then the maps IT (A) I prime

T (A) T (Aor) rarr T (A)or defined in (220) are given by

IT (A) = I primeT (A) = oplusN

n=1c(Aln)

or(Arn)

or (259)

Therefore by (221) T (A) = primeT (A) is equivalent to T (A) = T (prime

A) Since byLemma 222 T is injective on morphisms this proves part (i) Part (ii) can be checkedin the same way for example the condition mT (A) T (A) = ζ idT (A) is easily checkedto be equivalent to T (m A A) = ζ T (idA) Proposition 225 Let A be a Frobenius algebra A in C Then R(A) is a Frobeniusalgebra in C2plusmn and

(i) A is symmetric iff R(A) is symmetric(ii) A is (normalised-)special iff R(A) is (normalised-)special

Proof Recall that the structure morphisms of the Frobenius algebra R(A) are equal tothose of (A times 1) otimes R(1) Using this equality part (i) and (ii) follow because R(1) issymmetric and normalised-special For example

m R(A) R(A) = [(m A A)times id1] otimes idR(1) = R(m A A) (260)

so that m R(A) R(A) = ζ idR(A) is equivalent to R(m A A) = ζ R(idA) which byLemma 222 is equivalent to m A A = ζ idA

The functor R has one additional property not shared by T namely R takes absolutelysimple algebras to absolutely simple algebras We will see explicitly in Sect 33 thatthis is not true for T

890 L Kong I Runkel

Lemma 226 For a C-algebra A the map

R HomA|A(A A) rarr HomR(A)|R(A)(R(A) R(A)) (261)

given by f rarr R( f ) is well-defined and an isomorphism

Proof Since R is a lax tensor functor R(A) is naturally a R(A)-bimodule It is easy tosee that R in (261) is a well-defined map R(A) is also naturally a R(1)-bimodule whichcan be identified with the induced R(1)-bimodule structure on (A times 1)otimes R(1) wherethe left R(1) action on (Atimes1)otimes R(1) is given by (idAtimes1 otimesm R(1))(cminus1

Atimes1R(1)otimes idR(1))We have the following natural isomorphisms

HomR(1)|R(1)(R(A) R(A))sim=minusrarr HomC2plusmn(A times 1 R(A))

χminus1

minusminusrarr HomC(A A) (262)

which by (253) are given by for f isin HomR(1)|R(1)(R(A) R(A))

f rarr f prime = f (idAtimes1 otimes ηR(1)) rarr f primeprime = ρ T ( f prime) (263)

and its inverse is given by for g isin HomC(A A)

g rarr gprime = R(g) δ rarr gprimeprime = (idAtimes1 otimes m R(1)) (gprime otimes idR(1)) (264)

where gprimeprime is indeed a R(1)-bimodule map due to the commutativity of R(1) It is easy tocheck that gprimeprime = R(g) in (264) Therefore R gives an isomorphism from HomC(A A)to HomR(1)|R(1)(R(A) R(A)) Moreover one verifies that R(g) is an R(A)-bimodulemap iff g is an A-bimodule map In other words R g rarr R(g) gives an isomorphism

HomA|A(A A)sim=minusrarr HomR(A)|R(A)(R(A) R(A))

Corollary 227 Let A be a C-algebra

(i) A is absolutely simple iff R(A) is absolutely simple(ii) Let A be in addition Frobenius Then A is simple and special iff R(A) is simple and

special

Proof Part (i) immediately follows from Lemma 226 The statement of part (ii) withoutthe qualifier lsquosimplersquo is proved in Proposition 225 But as in the proof of (iii)rArr(i) inLemma 211 a special Frobenius algebra in a semi-simple category has a semi-simplecategory of bimodules and for a semi-simple C-linear category simple and absolutelysimple are equivalent Part (ii) then follows from part (i)

The following lemma will be needed in Sect 32 below to discuss the properties ofCardy algebras

Lemma 228 Let A be a Frobenius algebra in C Then (δA)lowast = ρA

Proof Recall from (250) that δA is a morphism A rarr T R(A) Since T and R are bothFrobenius functors T R(A) is a Frobenius algebra in C Substituting the definitions aftera short calculation one finds

Cardy Algebras and Sewing Constraints I 891

εT R(A) mT R(A) = Dim(C)oplusiisinI

A Uori Ui A U or

ı Uı

(265)

Substituting this in the definition of (δA)lowast gives again after a short calculation the mor-

phism ρA At an intermediate step one uses that the part of the morphism (δA)lowast which

is made up of Ui and Uı ribbons their duals and the basis morphisms λ(iı)0 andϒ(iı)0can be replaced by 1

dim Uimiddot dUi

3 Cardy Algebras

In this section we start by investigating the properties of Frobenius algebras which sat-isfy the so-called modular invariance condition We then give two definitions of a Cardyalgebra and prove their equivalence Finally in Sect 33 we study the properties of thesealgebras and state our main results

We fix a modular tensor category C Recall that C2plusmn is an abbreviation for C+ Cminus

31 Modular invariance In C2plusmn we define the object K and the morphism ω K rarr Kas

K =oplus

i jisinIUi times U j ω =

sumi jisinI

dim Ui dim U j

Dim C idUi timesU j (31)

They have the property (see eg [BK Cor 3111])

ω

K

(Uk times Ul )or

(Uk times Ul )or

Ui times U j

Ui times U j

= δik δ jlDim C

dim Ui dim U jbUi timesU j dUi timesU j (32)

892 L Kong I Runkel

Definition 31

(i) Let A B be objects of C2plusmn A morphism f A otimes B rarr B is called S-invariant iff

A

B

W

W

f =

A

B

W

W

f

K

ω(33)

holds for all for W isin C2plusmn(ii) A C2plusmn-algebra (Aclmcl ηcl) is called modular invariant iff θAcl = idAcl and mcl

is S-invariant

Lemma 32 The morphism f A otimes B rarr B is S-invariant if and only if

A

B

Ui times U j

Ui times U j

f = Dim Cdim Ui dim U j

sumα

A

B

B

f

Ui times U j

Ui times U j

α

α

(34)

holds for all i j isin I

Proof Condition (33) holds for all W iff it holds for all W = Ui times U j i j isin I so itis enough to show that the right-hand side of (33) with W = Ui times U j is equal to theright-hand side of (34) Recall the notation for basis morphisms in (236) Starting from(33) write

idBor otimes idUi timesU j =⎛⎝sum

klα

(b(ktimeslα)

B

)or (

bB(ktimeslα)

)or⎞⎠ otimes idUi timesU j (35)

and then apply (32) The graphical representation of the resulting morphism can bedeformed to give (34) Remark 33 As shown in [K3 Sect 61] the modular invariance condition of a C2plusmn-alge-bra exactly coincides with the modular invariance condition for torus 1-point correlationfunctions of a genus-01 closed CFT In particular the condition θAcl = idAcl is equiv-alent to invariance under the modular transformation T τ rarr τ + 1 and the condition(34) with f = mcl is equivalent to invariance under S τ rarr minus 1

τ Combining the

modular invariance condition with the genus zero properties of a genus-01 closed CFTresults in a modular invariant commutative symmetric Frobenius algebra in C2plusmn

Cardy Algebras and Sewing Constraints I 893

Let Acl be a modular invariant C2plusmn-algebra Evaluating (34) for f = mcl composingit with ηcl otimes idUi timesU j and taking the trace implies the following identity

Zi j = 1

DimCAclUi timesU j where

(36)

Zi j = dimC HomC2plusmn(UitimesU j Acl)

Decomposing Acl into simple objects this gives

Zi j =sum

klisinISik Zkl Sminus1

l j where Si j = si jradic

DimC (37)

which in CFT terms is of course nothing but the invariance of the torus partition functionunder the modular S-transformation

The following theorem gives a simple criterion for modular invariance

Theorem 34 Let Acl be a haploid commutative symmetric Frobenius algebra in C2plusmn

(i) If Acl is modular invariant then dim Acl = Dim C(ii) If dim Acl = Dim C then Acl is special and modular invariant

Proof Part (i) Since Acl is haploid for i = j = 0 Equation (36) reduces to 1 =dim AclDimCPart (ii) By the same reasoning as in the proof of (iii)rArr(i) in Lemma 211 one showsthat Acl is special Thus mcl cl = ζclidAcl for some ζcl = 0

By [KO Thm 45] the category (C2plusmn)locAcl

of local Acl-modules is again a mod-

ular tensor category and Dim (C2plusmn)locAcl

= (Dim C dim Acl)2 (see [Fr Prop 321 amp

Rem 323] for the same statement in the notation used here) Thus by assumption wehave Dim(C2plusmn)loc

Acl= 1 It then follows from [ENO1 Thm 23] that up to isomorphism

(C2plusmn)locAcl

has a unique simple object (namely the tensor unit) In other words every simplelocal Acl-module is isomorphic to Acl (seen as a left-module over itself)

We have the following isomorphisms between morphism spaces [FS Prop 47 amp 411]

HomAcl(Acl otimes (Ui times U j ) Acl) sim= HomC2plusmn(Ui times U j Acl)

HomAcl(Acl Acl otimes (Ui times U j )) sim= HomC2plusmn(AclUi times U j )

Using these to transport the bases (236) from the right to the left we obtain bases bα(i j)αof HomAcl(Acl otimes (Ui times U j ) Acl) and b(i j)

β β of HomAcl(Acl Acl otimes (Ui times U j )) Thesecan be expressed graphically as

894 L Kong I Runkel

bα(i j) =

Acl

Acl

Acl

Ui times U j

α

b(i j)β = DimC

dim Ui dim U j

1

ζcl

Acl

Ui times U jAcl

Acl

β

(38)

where the nonzero factor in b(i j)β is included for convenience Notice that bα(i j) b(i j)

β isa left Acl-module map Since Acl is simple as a left module over itself we have

bα(i j) b(i j)β = λαβ idAcl (39)

for some λαβ isin C By computing tr(bα(i j) b(i j)β ) it is easy to verify that λαβ = δαβ

We will now prove the following identity

1

ζcl

Acl

Acl

Ui times U j

Ui times U j

Acl

=sumα

DimCdim Ui dim U j

1

ζcl

Acl

Ui times U j

Ui times U j

Acl

Acl

Acl

α

α

(310)

One checks that the left-hand side of this equation is an idempotent which we denote byζminus1

cl PlAcl(Ui timesU j ) cf [Fr Sect 31] By [Fr Prop 41] the image Im(ζminus1

cl PlAcl(Ui timesU j ))

is a local Acl-module and hence isomorphic to AoplusNcl for some N isin Zge0

All left-module morphisms from Acl to Aclotimes(Ui timesU j ) are linear combinations of the

b(i j)β Furthermore one verifies that ζminus1

cl PlAcl(Ui timesU j )b(i j)

β = b(i j)β Therefore the b(i j)

β

describe precisely the image of the idempotent ie ζminus1cl Pl

Acl(Ui timesU j ) = sum

α b(i j)α bα(i j)

which is nothing but (310) Composing (310) with ζcl middot εcl otimes idUi timesU j from the left (iefrom the top) produces (34)

In addition since Acl is commutative symmetric Frobenius it satisfies θAcl = idAcl

[Fr Prop 225] Altogether this shows that Acl is modular invariant Remark 35 As we were writing this paper we heard that the results in Theorem 34were obtained independently by Kitaev and Muumlger [Ki]

Remark 36 Setting i = j = 0 in (36) gives the identity dim Acl = Z00 Dim C [KRProp 23] Combining this with Theorem 34 (ii) one may wonder if a general modularinvariant commutative symmetric Frobenius algebra Acl in C2plusmn is isomorphic to a directsum of simple such algebras However this is not so For example one can take the com-mutative symmetric Frobenius algebra Acl = C[x]x2 in the category of vector spacesequipped with the non-degenerate trace ε(ax + b) = a In this case the modular invari-ance condition holds automatically but Acl is clearly not a direct sum of two algebras

Cardy Algebras and Sewing Constraints I 895

For a general modular tensor category C the algebra C[x]x2 R(1) understood as analgebra in C2plusmn via the braided monoidal isomorphism Vect f (C) C2plusmn rarr C2plusmn providesanother counter-example

32 Two definitions Define a morphism PlA A rarr A for a Frobenius algebra A in C

or C2plusmn as follows [Fr Sect 24]

PlA =

A

A

A

m

(311)

If A is also commutative and obeys m A A = ζA idA we have PlA = ζA idA In partic-

ular this holds if A is commutative and special Using the fact that the Frobenius algebraR(1) is commutative and normalised-special one can check that Pl

R(A) R(A) rarr R(A)takes the following form

PlR(A) =

oplusiisinI

A

A

Uori

Uori

A

m

times

Ui

Ui

(312)

With these ingredients we can now give the first definition of a Cardy C|C2plusmn-algebrawhich was introduced in [K3 Def 514] cf Remark 315 below

Definition 37 (CardyC|C2plusmn-algebra I) A CardyC|C2plusmn-algebra is a triple (Aop|Acl ιcl-op)where (Aclmcl ηclcl εcl) is a modular invariant commutative symmetric Frobe-nius C2plusmn-algebra (Aopmop ηopop εop) is a symmetric Frobenius C-algebra andιcl-op Acl rarr R(Aop) an algebra homomorphism such that the following conditionsare satisfied

(i) Centre condition

Acl R(Aop)

R(Aop)

R(Aop)

ιcl-op

m R(Aop)

=

Acl

ιcl-op

R(Aop)

R(Aop)

R(Aop)

m R(Aop)

(313)

896 L Kong I Runkel

(ii) Cardy condition

ιcl-op ιlowastcl-op =

R(Aop)

R(Aop)

R(Aop)

(314)

Remark 38

(i) The name ldquoCardy C|C2plusmn-algebrardquo in Definition 37 was chosen because many ofthe important ingredients were first studied by Cardy the modular invariance ofthe closed theory [C1] the consistency of the annulus amplitude [C2] and thebulk-boundary OPE [CL] On the other hand the boundary-boundary OPE andthe OPE analogue of the centre condition were first considered in [Lw]

(ii) One can easily see that in the special case that C is the category Vect f (C) of finite-dimensional C-vector spaces a Cardy C|C2plusmn-algebra gives exactly the algebraicformulation of two-dimensional open-closed topological field theory over C (cfRemark 614 in [K3]) see [Lz Sect 48] [Mo Thm 11] [AN Thm 45] [LPCor 43] [MS Sect 22] When passing to a general modular tensor category Cthere are two important differences to the two-dimensional topological field the-ory Firstly the algebras Acl and Aop now live in different categories which inparticular affects the formulation of the centre condition and the Cardy conditionSecondly the modular invariance condition has to be imposed on Acl In the caseC = Vect f (C) modular invariance holds automatically

Definition 39 A homomorphism of Cardy C|C2plusmn-algebras (A(1)op |A(1)cl ι(1)cl-op) rarr (A(2)op

|A(2)cl ι(2)cl-op) is a pair ( fop fcl) of Frobenius algebra homomorphisms fop A(1)op rarr A(2)op

and fcl A(1)cl rarr A(2)cl such that the diagram

A(1)cl

fcl

ι(1)cl-op

A(2)cl

ι(2)cl-op

R(A(1)op )

R( fop) R(A(2)op )

(315)

commutes

Remark 310 Since a homomorphism of Frobenius algebras is invertible (cf Lemma218 (iv)) a homomorphism of Cardy algebras is always an isomorphism

For a homomorphism ( fop fcl) of Cardy C|C2plusmn-algebras using the commutativity of(315) and the fact that fcl and fop are both algebra and coalgebra homomorphisms itis easy to show that (315) commutes iff

R(A(1)op )R( fop)

(ι(1)cl-op)

lowast

R(A(2)op )

(ι(2)cl-op)

lowast

A(1)cl

fcl A(2)cl

(316)

Cardy Algebras and Sewing Constraints I 897

commutesLet (Aop|Acl ιcl-op) be a Cardy C|C2plusmn-algebra Define the morphism

ιcl-op = χminus1(ιcl-op) T (Acl) minusrarr Aop (317)

Decompose Acl as Acl = oplusNn=1Cl

n times Crn such that Cl

1 times Cr1 = ηcl(1 times 1) We use ι(n)cl-op

to denote the restriction of ιcl-op to Cln times Cr

n and ι(n)cl-op to denote the restriction of ιcl-op

to Cln otimes Cr

n We introduce the following graphical notation

ι(n)cl-op =

Cln Cr

n

Aop

(318)

By (243) ιcl-op can be expressed in terms of ιcl-op as follows

ιcl-op =Noplus

n=1

oplusiisinI

sumα

Cln

Crn

Aop

α

Uori

times α

Crn

Ui

(319)

Lemma 311 The centre condition (313) is equivalent to the following condition in C

Cln Cr

n

Aop

Aop

Aop

=

Cln Cr

n

Aop

Aop

Aop

for n = 1 N (320)

Proof First insert (319) and the definition (217) of m R(Aop) into (313) Then apply thecommutativity of R(1) to the left hand side of (313) The equivalence between (313)and (320) follows immediately Remark 312 The centre condition (320) is very natural from the open-closed confor-mal field theory point of view Correlators on the upper half plane are expressed in termsof conformal blocks on the full complex plane The objects Cl

n and Crn are associated to

the field insertion at a point z in the upper half plane and at the complex conjugate pointz in the lower half plane respectively The object Aop corresponds to a field inserted ata point r on the real axis The centre condition (320) simply says that the correlationfunctions in the disjoint domains |z| gt r gt 0 and r gt |z| gt 0 are analytic continuationsof each other see [K2 Prop 118]

898 L Kong I Runkel

Recall that we define ιlowastcl-op Aop rarr T (Acl) as in (230) We introduce the graphicalnotation

ιlowastcl-op =Noplus

n=1

Cln Cr

n

Aop

=Noplus

n=1

Cln Cr

n

Aop

Aop T (minus1Acl) (321)

where the second equality follows from (215) (221) and (259)

Lemma 313 The Cardy condition (314) is equivalent to the following identity in C

Noplusn=1

DimCdim Ui

sumα

Cln

Crn

Crn

α

α

Aop

Aop Uori

Uori

=Aop

Aop

Aop

Uori

Uori

op

mop for all i isin I (322)

Proof By (244) (312) and (319) it is easy to see that (322) is equivalent to thefollowing identity

ιcl-op χ(ιlowastcl-op) = PlR(Aop)

(323)

Therefore it is enough to show that

χ(ιlowastcl-op) = ιlowastcl-op (324)

We have

χminus1(ιlowastcl-op)(1)= T (ιlowastcl-op) δ (2)= T (ιcl-op)

lowast δlowastlowast (3)= (ρ T (ιcl-op)

)lowast (4)= (ιcl-op

)lowast (325)

In step (1) we use the expression (253) for χminus1 step (2) follows from Lemma 218 (v)and Lemma 219 Step (3) is Lemma 218 (i) and Lemma 228 and finally step (4)amounts to substituting (253) and (317) Acting with χ on both sides of the aboveequality produces (324)

Combining Lemmas 311 and 313 and Proposition 216 we obtain the followingequivalent definition of Cardy C|C2plusmn-algebra (recall the graphical notation (318) forιcl-op and (321) for ιlowastcl-op)

Cardy Algebras and Sewing Constraints I 899

Definition 314 (CardyC|C2plusmn-algebra II) A CardyC-algebra is a triple (Aop|Acl ιcl-op)where Acl is a commutative symmetric Frobenius C2plusmn-algebra satisfying property (34)with f = mcl Aop is a symmetric Frobenius C-algebra and ιcl-op T (Acl) rarr Aop isan algebra homomorphism satisfying the conditions (320) and (322)

Remark 315 Up to a choice of normalisation Definition 314 is the same as the orig-inal one in [K3 Def 613] The difference between the two definitions is the factorDim C dim Ui on the left hand side of (322) which in [K3 Def 613] is given byradic

Dim C dim Ui The two definitions are related by rescaling the coproduct cl andcounit εcl of Acl by 1

radicDim C and

radicDim C respectively We chose the convention in

(322) to remove all dimension factors from the expression (314) for the Cardy condition

33 Uniqueness and existence theorems In this subsection we investigate the structureof Cardy algebras We start with the following proposition which when combined withthe results of part II provides an alternative proof of [Fj Prop 422]

Proposition 316 Let (Aop|Acl ιcl-op) be a Cardy C|C2plusmn-algebra If Acl is simple anddim Aop = 0 then Aop is simple and special

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis therefore haploid Restricting the Cardy condition (322) to the case Ui = 1 andcomposing both sides with εop from the left we see that εop kills all terms associated toU j times 1 isin Acl in the sum except for a single 1 times 1 term Thus we obtain the followingidentity

β εop = dAop (mop otimes idAorop) (idAop otimes bAop) (326)

where β isin C Composing with ηop from the right in turn implies that βεop ηop =dim Aop which is nonzero by assumption Thus also β = 0 and εop is a nonzero multipleof the morphism on the right hand side of (326) By [FRS Lem 311] Aop is special

Since Aop is a special Frobenius algebra Aop is semi-simple as an Aop-bimodule(apply [FS Prop 524] to Aop tensored with its opposite algebra) Suppose Aop is not

simple so that we can write Aop = A(1)op oplus A(2)op for nonzero Aop-bimodules A(1)op and A(2)op We denote the canonical embeddings and projections associated to this decompositionas ι12 and π12 We have the identities

mop (ι1 otimes ι2) = 0 εop ηop =2sum

i=1

εop ιi πi ηop (327)

The first identity follows since π1 mop (ι1 otimes ι2) = 0 (as mop gives the left action of

Aop on Aop and hence it preserves A(2)op ) and similarly π2 mop (ι1 otimes ι2) = 0 Thesecond identity is just the completeness of ι12 π12

Since εop ηop = 0 without losing generality we can assume εop ι1 π1 ηop = 0Using that π2 is a bimodule map we compute

π2 [LHS of (322)]Ui =1 ι1 π1 ηop = π2 [RHS of (322)]Ui =1 ι1 π1 ηop

= PlA(2)op

π2 ι1 π1 ηop = 0 (328)

900 L Kong I Runkel

On the other hand using that Acl is haploid that ιcl-op is an algebra map and thatι lowastcl-op is a coalgebra map one can check that the left-hand side of (328) is equal toλ(εop ι1 π1 ηop) π2 ηop for some λ = 0 This implies that π2 ηop = 0 Thusηop = sum

i ιi πi ηop = ι1 π1 ηop Hence we have

0 = π2 ι2 = π2 mop (ηop otimes ι2) = π2 mop ((ι1 π1 ηop)otimes ι2) (329)

However the right-hand side is zero by (327) This is a contradiction and hence Aopmust be simple

To formulate the next theorem we need the notion of the full centre of an algebra [FjDef 49] Recall that an algebra A in a braided tensor category has a left centre and aright centre [VZO] both of which are sub-algebras of A Of these two we will onlyneed the left centre The following definition is [Fr Def 231] which in our setting isequivalent to that of [VZO]

Definition 317 Let A be a symmetric special Frobenius algebra such that m A A =ζA idA

(i) The left centre Cl(A) of A is the image of the idempotent ζminus1A Pl

A(ii) The full centre Z(A) is Cl(R(A))

That ζminus1A Pl

A is an idempotent follows from [FRS Lem 52] when keeping track ofthe factors ζA ([FRS] assumes normalised-special ie ζA = 1) Note that Cl(A) is againan object of C while Z(A) is an object of C2plusmn Let el

A Cl(A) rarr A be the embeddingof Cl(A) into A The left centre is in fact the maximal subobject of A such that

m A cAA (elA otimes idA) = m A (330)

see [Fr Lem 232] This observation explains the name left centre and also makes theconnection to [O Def 15]

The full centre is by definition the image of the idempotent ζminus1A Pl

R(A) R(A) rarrR(A) Since C2plusmn is abelian the idempotent splits and we obtain the embedding andrestriction morphisms

e Z(A) rarr R(A) and r R(A) Z(A) (331)

which obey r e = idZ(A) and e r = ζminus1A Pl

R(A) It follows from Proposition 225 and

[Fr Prop 237] that Z(A) is a commutative symmetric Frobenius algebra in C2plusmn withstructure morphisms5

m Z(A) = r m R(A) (e otimes e) ηZ(A) = r ηR(A)

Z(A) = ζA middot (r otimes r) R(A) e εZ(A) = ζminus1A middot εR(A) e

(332)

Moreover if A is simple then Z(A) is simple and if A is simple and dim A = 0 thenZ(A) is simple and special The normalisation of the counit is such that

εZ(A) ηZ(A) = ζminus2A dim A DimC (333)

5 The normalisation of product and unit is the standard one The factors in the coproduct and counit haveto be included in order for (A|Z(A) e) to be a Cardy algebra see Theorem 318 below The normalisation ofthe counit enters the Cardy condition (314) through the definition of ( middot )lowast

Cardy Algebras and Sewing Constraints I 901

Theorem 318 Let A be a special symmetric Frobenius C-algebra Then (A|Z(A) e)is a Cardy C|C2plusmn-algebra

The proof of this theorem makes use the following two lemmas

Lemma 319 e Z(A) rarr R(A) is an algebra map and elowast = ζA middot r

Proof It follows from [Fr Lem 229] (or by direct calculation using in particularm R(A) R(A) = ζAidR(A)) that

m R(A) (e otimes e) = ζminus1A middot Pl

R(A) m R(A) (e otimes e) (334)

Substituting e r = ζminus1A Pl

R(A) shows that e is compatible with multiplication For theunit one finds

e ηZ(A) = e r ηR(A) = ζminus1A Pl

R(A) ηR(A) = ηR(A) (335)

Thus e is an algebra map For the second statement one computes

elowast (1)= ζA

R(A)

r

R(A)

R(A)

Z(A)

R(A)

Z(A)

r

e

e

(2)=

R(A)

r

R(A)

R(A)

R(A)

Z(A)

(3)=

R(A)

r

R(A)

Z(A)

(4)= ζA middot r (336)

where in (1) the definitions (230) and (332) have been substituted step (2) is e r = ζminus1

A PlR(A) step (3) uses that R(A) is symmetric Frobenius and step (4) is again

e r = ζminus1A Pl

R(A) Lemma 320 Let A be a symmetric Frobenius algebra in C The morphism

PlR(A) m R(A)

(Pl

R(A) otimes PlR(A)

) R(A)otimes R(A) minusrarr R(A) (337)

is S-invariant

The proof of this lemma is a slightly lengthy explicit calculation and has been deferredto Appendix A2

902 L Kong I Runkel

Proof of Theorem 318 That e is an algebra map was proved in Lemma 319 The centrecondition (313) holds by property (330) of the left centre The Cardy condition (314)also is an immediate consequence of Lemma 319

ιcl-op ιlowastcl-op = e (ζA r) = PlR(A) (338)

The full centre Z(A) is a commutative symmetric Frobenius algebra It remains to provemodular invariance That θZ(A) = idZ(A) is implied by commutativity and symmetryof Z(A) [Fr Prop 225] The S-invariance condition (33) follows from Lemma 320In (337) substitute Pl

R(A) = ζA e r and then put the resulting morphism into (33)Compose the resulting equation with e otimes idW from the right (ie from the bottom) andsubstitute the definition (332) of m Z(A) This results in the statement that m Z(A) isS-invariant

The following theorem is analogous to [LR Prop 29] and [Fj Thm 426] whichroughly speaking answer the question under which circumstances the restriction of atwo-dimensional conformal field theory to the boundary already determines the entireconformal field theory The first work is set in Minkowski space and uses operator alge-bras and subfactors while the second work is set in Euclidean space and uses modulartensor categories

Theorem 321 Let (A|Acl ιcl-op) be a Cardy C|C2plusmn-algebra such that dim A = 0 andAcl is simple Then A is special and (A|Acl ιcl-op) sim= (A|Z(A) e) as Cardy algebras

Proof By Proposition 316 A is simple and special Since Acl is simple the algebramap ιcl-op Acl rarr R(A) is either zero or a monomorphism But ιcl-op ηcl = ηR(A)and so ιcl-op = 0 Thus ιcl-op is monic By Lemma 218 (ii) ζminus1

A ιlowastcl-op is epi The Cardycondition (314) implies

ιcl-op ζminus1A ιlowastcl-op = ζminus1

A PlR(A) = e r (339)

Composing this with e r from the left yields e r ιcl-op ζminus1A ιlowastcl-op = e r =

ιcl-op ζminus1A ιlowastcl-op Since ζminus1

A ιlowastcl-op is epi we have

e r ιcl-op = ιcl-op (340)

Actually (340) also follows from (313) and specialness of R(A) We will prove that

( fop fcl) (A|Acl ιcl-op) minusrarr (A|Z(A) e) where fop = idA fcl = r ιcl-op

(341)

is an isomorphism of Cardy algebrasfcl is an algebra map Compatibility with the units follows since ιcl-op is an algebramap

fcl ηcl = r ιcl-op ηcl = r ηR(A) = ηZ(A) (342)

Compatibility with the multiplication also follows since ιcl-op is an algebra map

m Z(A) ( fcl otimes fcl) = r m R(A) (e otimes e) (r otimes r) (ιcl-op otimes ιcl-op)

= r m R(A) (ιcl-op otimes ιcl-op) = r ιcl-op mcl = fcl mcl (343)

where in the second step we used (340)

Cardy Algebras and Sewing Constraints I 903

fcl is an isomorphism As above since fcl is an algebra map and since Acl is simple fcl

has to be monic By Lemma 319 rlowast = ζminus1A e Thus f lowast

cl = ιlowastcl-op rlowast = ζminus1A ιlowastcl-op e and

fcl f lowastcl = r ιcl-op ζminus1

A ιlowastcl-op e = r e r e = idZ(A) (344)

and so fcl is also epi and hence isofcl is a coalgebra map Since fcl is an algebra map so is f minus1

cl By (344) f minus1cl = f lowast

cland by Lemma 218 (iii) this implies that fcl is a also coalgebra mapThe diagram (315) commutes Commutativity of (315) is equivalent to e fcl = ιcl-opwhich holds by (340)

Let A be a special symmetric Frobenius algebra So far we have seen that (A|Z(A) e)is a Cardy algebra and that all Cardy algebras with Aop = A and simple Acl are of thisform It is now natural to ask if every simple Acl does occur as part of a Cardy algebraThe following theorem provides an affirmative answer Recall that for an A-left moduleM the object Mor otimesA M is an algebra (see eg [KR Lem 42])

Theorem 322 If Acl is a simple modular invariant commutative symmetric FrobeniusC2plusmn-algebra then there exist a simple special symmetric Frobenius C-algebra A and amorphism ιcl-op Acl rarr R(A) such that

(i) Acl sim= Z(A) as Frobenius algebras(ii) (A|Acl ιcl-op) is a Cardy C|C2plusmn-algebra

(iii) T (Acl) sim= oplusκisinJ Morκ otimesA Mκ as algebras where MκκisinJ is a set of representa-

tives of the isomorphism classes of simple A-left modules

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis haploid It then follows from Theorem 34 that Acl is special By Proposition 224T (Acl) is a special symmetric Frobenius algebra in C Thus T (Acl) = oplusi Ai where theAi are simple symmetric Frobenius algebras We will show that at least one of the Aiis special Since T (Acl) is special we have mT (Acl) T (Acl) = ζ idT (Acl) for someζ isin C

times Restricting this to the summand Ai shows mi i = ζ idAi FurthermoreεT (Acl) ηT (Acl) = ξ id1 for some ξ isin C

times But εT (Acl) ηT (Acl) = sumi εi ηi and so

at least one of the εi ηi has to be nonzero Therefore at least one of the Ai is speciallet A equiv Ai be this summand We denote the embedding A rarr T (Acl) by e0 and therestriction T (Acl) A by r0 Notice that r0 is an algebra homomorphism Define

ιcl-op = χ (r0) Acl minusrarr R(A) (345)

By Proposition 216 ιcl-op is an algebra homomorphism Next we verify the centre con-dition (313) or rather its equivalent form (320) By substituting the definitions onecan convince oneself that the commutativity mcl cAclAcl = mcl of Acl in C2plusmn implies thecondition mT (Acl) = mT (Acl) in C see [K2 Prop 36] Here T (Acl)otimes T (Acl) rarrT (Acl)otimes T (Acl) is given by

=oplusmn

(idCl

notimes cCl

m Crnotimes idCr

m

)

(cCl

m Clnotimes cminus1

Crn C

rm

)

(idCl

motimes cminus1

Cln C

rm

otimes idCrn

)

(346)

and we decomposed Acl as Acl = oplusnCln times Cr

n As a consequence we obtain the identity

r0 mT (Acl) (idAcl otimes e0) = r0 mT (Acl) (idAcl otimes e0) (347)

904 L Kong I Runkel

Using that r0 is an algebra map and that by definition ιcl-op = r0 we obtain (320)In order to show that (A|Acl ιcl-op) is a Cardy algebra it remains to show that the

Cardy condition (314) is satisfied We will demonstrate this via a detour by first provingthat Acl sim= Z(A) as Frobenius algebras

Recall the notations e and r given in (331) Using the centre condition (320) onecan check that Pl

R(A) ιcl-op = m R(A) R(A) ιcl-op By specialness of A we have

m A A = ζA idA and so together with e r = ζminus1A Pl

R(A) we get

e r ιcl-op = ιcl-op (348)

Next consider the morphism

fcl = r ιcl-op Acl minusrarr Z(A) (349)

By the same derivation as in (342) and (343) one sees that fcl is an algebra map Inparticular fcl ηcl = ηZ(A) = 0 and so fcl = 0 Since Acl is simple fcl has to be amonomorphism

By the same argument as used in the proof of Theorem 34 (ii) up to isomorphismAcl is the unique simple local Acl-(left-)module The algebra monomorphism fcl turnsZ(A) into an Acl-module Since Z(A) is commutative it is local as an Acl-module andso Z(A) sim= AoplusN

cl for some N ge 1 By construction A is a simple special symmetricFrobenius algebra Proposition 225 and Corollary 227 show that R(A) inherits all theseproperties and thus Z(A) is simple (see the comment below Eq (332)) By Theorem318 Z(A) is modular invariant and then by Theorem 34 (i) dim Z(A) = Dim C Thisimplies that N = 1 in Z(A) sim= AoplusN

cl and so fcl is in fact an isomorphismSince Acl and Z(A) are both haploid we have εZ(A) fcl = ξ εcl for some ξ isin C

timesThe counit uniquely determines the Frobenius structure on Acl and Z(A) (see eg [FRSLemma 37]) so that fcl is a coalgebra isomorphism iff ξ = 1 To compute ξ we composethe above identity with ηcl from the right Defining ζcl via εcl ηcl = ζminus1

cl Dim C middot id1 andusing (333) gives ξ = dim A ζclζ

2A By rescaling the comultiplication and the counit

of A and consequently changing ζA we can always achieve ξ = 1 This proves part (i)of the theorem

Equation (348) implies that ιcl-op = e fcl Since fcl is an isomorphism of Frobeniusalgebras by Lemmas 218 and 319 we have

ιcl-op ιlowastcl-op = e fcl f lowastcl elowast = ζA e r = Pl

R(A) (350)

Thus (A|Acl ιcl-op) is a Cardy algebra This proves part (ii) of the theoremPart (iii) can be seen as follows By [KR Prop 43] T Z(A) sim= oplusκisinJ Mor

κ otimesA Mκ asalgebras Together with the observation that T ( fcl) T (Acl) rarr T Z(A) is an isomor-phism of algebras this proves part (iii) Remark 323 Part (i) of Theorem 322 was announced by Muumlger [Muuml2] We provide anindependent proof in the setting of Cardy algebras

The above theorem together with Lemma 211 and Theorem 34 shows that a sim-ple commutative symmetric Frobenius C2plusmn-algebra Acl with dim Acl = Dim C is alwayspart of a Cardy algebra (Aop|Acl ιcl-op) for some simple special symmetric Frobeniusalgebra Aop in C However the above proof also illustrates that Aop is not unique Thisraises the question how two Cardy algebras with a given Acl can differ This question isanswered by [KR Thm 11] which in the present framework can be restated as follows

Cardy Algebras and Sewing Constraints I 905

Theorem 324 If (A(i)op |A(i)cl ι(i)cl-op) i = 1 2 are two Cardy C|C2plusmn-algebras such that

A(i)cl is simple and dim A(i)op = 0 for i = 1 2 then A(1)clsim= A(2)cl as algebras if and only

if A(1)op and A(2)op are Morita equivalent

Proof Theorem 11 in [KR] is stated for A(i)op being non-degenerate algebras and A(i)cl =Z(A(i)op ) for i = 1 2 By Proposition 316 A(i)op are simple and special for i = 1 2

Then by [KR Lem 21] A(i)op are non-degenerate algebras By Theorem 321 we have

A(i)clsim= Z(A(i)op ) as Frobenius algebras Finally by [KR Thm 11] Z(A(1)op ) sim= Z(A(2)op )

as algebras iff A(1)op and A(2)op are Morita equivalent Let Cmax(C2plusmn) be the set of equivalence classes [B] of simple modular invariant

commutative symmetric Frobenius algebras B in C2plusmn Two such algebras B and B prime areequivalent if B and B prime are isomorphic as algebras (but not necessarily as Frobenius alge-bras) Let Msimp(C) be the set of Morita classes of simple special symmetric Frobeniusalgebras in C Define the map z Msimp(C) rarr Cmax(C2plusmn) by z A rarr [Z(A)] whereA denotes the Morita class of A From Theorem 322 (i) and [KR Thm 11] we learn

Corollary 325 The map z Msimp(C) rarr Cmax(C2plusmn) is a bijection

A Appendix

A1 Proof of Lemma 27 We will show that if (F ψ F2 ψ

F0 ) is a colax tensor functor

from C1 to C2 then (G φG2 φ

G0 ) is a lax tensor functor from C2 to C1 Applying this

result to the opposed categories then gives the converse statementWe need to show that φG

0 and φG2 make the diagrams (21) and (22) commute We

first prove the commutativity of (21) Consider the following diagram

G(A)otimes (G(B)otimes G(C))

GψF2 δ

idG(A)otimesφG2 G(A)otimes G(B otimes C)

GψF2 δ

G(FG(A)otimes F(G(B)otimes G(C)))

G(F(idG(A))otimesF(φG2 ))

G(idFG(A)otimesψ F2 )

G(FG(A)otimes FG(B otimes C))

G(ρAotimesρBotimesC )

G(FG(A)otimes (FG(B)otimes FG(C)))

G(ρAotimes(ρBotimesρC )) G(A otimes (B otimes C))

(A1)

The top subdiagram is commutative because of the naturality of Gψ F2 δ The commu-

tativity of the bottom subdiagram follows from the following identities

(ρB otimes ρC ) ψ F2 = (ρB otimes ρC ) ψ F

2 ρF Fδ

= ρBotimesC FG(ρB otimes ρC ) FG(ψ F2 ) Fδ

= ρBotimesC F(φG2 ) (A2)

as a map F(G(B) otimes G(C)) rarr B otimes C The commutativity of (A1) implies that thecomposition of maps in the left column in (21) can be replaced by

G(ρA otimes (ρB otimes ρC )) G((idFG(A) otimes ψ F

2 ) ψ F2

) δ (A3)

906 L Kong I Runkel

Similarly we can show that the composition of maps in the right column in (21) can bereplaced by

G((ρA otimes ρB)otimes ρC ) G((ψ F

2 otimes idFG(C)) ψ F2

) δ (A4)

Using the commutativity of (23) it is easy to see that (21) with the left and right col-umns of (21) replaced by (A3) and (A4) respectively is commutative Hence (21) iscommutative

Now we prove the commutativity of the first diagram in (22)

φG2 (φG

0 otimes idG(A))

(1)= G(ρ12 otimes ρA) Gψ F2 δ

[(Gψ F

0 δ11)otimes idG(A)

]

(2)= G(ρ12 otimes ρA) Gψ F2 G F(Gψ0 otimes idG(A)) G F(δ11 otimes idG(A)) δ

(3)= G(ρ12 otimes ρA) G(FG(ψ F0 )otimes idFG(A)) Gψ F

2 G F(δ11 otimes idG(A)) δ(4)= G(id12 otimes ρA) G

([ψ F

0 ρF(11) (Fδ)11] otimes idFG(A)

) Gψ F

2 δ(5)= G(id12 otimes ρA) G(ψ F

0 otimes idFG(A)) Gψ F2 δ

(6)= G(id12 otimes ρA) G(lminus1FG(A)) G F(lG(A)) δ

(7)= G(lminus1A ) GρA δG lG(A)

(8)= G(lminus1A ) lG(A) (A5)

where in step (1) we substituted the definition of φG0 φ

G2 given in (210) in step (2)

we used the naturality of δ in step (3) we used the naturality of Gψ F2 in step (4) we

switched the position between Gρ12 and G FG(ψ F0 ) and the position between Gψ2 and

G F(δ12 otimes idG(A)) using the naturality of ρ and FψG2 respectively in step (5) we applied

the second identity in (28) in step (6) we used (24) in step (7) we used the naturalityof lminus1 and δ in step (8) we used the first identity in (28)

The proof of the commutativity of the second diagram in (22) is similar Thus wehave shown that G is a lax tensor functor

A2 Proof of Lemma 320 To prepare the proof recall that for a given object B isin Cthe modular group P SL(2Z) acts on the space oplusi HomC(B otimes Ui Ui ) see eg [BKSect 31] and [K3 Eq (455)] We will only need the action of S and Sminus1 Let f isinoplusi HomC(B otimes Ui Ui ) Then

S oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A6)

Cardy Algebras and Sewing Constraints I 907

Sminus1 oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A7)

By Lemma 32 to establish that (337) is S-invariant it is enough to prove the identity(34) when f is given by (337) Using (A6) and (A7) we can see that Eq (34) simplysays that oplusi j [RHS of (34)] is invariant under the action of S times S Consider the elementg of oplus jkisinIHomC2plusmn(R(A)otimes (Uor

j times Uk)Uorj times Uk) given by

g =oplusjkisinI

sumα

R(A)

R(A)

R(A)

R(A)

Uorj times Uk

Uorj times Uk

α

α

(A8)

By the above arguments proving S-invariance of (337) is equivalent to proving invari-ance of g under the action of S times S

For i isin I we denote by gi the component of g in(oplus jisinIHomC(A otimes Uor

i otimes Uorj U

orj )

)otimes (opluskisinIHomC(Ui otimes UkUk))

We view the second Hom-space in the above tensor product as a Hom-space in C+ insteadof Cminus It is enough to show that gi is invariant under the action of S times Sminus1 Note thatthe action of Sminus1 in C+ is equivalent to that of S in Cminus

The morphism gi can be canonically identified with a bilinear pairing

( middot middot )i ⎛⎝oplus

jisinIHomC(Uor

j A otimes Uori otimes Uor

j )

⎞⎠

times(oplus

kisinIHomC(UkUi otimes Uk)

)minusrarr C (A9)

as follows For h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) we set

(h1 h2)i = (dim U j dim Uk)minus1 trUor

j timesUk[gi (h1 times h2)] (A10)

908 L Kong I Runkel

When substituting the explicit form of the product m R(A) of R(A) = (A times 1) otimes R(1)after a short calculation one finds

(h1 h2)i =sumα

1

dim U j

A

A

A

A

A A

A α

α

h1

h2

U jUk

Uorj

Ui

Uk (A11)

Here the top morphism PlR(A) has been simplified with the help of the identity

PR(A) m R(A) (PR(A) otimes PR(A))

=(oplus

iisinI((m A A)otimes idUor

i)times idUi

) m R(A) (PR(A) otimes PR(A)) (A12)

which can be checked by direct calculation along the same lines as in the proof of [FrLem 310]

The action of the modular transformation S on oplusiisinI(B otimes Ui Ui ) for B isin C nat-urally induces an action on oplusiisinI(Ui B otimes Ui ) [K3 Prop 514] which we denote bySlowast In the present case we get an action of Slowast on oplus jisinIHomC(Uor

j A otimes Uori otimes Uor

j ) and

opluskisinIHomC(UkUi otimes Uk) Then to show gi is invariant under the action of S times Sminus1

amounts to showing that

(h1 h2)i = ((Sminus1)lowast h1 Slowast h2)i (A13)

Cardy Algebras and Sewing Constraints I 909

for all h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) We have

((Sminus1)lowast h1 Slowast h2)i =sum

mnα

dim Un

DimC

A

A

A

A

A A

A

Uk

α

α

h1

h2

Ui

U j

UmUn

(A14)

Now drag the upper vertex indexed by α in the above graph along its Uorm -leg until it

meets the lower vertex also indexed by α then sum over α and m This gives

((Sminus1)lowast h1 Slowast h2)i =sum

n

dim Un

DimC

A

A

A

A

A A

A

Uk

Un

h1

h2

Ui

U j

(A15)

910 L Kong I Runkel

If we just look at the neighbourhood of the Un-loop in the above graph we see thefollowing subgraphs

sumn

dim Un

DimC

A Aor

Uorj U j

Uk Uork

Un=

sumα

1

dim U j

A Aor Uk Uork

Uorj U j

αα

(A16)

where we have applied [BK Cor 3111] Substituting this subgraph back to the originalgraph in (A15) we obtain

((Sminus1)lowast h1 Slowast h2)i =sumα

1

dim U j

A

A

A

A

A A

A

A

A

Uorj

α

α

h1

h2m A

Ui

U jUk

Uk

(A17)

The graph in (A17) is equal to that in (A11) In order to see this we first drag theldquobubblerdquo (m A A) along A lines and through the m A vertex (because m A A is abimodule map) until it reaches the lower-left leg of the upper vertex indexed by α Thendrag the m A vertex along the (red) dotted line in the above graph Finally we applythe associativity of A (A12) and [Fr Lem 311] Then we see that the graph in (A17)exactly matches with the one in (A11)

Acknowledgements We would like to thank the organisers of the Oberwolfach Arbeitsgemeinschaft ldquoAlge-braic structures in conformal field theoriesrdquo (April 2007) where this work was started for an inspiring meetingWe would further like to thank the Hausdorff Institute for Mathematics in Bonn and the organisers of the stim-ulating meeting ldquoGeometry and Physicsrdquo (May 2008) We are indebted to Alexei Davydov Jens FjelstadJuumlrgen Fuchs Yi-Zhi Huang Alexei Kitaev Urs Schreiber Christoph Schweigert Stephan Stolz and PeterTeichner for helpful discussions andor comments on a draft of this paper The research of IR was partiallysupported by the EPSRC First Grant EPE0050471 the PPARC rolling grant PPC5071451 and the MarieCurie network lsquoSuperstring Theoryrsquo (MRTN-CT-2004-512194)

Cardy Algebras and Sewing Constraints I 911

References

[AN] Alexeevski A Natanzon SM Noncommutative two-dimensional topological field theories andhurwitz numbers for real algebraic curves Sel Math New Ser 12 307ndash377 (2006)

[Bi] Bichon J Cosovereign hopf algebras J Pure Appl Alg 157 121ndash133 (2001)[BK] Bakalov B Kirillov AA Lectures on Tensor Categories and Modular Functors Providence

RI Amer Math Soc 2001[C1] Cardy JL Operator content of two-dimensional conformal invariant theories Nucl Phys

B 270 186ndash204 (1986)[C2] Cardy JL Boundary conditions fusion rules and the verlinde formula Nucl Phys B 324

581ndash596 (1989)[CL] Cardy JL Lewellen DC Bulk and boundary operators in conformal field theory Phys Lett

B 259 274ndash278 (1991)[DMZ] Dong C-Y Mason G Zhu Y-C Discrete series of the Virasoro algebra and the moonshine

module In Algebraic Groups and Their Generalizations Quantum and infinite-dimensional Meth-ods Proc Symp Pure Math 56 Part 2 Providence RI Amer MathSoc 1994 pp 295ndash316

[DP] Day B Pastro C Note on frobenius monoidal functors New York J Math 14 733ndash742 (2008)[ENO1] Etingof PI Nikshych D Ostrik V On fusion categories Ann Math 162 581ndash642 (2005)[ENO2] Etingof PI Nikshych D Ostrik V An analogue of radfordrsquos s4 formula for finite tensor

categories Int Math Research Notices 54 2915ndash2933 (2004)[Fe] Felder G Froumlhlich J Fuchs J Schweigert C Correlation functions and boundary conditions in

rational conformal field theory and three-dimensional topology Comp Math 131 189ndash237 (2002)[FHL] Frenkel IB Huang Y-Z Lepowsky J On axiomatic approaches to vertex operator algebras

and modules Mem Amer Math Soc 104 (1993)[Fj] Fjelstad J Fuchs J Runkel I Schweigert C Uniqueness of openclosed rational cft with

given algebra of open states Adv Theor Math Phys 12 1283ndash1375 (2008)[FK] Froumlhlich J King C The chern-simons theory and knot polynomials Commun Math

Phys 126 167ndash199 (1989)[Fr] Froumlhlich J Fuchs J Runkel I Schweigert C Correspondences of ribbon categories Adv

Math 199 192ndash329 (2006)[FRS] Fuchs J Runkel I Schweigert C Tft construction of rcft correlators I partition functions Nucl

Phys B 646 353ndash497 (2002)[FS] Fuchs J Schweigert C Category theory for conformal boundary conditions Fields Inst Com-

mun 39 25ndash71 (2003)[H1] Huang Y-Z Two-dimensional conformal geometry and vertex operator algebras Progress in

Mathematics Vol 148 Boston Birkhaumluser 1997[H2] Huang Y-Z Rigidity and modularity of vertex tensor categories Comm Contemp Math 10

871ndash911 (2008)[HK1] Huang Y-Z Kong L Open-string vertex algebra category and operads Commun Math

Phys 250 433ndash471 (2004)[HK2] Huang Y-Z Kong L Full field algebras Commun Math Phys 272 345ndash396 (2007)[HK3] Huang Y-Z Kong L Modular invariance for conformal full field algebras httparxivorgabs

math0609570v2[mathQA] 2006[HL] Huang Y-Z Lepowsky J Tensor products of modules for a vertex operator algebra and vertex

tensor categories In Lie Theory and Geometry in honor of Bertram Kostant ed R BrylinskiJ-L Brylinski V Guillemin V Kac Boston Birkhaumluser 1994 pp 349ndash383

[JS] Joyal A Street R Braided tensor categories Adv Math 102 20ndash78 (1993)[K1] Kong L Full field algebras operads and tensor categories Adv Math 213 271ndash340 (2007)[K2] Kong L Open-closed field algebras Commun Math Phys 280 207ndash261 (2008)[K3] Kong L Cardy condition for open-closed field algebras Commun Math Phys 283 25ndash92 (2008)[Ki] Kitaev A Private communication[KO] Kirillov AA Ostrik V On q-analog of mckay correspondence and ade classification of sl(2)

conformal field theories Adv Math 171 183ndash227 (2002)[KR] Kong L Runkel I Morita classes of algebras in modular tensor categories Adv Math 219

1548ndash1576 (2008)[Ld] Lauda AD Frobenius algebras and ambidextrous adjunctions Theo Appl Cat 16 84ndash

122 (2006)[Li1] Li H-S Regular representations of vertex operator algebras Commun Contemp Math 4

639ndash683 (2002)[Li2] Li H-S Regular representations and huang-lepowsky tensor functors for vertex operator

algebras J Alge 255 423ndash462 (2002)

Cardy Algebras and Sewing Constraints I 873

remarks in the main text are the only places where we make reference to vertex operatoralgebras The reader who is not familiar with this structure is invited to skip ahead

All types of field algebras occurring below are called self-dual if they are endowedwith non-degenerate invariant bilinear forms

A genus-0 closed CFT is equivalent to an algebra over a partial dioperad consisting ofspheres with arbitrary in-coming and out-going punctures The dioperad structure allowsto compose one in-going and one out-going puncture of distinct spheres so that the resultis again a sphere Such an algebra with additional natural properties is canonically equiv-alent to a so-called self-dual conformal full field algebra [HK2K1] A conformal fullfield algebra contains chiral and anti-chiral parts the easiest nontrivial example is givenby V otimesV where V is a vertex operator algebra A conformal full field algebra containingV otimes V as a subalgebra is called a conformal full field algebra over V otimes V When V isrational the category of self-dual conformal full field algebras over V otimesV is isomorphicto the category of commutative symmetric Frobenius algebras in CV otimesV [K1 Thm 415]

Similarly a genus-0 open CFT is an algebra over a partial dioperad consisting ofdisks with an arbitrary number of in-coming and out-going boundary punctures Suchan algebra with additional natural properties is canonically equivalent to a self-dualopen-string vertex operator algebra as defined in [HK1] A vertex operator algebra V isnaturally an open-string vertex operator algebra An open-string vertex operator algebracontaining V as a subalgebra in its meromorphic centre is called an open-string vertexoperator algebra over V When V is rational the category of self-dual open-string vertexoperator algebras over V is isomorphic to the category of symmetric Frobenius algebrasin CV see [HK1 Thm 43] and [K3 Thm 610]

Finally a genus-0 open-closed CFT is an algebra over the Swiss-cheese partial dio-perad which consists of disks with both interior punctures and boundary puncturesand is equipped with an action of the partial spheres dioperad Such an algebra can beconstructed from a so-called self-dual open-closed field algebra [K2] It consists of aself-dual conformal full field algebra Acl a self-dual open-string vertex operator algebraAop and interactions between Acl and Aop satisfying certain compatibility conditionsNamely if Acl is defined over V otimes V and Aop over V one requires that the boundarycondition on a disc is V -invariant in the sense that both the chiral copy V otimes 1 and theanti-chiral copy 1 otimes V of V in Acl give the copy of V in Aop in the limit of the insertionpoint approaching a point on the boundary of the disc [K2 Def 125] An open-closedfield algebra with V -invariant boundary condition is called an open-closed field algebraover V When V is rational the category of self-dual open-closed field algebras overV is isomorphic to the category of triples (Aop|Acl ιcl-op) where Acl is a commuta-tive symmetric Frobenius CV otimesV -algebra Aop a symmetric Frobenius CV -algebra andιcl-op an algebra homomorphism T (Acl) rarr Aop satisfying a centre condition (given in(320) below) see [K2 Thm 314] and [K3 Sect 62] Here T CV otimesV rarr CV is theHuang-Lepowsky tensor product functor [HL]

The genus-1 theory does not provide new data as it is determined by taking tracesof genus-0 correlators but it does provide two additional consistency conditions themodular invariance condition for one-point correlators on the torus [So] and the Cardycondition for boundary-two-point correlators on the annulus [C2Lw] Their categor-ical formulations have been worked out in [HK3K3] Adding them to the axioms ofa self-dual open-closed field algebra over V finally results in the notion of a CardyCV |CV otimesV -algebra One can prove that the category of self-dual open-closed field alge-bras over a rational vertex operator algebra V satisfying the two genus-1 consistencyconditions is isomorphic to the category of Cardy CV |CV otimesV -algebras [K3 Thm 615]

874 L Kong I Runkel

If V is rational then so is V otimes V [DMZHK2] Thus both CV and CV otimesV aremodular tensor categories In fact CV otimesV sim= CV (CV )minus (see [FHL Thm 474] and[DMZ Thm 27]) where the minus sign relates to the particular braiding used for CV otimesV Namely for a given modular tensor category D we denote by Dminus the modular tensorcategory obtained from D by inverting braiding and twist We will also sometimes writeD+ for D The product amounts to taking direct sums of pairs of objects and tensorproducts of morphisms spaces The definition of a Cardy algebra can be stated in a waythat no longer makes reference to the vertex operator algebra V and therefore makessense in an arbitrary modular tensor category C Abbreviating C2plusmn equiv C+ Cminus this leadsto the definition of a Cardy C|C2plusmn-algebra

The relation to genus-01 open-closed CFT outlined above is the main motivation forour interest in Cardy C|C2plusmn-algebras In part I of this work we investigate how much onecan learn about Cardy algebras in the categorical setting and without the assumptionthat the modular tensor category C is given by CV for some V We briefly summariseour approach and results below

In Sect 21ndash23 we recall some basic notions we will need such as (co)lax tensorfunctors Frobenius functors and modular tensor categories In Sect 24 we study thefunctor T C2plusmn rarr C which is defined by the tensor product on C via T (oplusi Ai times Bi ) =oplusi Ai otimes Bi for Ai Bi isin C Using the braiding of C one can turn T into a tensor functorA tensor functor is automatically also a Frobenius functor and so takes a Frobeniusalgebra A in its domain category to a Frobenius algebra F(A) in its target category

An important object in this work is the functor R C rarr C2plusmn also defined in Sect 24We show that R is left and right adjoint to T As a consequence R is automatically alax and colax tensor functor but it is in general not a tensor functor However we willshow that it is still a Frobenius functor and so takes Frobenius algebras in C to Frobeniusalgebras in C2plusmn In fact it also preserves the properties simple special and symmetricof a Frobenius algebra In the case C = CV the functor R CV rarr CV otimesV was firstconstructed in [Li1Li2] using techniques from vertex operator algebras This motivatedthe present construction and notation The functor R was also considered in a slightlydifferent context in [ENO2]

The above results imply that R and T form an ambidextrous adjunction and we willuse this adjunction to transport algebraic structures between C and C2plusmn For examplethe algebra homomorphism ιcl-op T (Acl) rarr Aop in C is transported to an algebrahomomorphism ιcl-op Acl rarr R(Aop) in C2plusmn This gives rise to an alternative definitionof a Cardy C|C2plusmn-algebra as a triple (Aop|Acl ιcl-op)

To prepare the definition of a Cardy algebra in Sect 31 we discuss the so-calledmodular invariance condition for algebras in C2plusmn (Definition 31 below) We show thatwhen Acl is simple the modular invariance condition can be replaced by an easier con-dition on the quantum dimension of Acl (namely the dimension of Acl has to be that ofthe modular tensor category C) see Theorem 34

In Sect 32 we give the two definitions of a Cardy algebra and prove their equiva-lence Section 33 contains our main results We first show that for each special symmet-ric Frobenius algebra A in C (see Sect 22 for the definition of special) one obtains aCardy algebra (A|Z(A) e) where Z(A) is the full centre of A (Theorem 318) Thefull centre [Fj Def 49] is a subobject of R(A) and e Z(A) rarr R(A) is the canonicalembedding Next we prove a uniqueness theorem (Theorem 321) which states that if(Aop|Acl ιcl-op) is a Cardy algebra such that dim Aop = 0 and Acl is simple then Aopis special and (Aop|Acl ιcl-op) is isomorphic to (Aop|Z(Aop) e) When combined withpart II of this work this result amounts to [Fj Thm 426] and provides an alternative

Cardy Algebras and Sewing Constraints I 875

(and shorter) proof Finally we show that for every simple modular invariant commu-tative symmetric Frobenius algebra Acl in C2plusmn there exists a simple special symmetricFrobenius algebra Aop and an algebra homomorphism ιcl-op Acl rarr R(Aop) such that(Aop Acl ιcl-op) is a Cardy algebra (Theorem 322) This theorem is closely related toa result announced in [Muuml2] and provides an independent proof in the framework ofCardy algebras

In physical terms these two theorems mean that a rational open-closed CFT with aunique closed vacuum state can be uniquely reconstructed from its correlators involvingonly discs with boundary punctures and that every closed CFT with unique vacuum andleftright rational chiral algebra V otimes V occurs as part of such an open-closed CFT

2 Preliminaries on Tensor Categories

In this section we review some basic facts of tensor categories and fix our conventionsand notations along the way

21 Tensor categories and (co)lax tensor functors In a tensor (or monoidal) categoryC with tensor product bifunctor otimes and unit object 1 for U VW isin C we denote the

associator U otimes (V otimes W )sim=minusrarr (U otimes V ) otimes W by αUVW the left unit isomorphism

1 otimes Usim=minusrarr U by lU and the right unit isomorphism U otimes 1

sim=minusrarr U by rU If C is braidedfor U V isin C we write the braiding isomorphism as cUV U otimes V rarr V otimes U

Let C1 and C2 be two tensor categories with units 11 and 12 respectively For simplic-ity we will often write otimes α l r for the data of both C1 and C2 Lax and colax tensorfunctors are defined as follows see eg [Y Ch I3] or [Ln Ch I12]

Definition 21 A lax tensor functor G C1 rarr C2 is a functor equipped with a mor-phism φG

0 12 rarr G(11) in C2 and a natural transformation φG2 otimes(G times G) rarr G otimes

such that the following three diagrams commute

G(A)otimes (G(B)otimes G(C))α

idG(A)otimesφG2

(G(A)otimes G(B))otimes G(C)

φG2 otimesidG(C)

G(A)otimes G(B otimes C)

φG2

G(A otimes B)otimes G(C)

φG2

G(A otimes (B otimes C))

G(α) G((A otimes B)otimes C)

(21)

12 otimes G(A)

φG0 otimesidG(A)

lG(A) G(A)

G(lminus1A )

G(11)otimes G(A)

φG2 G(11 otimes A)

G(A)otimes 12

idG(A)otimesφG0

rG(A) G(A)

G(rminus1A )

G(A)otimes G(11)

φG2 G(A otimes 11)

(22)

Definition 22 A colax tensor functor is a functor F C1 rarr C2 equipped with a mor-phismψ F

0 F(11) rarr 12 in C2 and a natural transformationψ F2 Fotimes rarr otimes(F times F)

such that the following three diagrams commute

876 L Kong I Runkel

F(A)otimes (F(B)otimes F(C))α (F(A)otimes F(B))otimes F(C)

F(A)otimes F(B otimes C)

idF(A)otimesψ F2

F(A otimes B)otimes F(C)

ψF2 otimesidF(C)

F(A otimes (B otimes C))F(α)

ψF2

F((A otimes B)otimes C)

ψF2

(23)

12 otimes F(A) F(A)lminus1F(A)

F(11)otimes F(A)

ψF0 otimesidF(A)

F(11 otimes A)ψF

2

F(lA)

F(A)otimes 12 F(A)rminus1

F(A)

F(A)otimes F(11)

idF(A)otimesψ F0

F(A otimes 11)

F(rA)

ψF2

(24)

We denote a lax tensor functor by (G φG2 φ

G0 ) or just G and a colax tensor functor

by (F ψ F2 ψ

F0 ) or F

Definition 23 A tensor functor T C1 rarr C2 is a lax tensor functor (T φT2 φ

T0 ) such

that φT0 φ

T2 are both isomorphisms

A tensor functor (T φT2 φ

T0 ) is automatically a colax tensor functor (T ψT

2 ψT0 )

with ψT0 = (φT

0 )minus1 and ψT

2 = (φT2 )

minus1In the next section we will discuss algebras in tensor categories The defining proper-

ties (21) and (22) of a lax tensor functor are analogues of the associativity the left-unitand the right-unit properties of an algebra Indeed a lax tensor functor G C1 rarr C2maps a C1-algebra to a C2-algebra Similarly (23) and (24) are analogues of the coasso-ciativity the left-counit and the right-counit properties of a coalgebra and a colax tensorfunctor F C1 rarr C2 maps a C1-coalgebra to a C2-coalgebra We will later make useof functors that take Frobenius algebras to Frobenius algebras This requires a strongercondition than being lax and colax and leads to the notion of a lsquofunctor with Frobeniusstructurersquo or lsquoFrobenius monoidal functorrsquo [SzDPP] which we will simply refer to asFrobenius functor

Definition 24 A Frobenius functor F C1 rarr C2 is a tuple F equiv (F φF2 φ

F0 ψ

F2 ψ

F0 )

such that (F φF2 φ

F0 ) is a lax tensor functor (F ψ F

2 ψF0 ) is a colax tensor functor

and such that the following two diagrams commute

F(A)otimes (F(B)otimes F(C))α (F(A)otimes F(B))otimes F(C)

φF2 otimesidF(C)

F(A)otimes F(B otimes C)

idF(A)otimesψ F2

φF2

F(A otimes B)otimes F(C)

F(A otimes (B otimes C))F(α) F((A otimes B)otimes C)

ψF2

(25)

Cardy Algebras and Sewing Constraints I 877

F(A)otimes (F(B)otimes F(C))

idF(A)otimesφF2

(F(A)otimes F(B))otimes F(C)αminus1

F(A)otimes F(B otimes C) F(A otimes B)otimes F(C)

ψF2 otimesidF(C)

φF2

F(A otimes (B otimes C))

ψF2

F((A otimes B)otimes C)F(αminus1)

(26)

Proposition 25 If (F φF2 φ

F0 ) is a tensor functor then F is a Frobenius functor with

ψ F0 = (φF

0 )minus1 and ψ F

2 = (φF2 )

minus1

Proof Since F is a tensor functor it is lax and colax If we replace ψ F2 by (φF

2 )minus1 in

(25) and (26) both commuting diagrams are equivalent to (21) which holds becauseF is lax Thus F is a Frobenius functor

The converse statement does not hold For example the functor R which we definein Sect 24 is Frobenius but not tensor

Let us recall the notion of adjunctions and adjoint functors [Ma Ch IV1]

Definition 26 An adjunction from C1 to C2 is a triple 〈FG χ〉 where F and G arefunctors

F C1 rarr C2 G C2 rarr C1

and χ is a natural isomorphism which assigns to each pair of objects A1 isin C1 A2 isin C2a bijective map

χA1A2 HomC2(F(A1) A2)sim=minusminusrarr HomC1(A1G(A2))

which is natural in both A1 and A2 F is called a left-adjoint of G and G is called aright-adjoint of F

For simplicity we will often abbreviate χA1A2 as χ Associated to each adjunction

〈FG χ〉 there are two natural transformations idC1

δminusrarr G F and FGρminusrarr idC2 where

idC1 and idC2 are identity functors given by

δA1 = χ(idF(A1)) ρA2 = χminus1(idG(A2)) (27)

for Ai isin Ci i = 1 2 They satisfy the following two identities

GδGminusrarr G FG

Gρminusrarr G = GidGminusminusrarr G F

Fδminusrarr FG FρFminusrarr F = F

idFminusminusrarr F (28)

We have for g F(A1) rarr A2 and f A1 rarr G(A2)

χ(g) = G(g) δA1 χminus1( f ) = ρA2 F( f ) (29)

For simplicity δA1 and ρA2 are often abbreviated as δ and ρ respectivelyLet 〈FG χ〉 be an adjunction from a tensor category C1 to a tensor category C2

and (F ψ F2 ψ

F0 ) a colax tensor functor from C1 to C2 We can define a morphism

878 L Kong I Runkel

φG0 11 rarr G(12) and a natural transformation φG

2 otimes (G times G) rarr G otimes by forA B isin C2

φG0 = χ(ψ F

0 ) = 11δ11minusrarr G F(11)

G(ψ F0 )minusminusminusminusrarr G(12)

φG2 = χ((ρA otimes ρB) ψ F

2 ) = G(A)otimes G(B)δminusrarr G F (G(A)otimes G(B))

G(ψ F2 )minusminusminusminusrarr G (FG(A)otimes FG(B))

G(ρAotimesρB )minusminusminusminusminusminusrarr G(A otimes B) (210)

where we have used the first identity in (29) Notice that φG2 is natural because it is a

composition of natural transformations One can easily show that ψ F0 and ψ F

2 can bere-obtained from φG

0 and φG2 as follows

ψ F0 = χminus1(φG

0 ) = F(11)FφG

0minusminusrarr FG(12)ρminusrarr 12

ψ F2 = χminus1(φG

2 (δ otimes δ)) = F(U otimes V )F(δotimesδ)minusminusminusminusrarr F (G F(U )otimes G F(V ))

FφG2minusminusrarr FG (F(U )otimes F(V ))

ρminusrarr F(U )otimes F(V ) (211)

for U V isin C1 The following result is standard for the sake of completeness we givea proof in Appendix A1

Lemma 27 (F ψ F2 ψ

F0 ) is a colax tensor functor iff (G φG

2 φG0 ) is a lax tensor

functor

22 Algebras in tensor categories An algebra in a tensor category C or a C-algebra isa triple A = (Am η) where A is an object of C m (the multiplication) is a morphismA otimes A rarr A such that m (m otimes idA) αAAA = m (idA otimes m) and η (the unit) is amorphism 1 rarr A such that m (idA otimes η) = idA rA and m (ηotimes idA) = idA lAIf C is braided and m cAA = m then A is called commutative

A left A-module is a pair (MmM ) where M isin C and m M is a morphism AotimesM rarr Msuch that mM(idAotimesm M ) = m M(m AotimesidM )αAAM and m M(ηAotimesidM ) = idMlM Right A-modules and A-bimodules are defined similarly

Definition 28 Let C be a tensor category and let A be an algebra in C

(i) A is called simple iff it is simple as a bimodule over itselfLet C be in addition k-linear for k a field

(ii) A is called absolutely simple iff the space of A-bimodule maps from A to itself isone-dimensional dimk HomA|A(A A) = 1

(iii) A is called haploid iff dimk Hom(1 A) = 1 [FS Def 43]

In the following we will assume that all tensor categories are strict to avoid spelling outassociators and unit constraints

A C-coalgebra A = (A ε) is defined analogously to a C-algebra ie A rarrA otimes A and ε A rarr 1 obey coassociativity and counit conditions

If C is braided and if A and B are C-algebras there are two in general non-isomorphicalgebra structures on A otimes B We choose A otimes B to be the C-algebra with multiplicationm AotimesB = (m Aotimesm B)(idAotimescminus1

ABotimesidB) and unitηAotimesB = ηAotimesηB Similarly if A and Bare C-coalgebras then A otimes B becomes a C-coalgebra if we choose the comultiplicationAotimesB = (idA otimes cAB otimes idB) (A otimesB) and the counit εAotimesB = εA otimes εB

Cardy Algebras and Sewing Constraints I 879

Definition 29 A Frobenius algebra A = (Am η ε) is an algebra and a coalge-bra such that the coproduct is an intertwiner of A-bimodules

(idA otimes m) (otimes idA) = otimes m = (m otimes idA) (idA otimes)

We will use the following graphical representation for the morphisms of a Frobeniusalgebra

m =A A

A

η =A

=A A

A

ε =A

(212)

A Frobenius algebra A in a k-linear tensor category for k a field is called special iffm = ζ idA and ε η = ξ id1 for nonzero constants ζ ξ isin k If ζ = 1 we call A nor-malised-special A Frobenius algebra homomorphism between two Frobenius algebrasis both an algebra homomorphism and a coalgebra homomorphism

A (strictly) sovereign tensor category is a tensor category equipped with a left and aright duality which agree on objects and morphisms (see eg [BiFS] for more details)We will write the dualities as

Uor U

= dU Uor otimes U rarr 1U Uor

= dU U otimes Uor rarr 1

U Uor

= bU 1 rarr U otimes UorUor U

= bU 1 rarr Uor otimes U

(213)

In terms of these we define the left and right dimension of an object U as

diml U = dU bU dimr U = dU bU (214)

both of which are elements of Hom(1 1)Let now C be a sovereign tensor category For a Frobenius algebra A in C we define

two morphisms

A =

A

Aor

primeA =

A

Aor

(215)

Definition 210 A Frobenius algebra A is symmetric iff A = primeA

The following lemma shows that under certain conditions we do not need to distin-guish the various notions of simplicity in Definition 28

Lemma 211 Let A be a commutative symmetric Frobenius algebra in a C-linear semi-simple sovereign braided tensor category C and suppose that diml A = 0 Then thefollowing are equivalent

880 L Kong I Runkel

(i) A is simple(ii) A is absolutely simple

(iii) A is haploid

Proof (ii)hArr(iii) A is haploid iff it is absolutely simple as a left module over itself[FS Eq (417)] Furthermore for a commutative algebra we have HomA(A A) =HomA|A(A A) and so A is haploid iff it is absolutely simple(i)rArr(ii) If A is simple then every nonzero element of HomA|A(A A) is invertibleHence this space forms a division algebra over C and is therefore isomorphic to C(iii)rArr(i) Since C is semi-simple and A is haploid also Hom(A 1) is one-dimensionalThe counit ε is a nonzero element in this space and so gives a basis This implies firstlythat ε η = 0 and secondly that there is a constant β isin C such that

β middot ε = dA (idAor otimes m) (bA otimes idA) (216)

Composing with η from the right yields β εη = diml A The right-hand side is nonzeroand so β = 0 By [FRS Lem 311] A is special We have already proved (ii)hArr(iii)and so A is absolutely simple A special Frobenius algebra in a semi-simple categoryhas a semi-simple category of bimodules (apply [FS Prop 524] to the algebra tensoredwith its opposite algebra) For semi-simple C-linear categories simple and absolutelysimple are equivalent2 Thus A is simple Remark 212 For a Frobenius algebra A the morphisms (215) are invertible and henceA sim= Aor In this case one has diml A = dimr A [FS Rem 363] and so we could havestated the above lemma equivalently with the condition dimr A = 0

Let F C1 rarr C2 be a lax tensor functor between two tensor categories C1 C2 and

let (Am A ηA) be an algebra in C1 Define morphisms F(A)otimes F(A)m F(A)minusminusminusrarr F(A) and

12ηF(A)minusminusminusrarr F(A) as

m F(A) = F(m A) φF2 ηF(A) = F(ηA) φF

0 (217)

Then (F(A)m F(A) ηF(A)) is an algebra in C2 [JS Prop 55] If f A rarr B is analgebra homomorphism between two algebras A B isin C1 then F( f ) F(A) rarr F(B)is also an algebra homomorphism If (MmM ) is a left (or right) A-module in C1 then(F(M) F(mM )φF

2 ) is a left (or right) F(A)-module if M has a A-bimodule structurethen F(M) naturally has a F(A)-bimodule structure

Similarly if (AA εA) is a coalgebra in C1 and F C1 rarr C2 is a colax tensor func-

tor then F(A) with coproduct F(A)F(A)minusminusminusrarr F(A) otimes F(A) and counit F(A)

εF(A)minusminusminusrarr 12given by

F(A) = ψ F2 F(A) εF(A) = ψ F

0 F(εA) (218)

is a coalgebra in C2 If f A rarr B is a coalgebra homomorphism between two coalgebrasA B isin C1 then F( f ) F(A) rarr F(B) is also a coalgebra homomorphism

2 To see this note that if U is simple then the C-vector space Hom(UU ) is a division algebra and henceHom(UU ) = C idU Conversely if U is not simple then U = U1 oplus U2 and Hom(UU ) contains at leasttwo linearly independent elements namely idU1 and idU2

Cardy Algebras and Sewing Constraints I 881

Proposition 2133 If F C1 rarr C2 is a Frobenius functor and (Am A ηAA εA) aFrobenius algebra in C1 then (F(A)m F(A) ηF(A)F(A) εF(A)) is a Frobenius alge-bra in C2

Proof One Frobenius property (m F(A)otimes idF(A)) (idF(A)otimesF(A)) = F(A) m F(A)follows from the commutativity of the following diagram (we spell out the associativityisomorphisms)

F(A)otimes F(A)idF(A)otimesF(A)

φF2

F(A)otimes F(A otimes A)idF(A)otimesψF

2

φF2

F(A)otimes (F(A)otimes F(A))

αF(A)F(A)F(A)

F(A otimes A)

F(idAotimesA)

F(m A)

F(A otimes (A otimes A))

F(αAAA)

(F(A)otimes F(A))otimes F(A)

φF2 otimesidF(A)

F((A otimes A)otimes A)

F(m AotimesidA)

ψF2 F(A otimes A)otimes F(A)

F(m A)otimesidF(A)

F(A)

F(A) F(A otimes A)ψF

2 F(A)otimes F(A)

(219)

The commutativity of the upper-left subdiagram follows from the naturalness of φF2 that

of the upper-right subdiagram follows from (25) that of the lower-left subdiagram fol-lows from the Frobenius properties of A and that of the lower-right subdiagram followsfrom the naturalness of ψ F

2 The proof of the other Frobenius property is similar Proposition 214 If F C1 rarr C2 is a tensor functor and A a Frobenius algebra in C1then

(i) F(A) has a natural structure of Frobenius algebra as given in Proposition 213(ii) If A is (normalised-)special so is F(A)

Proof Part (i) follows from Propositions 25 and 213 Part (ii) is a straightforwardverification of the definition using ψ F

2 = (φF2 )

minus1 and ψ F0 = (φF

0 )minus1

Let C1 C2 be sovereign tensor categories and F C1 rarr C2 a Frobenius functor Wedefine two morphisms IF(Aor) I prime

F(Aor) F(Aor) rarr F(A)or for a Frobenius algebra A inC1 as follows

IF(Aor) = ((ψ F0 F(dA) φF

2 )otimes idF(A)or) (idF(Aor) otimes bF(A))

I primeF(Aor) = (idF(A)or otimes (ψ F

0 F(dA) φF2 )) (bF(A) otimes idF(Aor))

(220)

It is easy to see that these are isomorphisms

Lemma 215 If F C1 rarr C2 is a Frobenius functor and A a Frobenius algebra in C1then

F(A) = IF(Aor) F(A) primeF(A) = I prime

F(Aor) F(primeA) (221)

3 After the preprint of the present paper appeared we noticed that this proposition is also proved in[DP Cor 5]

882 L Kong I Runkel

Proof We only prove the first equality the second one can be seen in the same way Bydefinition we have

IF(Aor) F(A)

= ((ψ F0 F(dA) φF

2 )otimes idF(A)or) (idF(Aor) otimes bF(A)) F(A)

=[(ψ F

0 F(dA) φF2 ) (F(A)otimes idF(A))

]otimes idF(A)or

(idF(A) otimes bF(A))

For the term inside the square brackets we find

ψ F0 F(dA) φF

2 (F(A)otimes idF(A)) = ψ F0 F(dA) F(A otimes idA) φF

2

= ψ F0 F(dA (A otimes idA)) φF

2 = ψ F0 F(εA m A) φF

2 (222)

On the other hand by definitionF(A) = [((ψ F0 F(εA) (F(m A)φF

2 ))otimes idF(A)or] (idF(A) otimes bF(A)) This demonstrates the first equality in (221) Proposition 216 Let F C1 rarr C2 be a tensor functor G C2 rarr C1 a functor〈FG χ〉 an adjunction A a C1-algebra and B a C2-algebra Then f A rarr G(B)is an algebra homomorphism if and only if f = χminus1( f ) F(A) rarr B is an algebrahomomorphism

Proof We need to show that

mG(B) ( f otimes f ) = f m A and f ηA = ηG(B) (223)

is equivalent to

m B ( f otimes f ) = f m F(A) and f ηF(A) = ηBs (224)

We first prove that the first identity in (223) is equivalent to the first identity in (224)For the left-hand side of the first identity in (223) we have the following equalities

mG(B) ( f otimes f )(1)= G(m B) φG

2 ( f otimes f )(2)= G(m B) G(ρ otimes ρ) G(ψ F

2 ) δ ( f otimes f )(3)= G(m B) G(ρ otimes ρ) G(ψ F

2 ) G F( f otimes f ) δ(4)= G(m B) G(ρ otimes ρ) G(F( f )otimes F( f )) G(ψ F

2 ) δ(5)= G

(m B (ρ otimes ρ) (F( f )otimes F( f )) ψ F

2

) δ

(6)= χ(

m B (ρ otimes ρ) (F( f )otimes F( f )) ψ F2

) (225)

where (1) is the definition of mG(B) in (217) (2) is the second identity in (210) (3)and (4) are naturality of δ and ψ F

2 respectively step (5) is functoriality of G and finallystep (6) is (29) For the right hand side of the first identity in (223) we get

f m A(1)= Gρ δG ( f m A)

(2)= Gρ G F( f m A) δ(3)= G(ρ F( f m A)) δ(4)= χ(ρ F( f ) F(m A)) (226)

Cardy Algebras and Sewing Constraints I 883

where (1) is the adjunction property (28) (2) is naturality of δ (3) functoriality of Gand (4) amounts to (29) and functoriality of F

On the other hand we see that the first equality in (224) is equivalent to

m B (ρ otimes ρ) (F( f )otimes F( f )) = ρ F( f ) F(m A) φF2 (227)

Using that φF2 is invertible with inverse (φF

2 )minus1 = ψ F

2 and that χ is an isomorphismit follows that the statement that (225) is equal to (226) is equivalent to the identity(227)

Now we prove that the second identity in (223) is equivalent to the second identity in(224) Using (217) and (210) we can write ηG(B) = G(ηB)φF

0 = G(ηB)G(ψ F0 )δ1

Together with (29) this shows that the second identity in (223) is equivalent to

f ηA = χ(ηB ψ F0 ) (228)

On the other hand the second identity in (224) is equivalent to

ρ F( f ) F(ηA) φF0 = ηB (229)

which by φF0 = (ψ F

0 )minus1 and (29) is further equivalent to (228)

Definition 217 Let (Am A ηAA εA) and (Bm B ηBB εB) be two Frobeniusalgebras in a tensor category C For f A rarr B we define f lowast B rarr A by

f lowast = ((εB m B)otimes idA) (idB otimes f otimes idA) (idB otimes (A ηA)) (230)

The following lemma is immediate from the definition of (middot)lowast and the properties ofFrobenius algebras We omit the proof

Lemma 218 Let C be a tensor category let A BC be Frobenius algebras in C andlet f A rarr B and g B rarr C be morphisms

(i) (g f )lowast = f lowast glowast(ii) f is a monomorphism iff f lowast is an epimorphism

(iii) f is an algebra map iff f lowast is a coalgebra map(iv) If f is a homomorphism of Frobenius algebras then f lowast f = idA and f f lowast =

idB(v) If C is sovereign and if A and B are symmetric then f lowastlowast = f

Let C and D be tensor categories and let F C rarr D be a Frobenius functor GivenFrobenius algebras A B in C and a morphism f A rarr B the next lemma shows how(middot)lowast behaves under F

Lemma 219 F( f lowast) = F( f )lowast

Proof The definition of the structure morphisms of the Frobenius algebra F(A) is givenin (217) and (218) Substituting these definitions gives

F( f )lowast =[(ψ F

0 F(εB) F(m B) φF2

)otimes idF(A)

] [

idF(B) otimes F( f )otimes idF(A)]

[idF(B) otimes

(ψ F

2 F(A) F(ηA) φF0

)]

= (ψ F0 otimes idF(A)) [

F (εB m B (idB otimes f ))otimes idF(A)]

(φF2 otimes idF(A)) (idF(B) otimes ψ F

2 )

[idF(B) otimes F(A ηA)

] (idF(B) otimes φF0 ) (231)

884 L Kong I Runkel

In the middle line of the last expression we can use the defining property (25) of F namely we substitute (φF

2 otimes idF(A)) (idF(B) otimes ψ F2 ) = ψ F

2 φF2 Then ψ F

2 can bemoved to the left and φF

2 to the right until they can be omitted against ψ F0 and φF

0 respectively using (22) and (24) This results in

F( f )lowast = F ((εB m B (idB otimes f ))otimes idA) F (idB otimes (A ηA)) (232)

which is nothing but F( f lowast)

23 Modular tensor categories Let C be a modular tensor category [TBK] ie anabelian semi-simple finite C-linear ribbon category with simple tensor unit 1 and a non-degeneracy condition on the braiding (to be stated in a moment) We denote the set ofequivalence classes of simple objects in C by I elements in I by i j k isin I and theirrepresentatives by Ui U j Uk We also set U0 = 1 and for an index k isin I we define kby Uk

sim= Uork

Since the tensor unit is simple we shall for modular tensor categories identifyHom(1 1) sim= C (cf footnote 2) Define numbers si j isin C by4

si j = UiU j (233)

They obey si j = s ji and s0i = dim Ui see eg [BK Sect 31] (In a ribbon cate-gory the left and right dimension (214) of Ui coincide and are denoted by dim Ui )The non-degeneracy condition on the braiding of a modular tensor category is that the|I|times|I|-matrix s should be invertible In fact [BK Thm 317]

sumkisinI

sik sk j = Dim C δij (234)

where Dim C = sumiisinI(dim Ui )

2 One can show (even in the weaker context of fusioncategories over C) that Dim C ge 1 [ENO1 Thm 23] In particular Dim C = 0 We fixonce and for all a square root

radicDim C of Dim C

Let us fix a basis λα(i j)kN k

i jα=1 in HomC(Ui otimesU j Uk) and the dual basis ϒ(i j)k

α N ki j

α=1

in HomC(UkUi otimesU j ) The duality of the bases means that λα(i j)k ϒ(i j)kβ = δαβ idUk

We also fix λ(0i)i = λ(i0)i = idUi We denote the basis vectors graphically as follows

λα(i j)k = α

Uk

Ui U j

ϒ(i j)kα = α

Uk

Ui U j

(235)

4 In the graphical notation used below we have given an orientation to the ribbons indicated by the arrowsFor example it is understood that this orientation determines which of the duality morphisms in (213) to use

Cardy Algebras and Sewing Constraints I 885

For V isin C we also choose a basis b(iα)V of HomC(VUi ) and the dual basis bV(iβ)

of HomC(Ui V ) for i isin I such that b(iα)V bV(iβ) = δαβ idUi We use the graphical

notation

b(iα)V = α

Ui

V

bV(iα) = α

Ui

V

(236)

Given two modular tensor categories C and D by C D we mean the tensor productof additive categories over C [BK Def 1115] ie the category whose objects are directsums of pairs V times W of objects V isin C and W isin D and whose morphism spaces are

HomCD(V times W V prime times W prime) = HomC(V V prime)otimesC HomC(WW prime) (237)

for pairs and direct sums of these if the objects are direct sums of pairsIf we replace the braiding and the twist in C by the antibraiding cminus1 and the antitwist

θminus1 respectively we obtain another ribbon category structure on C In order to distin-guish these two distinct structures we denote (C c θ) and (C cminus1 θminus1) by C+ and Cminusrespectively As in the Introduction we will abbreviate

C2plusmn = C+ Cminus (238)

Note that a set of representatives of the simple objects in C2plusmn is given by Ui times U j fori j isin I

For the remainder of Sect 2 we fix a modular tensor category C

24 The functors T and R The tensor product bifunctor otimes can be naturally extendedto a functor T C2plusmn rarr C Namely T (oplusN

i=1Vi times Wi ) = oplusNi=1Vi otimes Wi for all Vi Wi isin C

and N isin N The functor T becomes a tensor functor as follows For φT0 1 rarr T (1 times 1)

take φT0 = id1 (or lminus1

1 in the non-strict case) Next notice that for U VW X isin C

T (U times V )otimes T (W times X)= (U otimes V )otimes (W otimes X)

T ((U times V )otimes (W times X))= (U otimes W )otimes (V otimes X)(239)

We define φT2 T (U times V )otimes T (W times X) rarr T ((U times V )otimes (W times X)) by

φT2 = idU otimes cminus1

W V otimes idX (240)

(In the non-strict case the appropriate associators have to be added) The above definitionof φT

2 can be naturally extended to a morphism φT2 T (M1 otimes M2) rarr T (M1)otimes T (M2)

for any pair of objects M1M2 in C2plusmn The following result can be checked by directcalculation [JS Prop 52]

Lemma 220 The triple (T φT2 φ

T0 ) gives a tensor functor

886 L Kong I Runkel

In particular (T φT2 φ

T0 ψ

T2 ψ

T0 ) where ψT

2 = (φT2 )

minus1 and ψT0 = (φT

0 )minus1 gives

a Frobenius functorDefine the functor R C rarr C2plusmn as follows for A isin C and f isin HomC(A B)

R(A) =oplusiisinI

(A otimes Uori )times Ui R( f ) =

oplusiisinI

( f otimes idUori)times idUi (241)

This functor was also considered in a slightly different context in [ENO2 Prop 23]The family of isomorphisms γ R

A = oplusiisinI DimCdim Ui

id(AotimesUori )timesUi

isin Aut(R(A)) defines a

natural isomorphism γ R R rarr ROur next aim is to show that R is left and right adjoint to T in other words R

and T form an ambidextrous adjunction (see eg [Ld] for a discussion of ambidextrousadjunctions) To this end we introduce two linear isomorphisms for A isin C and M isin C2plusmn

χ HomC(T (M) A) minusrarr HomC2plusmn(M R(A))

χ HomC(A T (M)) minusrarr HomC2plusmn(R(A)M)(242)

If we decompose M as M = oplusNn=1 Ml

n times Mrn then χ and χ are given by

χ Noplus

n=1

fn

Mln Mr

n

A

rarrNoplus

n=1

oplusiisinI

sumα

fn

Mln

Mrn

A

α

Uori

times α

Mrn

Ui

(243)

and

χ Noplus

n=1 gn

Mln Mr

n

A

rarrNoplus

n=1

oplusiisinI

sumα gn

Mln

Mrn

A

α

Uori

times α

Ui

Mrn

DimCdim Ui

(244)

Notice that χ and χ are independent of the choice of basis

Theorem 221 〈T R χ〉 and 〈R T χminus1〉 are adjunctions ie R is both left and rightadjoint of T

Proof Write M as M = oplusNn=1 Ml

n times Mrn The isomorphism χ amounts to the following

composition of natural isomorphisms

HomC(T (M) A) = oplusnHomC(Mln otimes Mr

n A)sim= oplusni HomC(Ml

n otimes Ui A)otimes HomC(Mrn Ui )

sim= oplusni HomC(Mln A otimes Uor

i )otimes HomC(Mrn Ui ) = HomC2plusmn(M R(A)) (245)

Cardy Algebras and Sewing Constraints I 887

Thus χ is natural Let (γ RA )

lowast HomC2plusmn(R(A)M) rarr HomC2plusmn(R(A)M) denote the

pull-back of γ RA The isomorphism χ is equal to the composition of (γ R

A )lowast and the

following sequence of natural isomorphisms

HomC(A T (M)) = oplusn HomC(AMln otimes Mr

n)

sim= oplusni HomC(AMln otimes Ui )otimes HomC(Ui Mr

n)

sim= oplusni HomC(A otimes Uori Ml

n)otimes HomC(Ui Mrn) = HomC2plusmn(R(A)M) (246)

We have proved that both χ and χ are natural isomorphisms There are four natural transformations associated to χ and χ namely

idC2plusmnδminusrarr RT

ρminusrarr idC2plusmn and idCδminusrarr T R

ρminusrarr idC (247)

defined by for A isin C M isin C2plusmn

δM = χ(idT (M)) ρA = χminus1(idR(A))

ρM = χ (idT (M)) δA = χminus1(idR(A))(248)

They can be expressed graphically as follows with M = oplusNn=1 Ml

n times Mrn

δM =oplusni

sumα

Mln

Mln Mr

n Uori

αtimes

Ui

Mrn

α

ρA =oplusiisinI

A

A

Uori Ui

(249)

ρM =oplusni

sumα

Mln

Mln

Mrn Uor

i

α times

Ui

Mrn

α DimCdim Ui

δA =oplusiisinI

A

A Uori Ui

dim UiDimC

Note that

ρM δM = Dim C middot idM and ρA δA = idA (250)

Lemma 222 The functors T and R as maps on the sets of morphisms have left inversesand thus are injective

Proof Let f A rarr B be a morphism in C We define a map Q R HomC2plusmn(R(A)

R(B)) rarr HomC(A B) by f prime rarr ρB T ( f prime) δA Then we have

Q R R( f ) = ρB T R( f ) δA = ρB δB f = f (251)

888 L Kong I Runkel

where we used naturality of δ and (250) in the second and third equalities respectivelySo Q R is a left inverse of R on morphisms Thus R is injective on morphisms Similarlylet g M rarr N be a morphism in C2plusmn We define a map QT HomC(T (M) T (N )) rarrHomC2plusmn(M N ) by gprime rarr (Dim C)minus1 middot ρN R(gprime) δM Then we have

QT T (g) = (Dim C)minus1 middot ρN RT (g) δM = (Dim C)minus1 middot ρN δN g = g (252)

So QT is a left inverse of T on morphisms Thus T is injective on morphisms Using (29) and (250) one can express the two inverse maps χminus1 χminus1 as follows

for f isin HomC2plusmn(M R(A)) and g isin HomC2plusmn(R(A)M)

χminus1( f ) = ρ T ( f ) χminus1(g) = T (g) δ (253)

By Proposition 25 and Lemma 27 R is both a lax and colax tensor functor In particularφR

0 1 times 1 rarr R(1) is given by

φR0 = χ (ψT

0 ) = R(ψT0 ) δ1 times 1 = id1times1 (254)

and φR2 R(A)otimes R(B) rarr R(A otimes B) by φR

2 = R(ρA otimes ρB) R(ψT2 ) δ which can

be expressed graphically as

φR2 =

oplusi jkisinI

sumα

A

A

Uori B

B

Uorj

Uork

α

times

Ui U j

α

Uk

(255)

Similarly ψ R0 R(1) rarr 1 times 1 is given by

ψ R0 = ρ1 R(φT

0 ) = Dim C id1times1 (256)

and ψ R2 R(A otimes B) rarr R(A) otimes R(B) by ψ R

2 = ρ R(φT2 ) R(δA otimes δB) which in

graphical notation reads

ψ R2 =

oplusi jkisinI

sumα

A

A Uori

B

B Uorj

Uork

α

times

Ui U j

α

Uk

dim Ui dim U j

dim Uk Dim C

(257)

If C has more than one simple object then R does not take the tensor unit of C to thetensor unit of C2plusmn and so is clearly not a tensor functor However we will show that R isstill a Frobenius functor This will imply that if A is a Frobenius algebra in C then

R(A) = (R(A)m R(A) ηR(A)R(A) εR(A)) (258)

Cardy Algebras and Sewing Constraints I 889

is a Frobenius algebra in C2plusmn where the structure morphisms were given in (217) and(218) In the case A = 1 it was proved in [Muuml1 Prop 41] (see also [Fr Lem 619]and [K1 Thm 52]) that (258) is a commutative simple symmetric normalised-specialFrobenius algebra in C2plusmn In fact given a Frobenius algebra A in C it is straightforwardto verify that the structure morphisms in (258) are precisely those of (A times 1)otimes R(1)cf Sect 22

Proposition 223 (R φR2 φ

R0 ψ

R2 ψ

R0 ) is a Frobenius functor

Proof Using the explicit graphical expression of φR2 φ

R0 ψ

R2 ψ

R0 it is easy to see that

the commutativity of the diagrams (25) and (26) are equivalent to the statement thatR(1) with structure morphisms as in (258) is a Frobenius algebra in C2plusmn The latterstatement is true by [Muuml1 Prop 41]

From Lemma 220 and Proposition 223 we see that T and R take Frobenius alge-bras to Frobenius algebras The following two propositions show how the properties ofFrobenius algebras are transported

Proposition 224 Let A be a Frobenius algebra in C2plusmn Then T (A) is a Frobenius algebrain C and

(i) A is symmetric iff T (A) is symmetric(ii) A is (normalised-)special iff T (A) is (normalised-)special

Proof For part (i) write A as a direct sum oplusNn=1 Al

n times Arn Then the maps IT (A) I prime

T (A) T (Aor) rarr T (A)or defined in (220) are given by

IT (A) = I primeT (A) = oplusN

n=1c(Aln)

or(Arn)

or (259)

Therefore by (221) T (A) = primeT (A) is equivalent to T (A) = T (prime

A) Since byLemma 222 T is injective on morphisms this proves part (i) Part (ii) can be checkedin the same way for example the condition mT (A) T (A) = ζ idT (A) is easily checkedto be equivalent to T (m A A) = ζ T (idA) Proposition 225 Let A be a Frobenius algebra A in C Then R(A) is a Frobeniusalgebra in C2plusmn and

(i) A is symmetric iff R(A) is symmetric(ii) A is (normalised-)special iff R(A) is (normalised-)special

Proof Recall that the structure morphisms of the Frobenius algebra R(A) are equal tothose of (A times 1) otimes R(1) Using this equality part (i) and (ii) follow because R(1) issymmetric and normalised-special For example

m R(A) R(A) = [(m A A)times id1] otimes idR(1) = R(m A A) (260)

so that m R(A) R(A) = ζ idR(A) is equivalent to R(m A A) = ζ R(idA) which byLemma 222 is equivalent to m A A = ζ idA

The functor R has one additional property not shared by T namely R takes absolutelysimple algebras to absolutely simple algebras We will see explicitly in Sect 33 thatthis is not true for T

890 L Kong I Runkel

Lemma 226 For a C-algebra A the map

R HomA|A(A A) rarr HomR(A)|R(A)(R(A) R(A)) (261)

given by f rarr R( f ) is well-defined and an isomorphism

Proof Since R is a lax tensor functor R(A) is naturally a R(A)-bimodule It is easy tosee that R in (261) is a well-defined map R(A) is also naturally a R(1)-bimodule whichcan be identified with the induced R(1)-bimodule structure on (A times 1)otimes R(1) wherethe left R(1) action on (Atimes1)otimes R(1) is given by (idAtimes1 otimesm R(1))(cminus1

Atimes1R(1)otimes idR(1))We have the following natural isomorphisms

HomR(1)|R(1)(R(A) R(A))sim=minusrarr HomC2plusmn(A times 1 R(A))

χminus1

minusminusrarr HomC(A A) (262)

which by (253) are given by for f isin HomR(1)|R(1)(R(A) R(A))

f rarr f prime = f (idAtimes1 otimes ηR(1)) rarr f primeprime = ρ T ( f prime) (263)

and its inverse is given by for g isin HomC(A A)

g rarr gprime = R(g) δ rarr gprimeprime = (idAtimes1 otimes m R(1)) (gprime otimes idR(1)) (264)

where gprimeprime is indeed a R(1)-bimodule map due to the commutativity of R(1) It is easy tocheck that gprimeprime = R(g) in (264) Therefore R gives an isomorphism from HomC(A A)to HomR(1)|R(1)(R(A) R(A)) Moreover one verifies that R(g) is an R(A)-bimodulemap iff g is an A-bimodule map In other words R g rarr R(g) gives an isomorphism

HomA|A(A A)sim=minusrarr HomR(A)|R(A)(R(A) R(A))

Corollary 227 Let A be a C-algebra

(i) A is absolutely simple iff R(A) is absolutely simple(ii) Let A be in addition Frobenius Then A is simple and special iff R(A) is simple and

special

Proof Part (i) immediately follows from Lemma 226 The statement of part (ii) withoutthe qualifier lsquosimplersquo is proved in Proposition 225 But as in the proof of (iii)rArr(i) inLemma 211 a special Frobenius algebra in a semi-simple category has a semi-simplecategory of bimodules and for a semi-simple C-linear category simple and absolutelysimple are equivalent Part (ii) then follows from part (i)

The following lemma will be needed in Sect 32 below to discuss the properties ofCardy algebras

Lemma 228 Let A be a Frobenius algebra in C Then (δA)lowast = ρA

Proof Recall from (250) that δA is a morphism A rarr T R(A) Since T and R are bothFrobenius functors T R(A) is a Frobenius algebra in C Substituting the definitions aftera short calculation one finds

Cardy Algebras and Sewing Constraints I 891

εT R(A) mT R(A) = Dim(C)oplusiisinI

A Uori Ui A U or

ı Uı

(265)

Substituting this in the definition of (δA)lowast gives again after a short calculation the mor-

phism ρA At an intermediate step one uses that the part of the morphism (δA)lowast which

is made up of Ui and Uı ribbons their duals and the basis morphisms λ(iı)0 andϒ(iı)0can be replaced by 1

dim Uimiddot dUi

3 Cardy Algebras

In this section we start by investigating the properties of Frobenius algebras which sat-isfy the so-called modular invariance condition We then give two definitions of a Cardyalgebra and prove their equivalence Finally in Sect 33 we study the properties of thesealgebras and state our main results

We fix a modular tensor category C Recall that C2plusmn is an abbreviation for C+ Cminus

31 Modular invariance In C2plusmn we define the object K and the morphism ω K rarr Kas

K =oplus

i jisinIUi times U j ω =

sumi jisinI

dim Ui dim U j

Dim C idUi timesU j (31)

They have the property (see eg [BK Cor 3111])

ω

K

(Uk times Ul )or

(Uk times Ul )or

Ui times U j

Ui times U j

= δik δ jlDim C

dim Ui dim U jbUi timesU j dUi timesU j (32)

892 L Kong I Runkel

Definition 31

(i) Let A B be objects of C2plusmn A morphism f A otimes B rarr B is called S-invariant iff

A

B

W

W

f =

A

B

W

W

f

K

ω(33)

holds for all for W isin C2plusmn(ii) A C2plusmn-algebra (Aclmcl ηcl) is called modular invariant iff θAcl = idAcl and mcl

is S-invariant

Lemma 32 The morphism f A otimes B rarr B is S-invariant if and only if

A

B

Ui times U j

Ui times U j

f = Dim Cdim Ui dim U j

sumα

A

B

B

f

Ui times U j

Ui times U j

α

α

(34)

holds for all i j isin I

Proof Condition (33) holds for all W iff it holds for all W = Ui times U j i j isin I so itis enough to show that the right-hand side of (33) with W = Ui times U j is equal to theright-hand side of (34) Recall the notation for basis morphisms in (236) Starting from(33) write

idBor otimes idUi timesU j =⎛⎝sum

klα

(b(ktimeslα)

B

)or (

bB(ktimeslα)

)or⎞⎠ otimes idUi timesU j (35)

and then apply (32) The graphical representation of the resulting morphism can bedeformed to give (34) Remark 33 As shown in [K3 Sect 61] the modular invariance condition of a C2plusmn-alge-bra exactly coincides with the modular invariance condition for torus 1-point correlationfunctions of a genus-01 closed CFT In particular the condition θAcl = idAcl is equiv-alent to invariance under the modular transformation T τ rarr τ + 1 and the condition(34) with f = mcl is equivalent to invariance under S τ rarr minus 1

τ Combining the

modular invariance condition with the genus zero properties of a genus-01 closed CFTresults in a modular invariant commutative symmetric Frobenius algebra in C2plusmn

Cardy Algebras and Sewing Constraints I 893

Let Acl be a modular invariant C2plusmn-algebra Evaluating (34) for f = mcl composingit with ηcl otimes idUi timesU j and taking the trace implies the following identity

Zi j = 1

DimCAclUi timesU j where

(36)

Zi j = dimC HomC2plusmn(UitimesU j Acl)

Decomposing Acl into simple objects this gives

Zi j =sum

klisinISik Zkl Sminus1

l j where Si j = si jradic

DimC (37)

which in CFT terms is of course nothing but the invariance of the torus partition functionunder the modular S-transformation

The following theorem gives a simple criterion for modular invariance

Theorem 34 Let Acl be a haploid commutative symmetric Frobenius algebra in C2plusmn

(i) If Acl is modular invariant then dim Acl = Dim C(ii) If dim Acl = Dim C then Acl is special and modular invariant

Proof Part (i) Since Acl is haploid for i = j = 0 Equation (36) reduces to 1 =dim AclDimCPart (ii) By the same reasoning as in the proof of (iii)rArr(i) in Lemma 211 one showsthat Acl is special Thus mcl cl = ζclidAcl for some ζcl = 0

By [KO Thm 45] the category (C2plusmn)locAcl

of local Acl-modules is again a mod-

ular tensor category and Dim (C2plusmn)locAcl

= (Dim C dim Acl)2 (see [Fr Prop 321 amp

Rem 323] for the same statement in the notation used here) Thus by assumption wehave Dim(C2plusmn)loc

Acl= 1 It then follows from [ENO1 Thm 23] that up to isomorphism

(C2plusmn)locAcl

has a unique simple object (namely the tensor unit) In other words every simplelocal Acl-module is isomorphic to Acl (seen as a left-module over itself)

We have the following isomorphisms between morphism spaces [FS Prop 47 amp 411]

HomAcl(Acl otimes (Ui times U j ) Acl) sim= HomC2plusmn(Ui times U j Acl)

HomAcl(Acl Acl otimes (Ui times U j )) sim= HomC2plusmn(AclUi times U j )

Using these to transport the bases (236) from the right to the left we obtain bases bα(i j)αof HomAcl(Acl otimes (Ui times U j ) Acl) and b(i j)

β β of HomAcl(Acl Acl otimes (Ui times U j )) Thesecan be expressed graphically as

894 L Kong I Runkel

bα(i j) =

Acl

Acl

Acl

Ui times U j

α

b(i j)β = DimC

dim Ui dim U j

1

ζcl

Acl

Ui times U jAcl

Acl

β

(38)

where the nonzero factor in b(i j)β is included for convenience Notice that bα(i j) b(i j)

β isa left Acl-module map Since Acl is simple as a left module over itself we have

bα(i j) b(i j)β = λαβ idAcl (39)

for some λαβ isin C By computing tr(bα(i j) b(i j)β ) it is easy to verify that λαβ = δαβ

We will now prove the following identity

1

ζcl

Acl

Acl

Ui times U j

Ui times U j

Acl

=sumα

DimCdim Ui dim U j

1

ζcl

Acl

Ui times U j

Ui times U j

Acl

Acl

Acl

α

α

(310)

One checks that the left-hand side of this equation is an idempotent which we denote byζminus1

cl PlAcl(Ui timesU j ) cf [Fr Sect 31] By [Fr Prop 41] the image Im(ζminus1

cl PlAcl(Ui timesU j ))

is a local Acl-module and hence isomorphic to AoplusNcl for some N isin Zge0

All left-module morphisms from Acl to Aclotimes(Ui timesU j ) are linear combinations of the

b(i j)β Furthermore one verifies that ζminus1

cl PlAcl(Ui timesU j )b(i j)

β = b(i j)β Therefore the b(i j)

β

describe precisely the image of the idempotent ie ζminus1cl Pl

Acl(Ui timesU j ) = sum

α b(i j)α bα(i j)

which is nothing but (310) Composing (310) with ζcl middot εcl otimes idUi timesU j from the left (iefrom the top) produces (34)

In addition since Acl is commutative symmetric Frobenius it satisfies θAcl = idAcl

[Fr Prop 225] Altogether this shows that Acl is modular invariant Remark 35 As we were writing this paper we heard that the results in Theorem 34were obtained independently by Kitaev and Muumlger [Ki]

Remark 36 Setting i = j = 0 in (36) gives the identity dim Acl = Z00 Dim C [KRProp 23] Combining this with Theorem 34 (ii) one may wonder if a general modularinvariant commutative symmetric Frobenius algebra Acl in C2plusmn is isomorphic to a directsum of simple such algebras However this is not so For example one can take the com-mutative symmetric Frobenius algebra Acl = C[x]x2 in the category of vector spacesequipped with the non-degenerate trace ε(ax + b) = a In this case the modular invari-ance condition holds automatically but Acl is clearly not a direct sum of two algebras

Cardy Algebras and Sewing Constraints I 895

For a general modular tensor category C the algebra C[x]x2 R(1) understood as analgebra in C2plusmn via the braided monoidal isomorphism Vect f (C) C2plusmn rarr C2plusmn providesanother counter-example

32 Two definitions Define a morphism PlA A rarr A for a Frobenius algebra A in C

or C2plusmn as follows [Fr Sect 24]

PlA =

A

A

A

m

(311)

If A is also commutative and obeys m A A = ζA idA we have PlA = ζA idA In partic-

ular this holds if A is commutative and special Using the fact that the Frobenius algebraR(1) is commutative and normalised-special one can check that Pl

R(A) R(A) rarr R(A)takes the following form

PlR(A) =

oplusiisinI

A

A

Uori

Uori

A

m

times

Ui

Ui

(312)

With these ingredients we can now give the first definition of a Cardy C|C2plusmn-algebrawhich was introduced in [K3 Def 514] cf Remark 315 below

Definition 37 (CardyC|C2plusmn-algebra I) A CardyC|C2plusmn-algebra is a triple (Aop|Acl ιcl-op)where (Aclmcl ηclcl εcl) is a modular invariant commutative symmetric Frobe-nius C2plusmn-algebra (Aopmop ηopop εop) is a symmetric Frobenius C-algebra andιcl-op Acl rarr R(Aop) an algebra homomorphism such that the following conditionsare satisfied

(i) Centre condition

Acl R(Aop)

R(Aop)

R(Aop)

ιcl-op

m R(Aop)

=

Acl

ιcl-op

R(Aop)

R(Aop)

R(Aop)

m R(Aop)

(313)

896 L Kong I Runkel

(ii) Cardy condition

ιcl-op ιlowastcl-op =

R(Aop)

R(Aop)

R(Aop)

(314)

Remark 38

(i) The name ldquoCardy C|C2plusmn-algebrardquo in Definition 37 was chosen because many ofthe important ingredients were first studied by Cardy the modular invariance ofthe closed theory [C1] the consistency of the annulus amplitude [C2] and thebulk-boundary OPE [CL] On the other hand the boundary-boundary OPE andthe OPE analogue of the centre condition were first considered in [Lw]

(ii) One can easily see that in the special case that C is the category Vect f (C) of finite-dimensional C-vector spaces a Cardy C|C2plusmn-algebra gives exactly the algebraicformulation of two-dimensional open-closed topological field theory over C (cfRemark 614 in [K3]) see [Lz Sect 48] [Mo Thm 11] [AN Thm 45] [LPCor 43] [MS Sect 22] When passing to a general modular tensor category Cthere are two important differences to the two-dimensional topological field the-ory Firstly the algebras Acl and Aop now live in different categories which inparticular affects the formulation of the centre condition and the Cardy conditionSecondly the modular invariance condition has to be imposed on Acl In the caseC = Vect f (C) modular invariance holds automatically

Definition 39 A homomorphism of Cardy C|C2plusmn-algebras (A(1)op |A(1)cl ι(1)cl-op) rarr (A(2)op

|A(2)cl ι(2)cl-op) is a pair ( fop fcl) of Frobenius algebra homomorphisms fop A(1)op rarr A(2)op

and fcl A(1)cl rarr A(2)cl such that the diagram

A(1)cl

fcl

ι(1)cl-op

A(2)cl

ι(2)cl-op

R(A(1)op )

R( fop) R(A(2)op )

(315)

commutes

Remark 310 Since a homomorphism of Frobenius algebras is invertible (cf Lemma218 (iv)) a homomorphism of Cardy algebras is always an isomorphism

For a homomorphism ( fop fcl) of Cardy C|C2plusmn-algebras using the commutativity of(315) and the fact that fcl and fop are both algebra and coalgebra homomorphisms itis easy to show that (315) commutes iff

R(A(1)op )R( fop)

(ι(1)cl-op)

lowast

R(A(2)op )

(ι(2)cl-op)

lowast

A(1)cl

fcl A(2)cl

(316)

Cardy Algebras and Sewing Constraints I 897

commutesLet (Aop|Acl ιcl-op) be a Cardy C|C2plusmn-algebra Define the morphism

ιcl-op = χminus1(ιcl-op) T (Acl) minusrarr Aop (317)

Decompose Acl as Acl = oplusNn=1Cl

n times Crn such that Cl

1 times Cr1 = ηcl(1 times 1) We use ι(n)cl-op

to denote the restriction of ιcl-op to Cln times Cr

n and ι(n)cl-op to denote the restriction of ιcl-op

to Cln otimes Cr

n We introduce the following graphical notation

ι(n)cl-op =

Cln Cr

n

Aop

(318)

By (243) ιcl-op can be expressed in terms of ιcl-op as follows

ιcl-op =Noplus

n=1

oplusiisinI

sumα

Cln

Crn

Aop

α

Uori

times α

Crn

Ui

(319)

Lemma 311 The centre condition (313) is equivalent to the following condition in C

Cln Cr

n

Aop

Aop

Aop

=

Cln Cr

n

Aop

Aop

Aop

for n = 1 N (320)

Proof First insert (319) and the definition (217) of m R(Aop) into (313) Then apply thecommutativity of R(1) to the left hand side of (313) The equivalence between (313)and (320) follows immediately Remark 312 The centre condition (320) is very natural from the open-closed confor-mal field theory point of view Correlators on the upper half plane are expressed in termsof conformal blocks on the full complex plane The objects Cl

n and Crn are associated to

the field insertion at a point z in the upper half plane and at the complex conjugate pointz in the lower half plane respectively The object Aop corresponds to a field inserted ata point r on the real axis The centre condition (320) simply says that the correlationfunctions in the disjoint domains |z| gt r gt 0 and r gt |z| gt 0 are analytic continuationsof each other see [K2 Prop 118]

898 L Kong I Runkel

Recall that we define ιlowastcl-op Aop rarr T (Acl) as in (230) We introduce the graphicalnotation

ιlowastcl-op =Noplus

n=1

Cln Cr

n

Aop

=Noplus

n=1

Cln Cr

n

Aop

Aop T (minus1Acl) (321)

where the second equality follows from (215) (221) and (259)

Lemma 313 The Cardy condition (314) is equivalent to the following identity in C

Noplusn=1

DimCdim Ui

sumα

Cln

Crn

Crn

α

α

Aop

Aop Uori

Uori

=Aop

Aop

Aop

Uori

Uori

op

mop for all i isin I (322)

Proof By (244) (312) and (319) it is easy to see that (322) is equivalent to thefollowing identity

ιcl-op χ(ιlowastcl-op) = PlR(Aop)

(323)

Therefore it is enough to show that

χ(ιlowastcl-op) = ιlowastcl-op (324)

We have

χminus1(ιlowastcl-op)(1)= T (ιlowastcl-op) δ (2)= T (ιcl-op)

lowast δlowastlowast (3)= (ρ T (ιcl-op)

)lowast (4)= (ιcl-op

)lowast (325)

In step (1) we use the expression (253) for χminus1 step (2) follows from Lemma 218 (v)and Lemma 219 Step (3) is Lemma 218 (i) and Lemma 228 and finally step (4)amounts to substituting (253) and (317) Acting with χ on both sides of the aboveequality produces (324)

Combining Lemmas 311 and 313 and Proposition 216 we obtain the followingequivalent definition of Cardy C|C2plusmn-algebra (recall the graphical notation (318) forιcl-op and (321) for ιlowastcl-op)

Cardy Algebras and Sewing Constraints I 899

Definition 314 (CardyC|C2plusmn-algebra II) A CardyC-algebra is a triple (Aop|Acl ιcl-op)where Acl is a commutative symmetric Frobenius C2plusmn-algebra satisfying property (34)with f = mcl Aop is a symmetric Frobenius C-algebra and ιcl-op T (Acl) rarr Aop isan algebra homomorphism satisfying the conditions (320) and (322)

Remark 315 Up to a choice of normalisation Definition 314 is the same as the orig-inal one in [K3 Def 613] The difference between the two definitions is the factorDim C dim Ui on the left hand side of (322) which in [K3 Def 613] is given byradic

Dim C dim Ui The two definitions are related by rescaling the coproduct cl andcounit εcl of Acl by 1

radicDim C and

radicDim C respectively We chose the convention in

(322) to remove all dimension factors from the expression (314) for the Cardy condition

33 Uniqueness and existence theorems In this subsection we investigate the structureof Cardy algebras We start with the following proposition which when combined withthe results of part II provides an alternative proof of [Fj Prop 422]

Proposition 316 Let (Aop|Acl ιcl-op) be a Cardy C|C2plusmn-algebra If Acl is simple anddim Aop = 0 then Aop is simple and special

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis therefore haploid Restricting the Cardy condition (322) to the case Ui = 1 andcomposing both sides with εop from the left we see that εop kills all terms associated toU j times 1 isin Acl in the sum except for a single 1 times 1 term Thus we obtain the followingidentity

β εop = dAop (mop otimes idAorop) (idAop otimes bAop) (326)

where β isin C Composing with ηop from the right in turn implies that βεop ηop =dim Aop which is nonzero by assumption Thus also β = 0 and εop is a nonzero multipleof the morphism on the right hand side of (326) By [FRS Lem 311] Aop is special

Since Aop is a special Frobenius algebra Aop is semi-simple as an Aop-bimodule(apply [FS Prop 524] to Aop tensored with its opposite algebra) Suppose Aop is not

simple so that we can write Aop = A(1)op oplus A(2)op for nonzero Aop-bimodules A(1)op and A(2)op We denote the canonical embeddings and projections associated to this decompositionas ι12 and π12 We have the identities

mop (ι1 otimes ι2) = 0 εop ηop =2sum

i=1

εop ιi πi ηop (327)

The first identity follows since π1 mop (ι1 otimes ι2) = 0 (as mop gives the left action of

Aop on Aop and hence it preserves A(2)op ) and similarly π2 mop (ι1 otimes ι2) = 0 Thesecond identity is just the completeness of ι12 π12

Since εop ηop = 0 without losing generality we can assume εop ι1 π1 ηop = 0Using that π2 is a bimodule map we compute

π2 [LHS of (322)]Ui =1 ι1 π1 ηop = π2 [RHS of (322)]Ui =1 ι1 π1 ηop

= PlA(2)op

π2 ι1 π1 ηop = 0 (328)

900 L Kong I Runkel

On the other hand using that Acl is haploid that ιcl-op is an algebra map and thatι lowastcl-op is a coalgebra map one can check that the left-hand side of (328) is equal toλ(εop ι1 π1 ηop) π2 ηop for some λ = 0 This implies that π2 ηop = 0 Thusηop = sum

i ιi πi ηop = ι1 π1 ηop Hence we have

0 = π2 ι2 = π2 mop (ηop otimes ι2) = π2 mop ((ι1 π1 ηop)otimes ι2) (329)

However the right-hand side is zero by (327) This is a contradiction and hence Aopmust be simple

To formulate the next theorem we need the notion of the full centre of an algebra [FjDef 49] Recall that an algebra A in a braided tensor category has a left centre and aright centre [VZO] both of which are sub-algebras of A Of these two we will onlyneed the left centre The following definition is [Fr Def 231] which in our setting isequivalent to that of [VZO]

Definition 317 Let A be a symmetric special Frobenius algebra such that m A A =ζA idA

(i) The left centre Cl(A) of A is the image of the idempotent ζminus1A Pl

A(ii) The full centre Z(A) is Cl(R(A))

That ζminus1A Pl

A is an idempotent follows from [FRS Lem 52] when keeping track ofthe factors ζA ([FRS] assumes normalised-special ie ζA = 1) Note that Cl(A) is againan object of C while Z(A) is an object of C2plusmn Let el

A Cl(A) rarr A be the embeddingof Cl(A) into A The left centre is in fact the maximal subobject of A such that

m A cAA (elA otimes idA) = m A (330)

see [Fr Lem 232] This observation explains the name left centre and also makes theconnection to [O Def 15]

The full centre is by definition the image of the idempotent ζminus1A Pl

R(A) R(A) rarrR(A) Since C2plusmn is abelian the idempotent splits and we obtain the embedding andrestriction morphisms

e Z(A) rarr R(A) and r R(A) Z(A) (331)

which obey r e = idZ(A) and e r = ζminus1A Pl

R(A) It follows from Proposition 225 and

[Fr Prop 237] that Z(A) is a commutative symmetric Frobenius algebra in C2plusmn withstructure morphisms5

m Z(A) = r m R(A) (e otimes e) ηZ(A) = r ηR(A)

Z(A) = ζA middot (r otimes r) R(A) e εZ(A) = ζminus1A middot εR(A) e

(332)

Moreover if A is simple then Z(A) is simple and if A is simple and dim A = 0 thenZ(A) is simple and special The normalisation of the counit is such that

εZ(A) ηZ(A) = ζminus2A dim A DimC (333)

5 The normalisation of product and unit is the standard one The factors in the coproduct and counit haveto be included in order for (A|Z(A) e) to be a Cardy algebra see Theorem 318 below The normalisation ofthe counit enters the Cardy condition (314) through the definition of ( middot )lowast

Cardy Algebras and Sewing Constraints I 901

Theorem 318 Let A be a special symmetric Frobenius C-algebra Then (A|Z(A) e)is a Cardy C|C2plusmn-algebra

The proof of this theorem makes use the following two lemmas

Lemma 319 e Z(A) rarr R(A) is an algebra map and elowast = ζA middot r

Proof It follows from [Fr Lem 229] (or by direct calculation using in particularm R(A) R(A) = ζAidR(A)) that

m R(A) (e otimes e) = ζminus1A middot Pl

R(A) m R(A) (e otimes e) (334)

Substituting e r = ζminus1A Pl

R(A) shows that e is compatible with multiplication For theunit one finds

e ηZ(A) = e r ηR(A) = ζminus1A Pl

R(A) ηR(A) = ηR(A) (335)

Thus e is an algebra map For the second statement one computes

elowast (1)= ζA

R(A)

r

R(A)

R(A)

Z(A)

R(A)

Z(A)

r

e

e

(2)=

R(A)

r

R(A)

R(A)

R(A)

Z(A)

(3)=

R(A)

r

R(A)

Z(A)

(4)= ζA middot r (336)

where in (1) the definitions (230) and (332) have been substituted step (2) is e r = ζminus1

A PlR(A) step (3) uses that R(A) is symmetric Frobenius and step (4) is again

e r = ζminus1A Pl

R(A) Lemma 320 Let A be a symmetric Frobenius algebra in C The morphism

PlR(A) m R(A)

(Pl

R(A) otimes PlR(A)

) R(A)otimes R(A) minusrarr R(A) (337)

is S-invariant

The proof of this lemma is a slightly lengthy explicit calculation and has been deferredto Appendix A2

902 L Kong I Runkel

Proof of Theorem 318 That e is an algebra map was proved in Lemma 319 The centrecondition (313) holds by property (330) of the left centre The Cardy condition (314)also is an immediate consequence of Lemma 319

ιcl-op ιlowastcl-op = e (ζA r) = PlR(A) (338)

The full centre Z(A) is a commutative symmetric Frobenius algebra It remains to provemodular invariance That θZ(A) = idZ(A) is implied by commutativity and symmetryof Z(A) [Fr Prop 225] The S-invariance condition (33) follows from Lemma 320In (337) substitute Pl

R(A) = ζA e r and then put the resulting morphism into (33)Compose the resulting equation with e otimes idW from the right (ie from the bottom) andsubstitute the definition (332) of m Z(A) This results in the statement that m Z(A) isS-invariant

The following theorem is analogous to [LR Prop 29] and [Fj Thm 426] whichroughly speaking answer the question under which circumstances the restriction of atwo-dimensional conformal field theory to the boundary already determines the entireconformal field theory The first work is set in Minkowski space and uses operator alge-bras and subfactors while the second work is set in Euclidean space and uses modulartensor categories

Theorem 321 Let (A|Acl ιcl-op) be a Cardy C|C2plusmn-algebra such that dim A = 0 andAcl is simple Then A is special and (A|Acl ιcl-op) sim= (A|Z(A) e) as Cardy algebras

Proof By Proposition 316 A is simple and special Since Acl is simple the algebramap ιcl-op Acl rarr R(A) is either zero or a monomorphism But ιcl-op ηcl = ηR(A)and so ιcl-op = 0 Thus ιcl-op is monic By Lemma 218 (ii) ζminus1

A ιlowastcl-op is epi The Cardycondition (314) implies

ιcl-op ζminus1A ιlowastcl-op = ζminus1

A PlR(A) = e r (339)

Composing this with e r from the left yields e r ιcl-op ζminus1A ιlowastcl-op = e r =

ιcl-op ζminus1A ιlowastcl-op Since ζminus1

A ιlowastcl-op is epi we have

e r ιcl-op = ιcl-op (340)

Actually (340) also follows from (313) and specialness of R(A) We will prove that

( fop fcl) (A|Acl ιcl-op) minusrarr (A|Z(A) e) where fop = idA fcl = r ιcl-op

(341)

is an isomorphism of Cardy algebrasfcl is an algebra map Compatibility with the units follows since ιcl-op is an algebramap

fcl ηcl = r ιcl-op ηcl = r ηR(A) = ηZ(A) (342)

Compatibility with the multiplication also follows since ιcl-op is an algebra map

m Z(A) ( fcl otimes fcl) = r m R(A) (e otimes e) (r otimes r) (ιcl-op otimes ιcl-op)

= r m R(A) (ιcl-op otimes ιcl-op) = r ιcl-op mcl = fcl mcl (343)

where in the second step we used (340)

Cardy Algebras and Sewing Constraints I 903

fcl is an isomorphism As above since fcl is an algebra map and since Acl is simple fcl

has to be monic By Lemma 319 rlowast = ζminus1A e Thus f lowast

cl = ιlowastcl-op rlowast = ζminus1A ιlowastcl-op e and

fcl f lowastcl = r ιcl-op ζminus1

A ιlowastcl-op e = r e r e = idZ(A) (344)

and so fcl is also epi and hence isofcl is a coalgebra map Since fcl is an algebra map so is f minus1

cl By (344) f minus1cl = f lowast

cland by Lemma 218 (iii) this implies that fcl is a also coalgebra mapThe diagram (315) commutes Commutativity of (315) is equivalent to e fcl = ιcl-opwhich holds by (340)

Let A be a special symmetric Frobenius algebra So far we have seen that (A|Z(A) e)is a Cardy algebra and that all Cardy algebras with Aop = A and simple Acl are of thisform It is now natural to ask if every simple Acl does occur as part of a Cardy algebraThe following theorem provides an affirmative answer Recall that for an A-left moduleM the object Mor otimesA M is an algebra (see eg [KR Lem 42])

Theorem 322 If Acl is a simple modular invariant commutative symmetric FrobeniusC2plusmn-algebra then there exist a simple special symmetric Frobenius C-algebra A and amorphism ιcl-op Acl rarr R(A) such that

(i) Acl sim= Z(A) as Frobenius algebras(ii) (A|Acl ιcl-op) is a Cardy C|C2plusmn-algebra

(iii) T (Acl) sim= oplusκisinJ Morκ otimesA Mκ as algebras where MκκisinJ is a set of representa-

tives of the isomorphism classes of simple A-left modules

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis haploid It then follows from Theorem 34 that Acl is special By Proposition 224T (Acl) is a special symmetric Frobenius algebra in C Thus T (Acl) = oplusi Ai where theAi are simple symmetric Frobenius algebras We will show that at least one of the Aiis special Since T (Acl) is special we have mT (Acl) T (Acl) = ζ idT (Acl) for someζ isin C

times Restricting this to the summand Ai shows mi i = ζ idAi FurthermoreεT (Acl) ηT (Acl) = ξ id1 for some ξ isin C

times But εT (Acl) ηT (Acl) = sumi εi ηi and so

at least one of the εi ηi has to be nonzero Therefore at least one of the Ai is speciallet A equiv Ai be this summand We denote the embedding A rarr T (Acl) by e0 and therestriction T (Acl) A by r0 Notice that r0 is an algebra homomorphism Define

ιcl-op = χ (r0) Acl minusrarr R(A) (345)

By Proposition 216 ιcl-op is an algebra homomorphism Next we verify the centre con-dition (313) or rather its equivalent form (320) By substituting the definitions onecan convince oneself that the commutativity mcl cAclAcl = mcl of Acl in C2plusmn implies thecondition mT (Acl) = mT (Acl) in C see [K2 Prop 36] Here T (Acl)otimes T (Acl) rarrT (Acl)otimes T (Acl) is given by

=oplusmn

(idCl

notimes cCl

m Crnotimes idCr

m

)

(cCl

m Clnotimes cminus1

Crn C

rm

)

(idCl

motimes cminus1

Cln C

rm

otimes idCrn

)

(346)

and we decomposed Acl as Acl = oplusnCln times Cr

n As a consequence we obtain the identity

r0 mT (Acl) (idAcl otimes e0) = r0 mT (Acl) (idAcl otimes e0) (347)

904 L Kong I Runkel

Using that r0 is an algebra map and that by definition ιcl-op = r0 we obtain (320)In order to show that (A|Acl ιcl-op) is a Cardy algebra it remains to show that the

Cardy condition (314) is satisfied We will demonstrate this via a detour by first provingthat Acl sim= Z(A) as Frobenius algebras

Recall the notations e and r given in (331) Using the centre condition (320) onecan check that Pl

R(A) ιcl-op = m R(A) R(A) ιcl-op By specialness of A we have

m A A = ζA idA and so together with e r = ζminus1A Pl

R(A) we get

e r ιcl-op = ιcl-op (348)

Next consider the morphism

fcl = r ιcl-op Acl minusrarr Z(A) (349)

By the same derivation as in (342) and (343) one sees that fcl is an algebra map Inparticular fcl ηcl = ηZ(A) = 0 and so fcl = 0 Since Acl is simple fcl has to be amonomorphism

By the same argument as used in the proof of Theorem 34 (ii) up to isomorphismAcl is the unique simple local Acl-(left-)module The algebra monomorphism fcl turnsZ(A) into an Acl-module Since Z(A) is commutative it is local as an Acl-module andso Z(A) sim= AoplusN

cl for some N ge 1 By construction A is a simple special symmetricFrobenius algebra Proposition 225 and Corollary 227 show that R(A) inherits all theseproperties and thus Z(A) is simple (see the comment below Eq (332)) By Theorem318 Z(A) is modular invariant and then by Theorem 34 (i) dim Z(A) = Dim C Thisimplies that N = 1 in Z(A) sim= AoplusN

cl and so fcl is in fact an isomorphismSince Acl and Z(A) are both haploid we have εZ(A) fcl = ξ εcl for some ξ isin C

timesThe counit uniquely determines the Frobenius structure on Acl and Z(A) (see eg [FRSLemma 37]) so that fcl is a coalgebra isomorphism iff ξ = 1 To compute ξ we composethe above identity with ηcl from the right Defining ζcl via εcl ηcl = ζminus1

cl Dim C middot id1 andusing (333) gives ξ = dim A ζclζ

2A By rescaling the comultiplication and the counit

of A and consequently changing ζA we can always achieve ξ = 1 This proves part (i)of the theorem

Equation (348) implies that ιcl-op = e fcl Since fcl is an isomorphism of Frobeniusalgebras by Lemmas 218 and 319 we have

ιcl-op ιlowastcl-op = e fcl f lowastcl elowast = ζA e r = Pl

R(A) (350)

Thus (A|Acl ιcl-op) is a Cardy algebra This proves part (ii) of the theoremPart (iii) can be seen as follows By [KR Prop 43] T Z(A) sim= oplusκisinJ Mor

κ otimesA Mκ asalgebras Together with the observation that T ( fcl) T (Acl) rarr T Z(A) is an isomor-phism of algebras this proves part (iii) Remark 323 Part (i) of Theorem 322 was announced by Muumlger [Muuml2] We provide anindependent proof in the setting of Cardy algebras

The above theorem together with Lemma 211 and Theorem 34 shows that a sim-ple commutative symmetric Frobenius C2plusmn-algebra Acl with dim Acl = Dim C is alwayspart of a Cardy algebra (Aop|Acl ιcl-op) for some simple special symmetric Frobeniusalgebra Aop in C However the above proof also illustrates that Aop is not unique Thisraises the question how two Cardy algebras with a given Acl can differ This question isanswered by [KR Thm 11] which in the present framework can be restated as follows

Cardy Algebras and Sewing Constraints I 905

Theorem 324 If (A(i)op |A(i)cl ι(i)cl-op) i = 1 2 are two Cardy C|C2plusmn-algebras such that

A(i)cl is simple and dim A(i)op = 0 for i = 1 2 then A(1)clsim= A(2)cl as algebras if and only

if A(1)op and A(2)op are Morita equivalent

Proof Theorem 11 in [KR] is stated for A(i)op being non-degenerate algebras and A(i)cl =Z(A(i)op ) for i = 1 2 By Proposition 316 A(i)op are simple and special for i = 1 2

Then by [KR Lem 21] A(i)op are non-degenerate algebras By Theorem 321 we have

A(i)clsim= Z(A(i)op ) as Frobenius algebras Finally by [KR Thm 11] Z(A(1)op ) sim= Z(A(2)op )

as algebras iff A(1)op and A(2)op are Morita equivalent Let Cmax(C2plusmn) be the set of equivalence classes [B] of simple modular invariant

commutative symmetric Frobenius algebras B in C2plusmn Two such algebras B and B prime areequivalent if B and B prime are isomorphic as algebras (but not necessarily as Frobenius alge-bras) Let Msimp(C) be the set of Morita classes of simple special symmetric Frobeniusalgebras in C Define the map z Msimp(C) rarr Cmax(C2plusmn) by z A rarr [Z(A)] whereA denotes the Morita class of A From Theorem 322 (i) and [KR Thm 11] we learn

Corollary 325 The map z Msimp(C) rarr Cmax(C2plusmn) is a bijection

A Appendix

A1 Proof of Lemma 27 We will show that if (F ψ F2 ψ

F0 ) is a colax tensor functor

from C1 to C2 then (G φG2 φ

G0 ) is a lax tensor functor from C2 to C1 Applying this

result to the opposed categories then gives the converse statementWe need to show that φG

0 and φG2 make the diagrams (21) and (22) commute We

first prove the commutativity of (21) Consider the following diagram

G(A)otimes (G(B)otimes G(C))

GψF2 δ

idG(A)otimesφG2 G(A)otimes G(B otimes C)

GψF2 δ

G(FG(A)otimes F(G(B)otimes G(C)))

G(F(idG(A))otimesF(φG2 ))

G(idFG(A)otimesψ F2 )

G(FG(A)otimes FG(B otimes C))

G(ρAotimesρBotimesC )

G(FG(A)otimes (FG(B)otimes FG(C)))

G(ρAotimes(ρBotimesρC )) G(A otimes (B otimes C))

(A1)

The top subdiagram is commutative because of the naturality of Gψ F2 δ The commu-

tativity of the bottom subdiagram follows from the following identities

(ρB otimes ρC ) ψ F2 = (ρB otimes ρC ) ψ F

2 ρF Fδ

= ρBotimesC FG(ρB otimes ρC ) FG(ψ F2 ) Fδ

= ρBotimesC F(φG2 ) (A2)

as a map F(G(B) otimes G(C)) rarr B otimes C The commutativity of (A1) implies that thecomposition of maps in the left column in (21) can be replaced by

G(ρA otimes (ρB otimes ρC )) G((idFG(A) otimes ψ F

2 ) ψ F2

) δ (A3)

906 L Kong I Runkel

Similarly we can show that the composition of maps in the right column in (21) can bereplaced by

G((ρA otimes ρB)otimes ρC ) G((ψ F

2 otimes idFG(C)) ψ F2

) δ (A4)

Using the commutativity of (23) it is easy to see that (21) with the left and right col-umns of (21) replaced by (A3) and (A4) respectively is commutative Hence (21) iscommutative

Now we prove the commutativity of the first diagram in (22)

φG2 (φG

0 otimes idG(A))

(1)= G(ρ12 otimes ρA) Gψ F2 δ

[(Gψ F

0 δ11)otimes idG(A)

]

(2)= G(ρ12 otimes ρA) Gψ F2 G F(Gψ0 otimes idG(A)) G F(δ11 otimes idG(A)) δ

(3)= G(ρ12 otimes ρA) G(FG(ψ F0 )otimes idFG(A)) Gψ F

2 G F(δ11 otimes idG(A)) δ(4)= G(id12 otimes ρA) G

([ψ F

0 ρF(11) (Fδ)11] otimes idFG(A)

) Gψ F

2 δ(5)= G(id12 otimes ρA) G(ψ F

0 otimes idFG(A)) Gψ F2 δ

(6)= G(id12 otimes ρA) G(lminus1FG(A)) G F(lG(A)) δ

(7)= G(lminus1A ) GρA δG lG(A)

(8)= G(lminus1A ) lG(A) (A5)

where in step (1) we substituted the definition of φG0 φ

G2 given in (210) in step (2)

we used the naturality of δ in step (3) we used the naturality of Gψ F2 in step (4) we

switched the position between Gρ12 and G FG(ψ F0 ) and the position between Gψ2 and

G F(δ12 otimes idG(A)) using the naturality of ρ and FψG2 respectively in step (5) we applied

the second identity in (28) in step (6) we used (24) in step (7) we used the naturalityof lminus1 and δ in step (8) we used the first identity in (28)

The proof of the commutativity of the second diagram in (22) is similar Thus wehave shown that G is a lax tensor functor

A2 Proof of Lemma 320 To prepare the proof recall that for a given object B isin Cthe modular group P SL(2Z) acts on the space oplusi HomC(B otimes Ui Ui ) see eg [BKSect 31] and [K3 Eq (455)] We will only need the action of S and Sminus1 Let f isinoplusi HomC(B otimes Ui Ui ) Then

S oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A6)

Cardy Algebras and Sewing Constraints I 907

Sminus1 oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A7)

By Lemma 32 to establish that (337) is S-invariant it is enough to prove the identity(34) when f is given by (337) Using (A6) and (A7) we can see that Eq (34) simplysays that oplusi j [RHS of (34)] is invariant under the action of S times S Consider the elementg of oplus jkisinIHomC2plusmn(R(A)otimes (Uor

j times Uk)Uorj times Uk) given by

g =oplusjkisinI

sumα

R(A)

R(A)

R(A)

R(A)

Uorj times Uk

Uorj times Uk

α

α

(A8)

By the above arguments proving S-invariance of (337) is equivalent to proving invari-ance of g under the action of S times S

For i isin I we denote by gi the component of g in(oplus jisinIHomC(A otimes Uor

i otimes Uorj U

orj )

)otimes (opluskisinIHomC(Ui otimes UkUk))

We view the second Hom-space in the above tensor product as a Hom-space in C+ insteadof Cminus It is enough to show that gi is invariant under the action of S times Sminus1 Note thatthe action of Sminus1 in C+ is equivalent to that of S in Cminus

The morphism gi can be canonically identified with a bilinear pairing

( middot middot )i ⎛⎝oplus

jisinIHomC(Uor

j A otimes Uori otimes Uor

j )

⎞⎠

times(oplus

kisinIHomC(UkUi otimes Uk)

)minusrarr C (A9)

as follows For h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) we set

(h1 h2)i = (dim U j dim Uk)minus1 trUor

j timesUk[gi (h1 times h2)] (A10)

908 L Kong I Runkel

When substituting the explicit form of the product m R(A) of R(A) = (A times 1) otimes R(1)after a short calculation one finds

(h1 h2)i =sumα

1

dim U j

A

A

A

A

A A

A α

α

h1

h2

U jUk

Uorj

Ui

Uk (A11)

Here the top morphism PlR(A) has been simplified with the help of the identity

PR(A) m R(A) (PR(A) otimes PR(A))

=(oplus

iisinI((m A A)otimes idUor

i)times idUi

) m R(A) (PR(A) otimes PR(A)) (A12)

which can be checked by direct calculation along the same lines as in the proof of [FrLem 310]

The action of the modular transformation S on oplusiisinI(B otimes Ui Ui ) for B isin C nat-urally induces an action on oplusiisinI(Ui B otimes Ui ) [K3 Prop 514] which we denote bySlowast In the present case we get an action of Slowast on oplus jisinIHomC(Uor

j A otimes Uori otimes Uor

j ) and

opluskisinIHomC(UkUi otimes Uk) Then to show gi is invariant under the action of S times Sminus1

amounts to showing that

(h1 h2)i = ((Sminus1)lowast h1 Slowast h2)i (A13)

Cardy Algebras and Sewing Constraints I 909

for all h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) We have

((Sminus1)lowast h1 Slowast h2)i =sum

mnα

dim Un

DimC

A

A

A

A

A A

A

Uk

α

α

h1

h2

Ui

U j

UmUn

(A14)

Now drag the upper vertex indexed by α in the above graph along its Uorm -leg until it

meets the lower vertex also indexed by α then sum over α and m This gives

((Sminus1)lowast h1 Slowast h2)i =sum

n

dim Un

DimC

A

A

A

A

A A

A

Uk

Un

h1

h2

Ui

U j

(A15)

910 L Kong I Runkel

If we just look at the neighbourhood of the Un-loop in the above graph we see thefollowing subgraphs

sumn

dim Un

DimC

A Aor

Uorj U j

Uk Uork

Un=

sumα

1

dim U j

A Aor Uk Uork

Uorj U j

αα

(A16)

where we have applied [BK Cor 3111] Substituting this subgraph back to the originalgraph in (A15) we obtain

((Sminus1)lowast h1 Slowast h2)i =sumα

1

dim U j

A

A

A

A

A A

A

A

A

Uorj

α

α

h1

h2m A

Ui

U jUk

Uk

(A17)

The graph in (A17) is equal to that in (A11) In order to see this we first drag theldquobubblerdquo (m A A) along A lines and through the m A vertex (because m A A is abimodule map) until it reaches the lower-left leg of the upper vertex indexed by α Thendrag the m A vertex along the (red) dotted line in the above graph Finally we applythe associativity of A (A12) and [Fr Lem 311] Then we see that the graph in (A17)exactly matches with the one in (A11)

Acknowledgements We would like to thank the organisers of the Oberwolfach Arbeitsgemeinschaft ldquoAlge-braic structures in conformal field theoriesrdquo (April 2007) where this work was started for an inspiring meetingWe would further like to thank the Hausdorff Institute for Mathematics in Bonn and the organisers of the stim-ulating meeting ldquoGeometry and Physicsrdquo (May 2008) We are indebted to Alexei Davydov Jens FjelstadJuumlrgen Fuchs Yi-Zhi Huang Alexei Kitaev Urs Schreiber Christoph Schweigert Stephan Stolz and PeterTeichner for helpful discussions andor comments on a draft of this paper The research of IR was partiallysupported by the EPSRC First Grant EPE0050471 the PPARC rolling grant PPC5071451 and the MarieCurie network lsquoSuperstring Theoryrsquo (MRTN-CT-2004-512194)

Cardy Algebras and Sewing Constraints I 911

References

[AN] Alexeevski A Natanzon SM Noncommutative two-dimensional topological field theories andhurwitz numbers for real algebraic curves Sel Math New Ser 12 307ndash377 (2006)

[Bi] Bichon J Cosovereign hopf algebras J Pure Appl Alg 157 121ndash133 (2001)[BK] Bakalov B Kirillov AA Lectures on Tensor Categories and Modular Functors Providence

RI Amer Math Soc 2001[C1] Cardy JL Operator content of two-dimensional conformal invariant theories Nucl Phys

B 270 186ndash204 (1986)[C2] Cardy JL Boundary conditions fusion rules and the verlinde formula Nucl Phys B 324

581ndash596 (1989)[CL] Cardy JL Lewellen DC Bulk and boundary operators in conformal field theory Phys Lett

B 259 274ndash278 (1991)[DMZ] Dong C-Y Mason G Zhu Y-C Discrete series of the Virasoro algebra and the moonshine

module In Algebraic Groups and Their Generalizations Quantum and infinite-dimensional Meth-ods Proc Symp Pure Math 56 Part 2 Providence RI Amer MathSoc 1994 pp 295ndash316

[DP] Day B Pastro C Note on frobenius monoidal functors New York J Math 14 733ndash742 (2008)[ENO1] Etingof PI Nikshych D Ostrik V On fusion categories Ann Math 162 581ndash642 (2005)[ENO2] Etingof PI Nikshych D Ostrik V An analogue of radfordrsquos s4 formula for finite tensor

categories Int Math Research Notices 54 2915ndash2933 (2004)[Fe] Felder G Froumlhlich J Fuchs J Schweigert C Correlation functions and boundary conditions in

rational conformal field theory and three-dimensional topology Comp Math 131 189ndash237 (2002)[FHL] Frenkel IB Huang Y-Z Lepowsky J On axiomatic approaches to vertex operator algebras

and modules Mem Amer Math Soc 104 (1993)[Fj] Fjelstad J Fuchs J Runkel I Schweigert C Uniqueness of openclosed rational cft with

given algebra of open states Adv Theor Math Phys 12 1283ndash1375 (2008)[FK] Froumlhlich J King C The chern-simons theory and knot polynomials Commun Math

Phys 126 167ndash199 (1989)[Fr] Froumlhlich J Fuchs J Runkel I Schweigert C Correspondences of ribbon categories Adv

Math 199 192ndash329 (2006)[FRS] Fuchs J Runkel I Schweigert C Tft construction of rcft correlators I partition functions Nucl

Phys B 646 353ndash497 (2002)[FS] Fuchs J Schweigert C Category theory for conformal boundary conditions Fields Inst Com-

mun 39 25ndash71 (2003)[H1] Huang Y-Z Two-dimensional conformal geometry and vertex operator algebras Progress in

Mathematics Vol 148 Boston Birkhaumluser 1997[H2] Huang Y-Z Rigidity and modularity of vertex tensor categories Comm Contemp Math 10

871ndash911 (2008)[HK1] Huang Y-Z Kong L Open-string vertex algebra category and operads Commun Math

Phys 250 433ndash471 (2004)[HK2] Huang Y-Z Kong L Full field algebras Commun Math Phys 272 345ndash396 (2007)[HK3] Huang Y-Z Kong L Modular invariance for conformal full field algebras httparxivorgabs

math0609570v2[mathQA] 2006[HL] Huang Y-Z Lepowsky J Tensor products of modules for a vertex operator algebra and vertex

tensor categories In Lie Theory and Geometry in honor of Bertram Kostant ed R BrylinskiJ-L Brylinski V Guillemin V Kac Boston Birkhaumluser 1994 pp 349ndash383

[JS] Joyal A Street R Braided tensor categories Adv Math 102 20ndash78 (1993)[K1] Kong L Full field algebras operads and tensor categories Adv Math 213 271ndash340 (2007)[K2] Kong L Open-closed field algebras Commun Math Phys 280 207ndash261 (2008)[K3] Kong L Cardy condition for open-closed field algebras Commun Math Phys 283 25ndash92 (2008)[Ki] Kitaev A Private communication[KO] Kirillov AA Ostrik V On q-analog of mckay correspondence and ade classification of sl(2)

conformal field theories Adv Math 171 183ndash227 (2002)[KR] Kong L Runkel I Morita classes of algebras in modular tensor categories Adv Math 219

1548ndash1576 (2008)[Ld] Lauda AD Frobenius algebras and ambidextrous adjunctions Theo Appl Cat 16 84ndash

122 (2006)[Li1] Li H-S Regular representations of vertex operator algebras Commun Contemp Math 4

639ndash683 (2002)[Li2] Li H-S Regular representations and huang-lepowsky tensor functors for vertex operator

algebras J Alge 255 423ndash462 (2002)

874 L Kong I Runkel

If V is rational then so is V otimes V [DMZHK2] Thus both CV and CV otimesV aremodular tensor categories In fact CV otimesV sim= CV (CV )minus (see [FHL Thm 474] and[DMZ Thm 27]) where the minus sign relates to the particular braiding used for CV otimesV Namely for a given modular tensor category D we denote by Dminus the modular tensorcategory obtained from D by inverting braiding and twist We will also sometimes writeD+ for D The product amounts to taking direct sums of pairs of objects and tensorproducts of morphisms spaces The definition of a Cardy algebra can be stated in a waythat no longer makes reference to the vertex operator algebra V and therefore makessense in an arbitrary modular tensor category C Abbreviating C2plusmn equiv C+ Cminus this leadsto the definition of a Cardy C|C2plusmn-algebra

The relation to genus-01 open-closed CFT outlined above is the main motivation forour interest in Cardy C|C2plusmn-algebras In part I of this work we investigate how much onecan learn about Cardy algebras in the categorical setting and without the assumptionthat the modular tensor category C is given by CV for some V We briefly summariseour approach and results below

In Sect 21ndash23 we recall some basic notions we will need such as (co)lax tensorfunctors Frobenius functors and modular tensor categories In Sect 24 we study thefunctor T C2plusmn rarr C which is defined by the tensor product on C via T (oplusi Ai times Bi ) =oplusi Ai otimes Bi for Ai Bi isin C Using the braiding of C one can turn T into a tensor functorA tensor functor is automatically also a Frobenius functor and so takes a Frobeniusalgebra A in its domain category to a Frobenius algebra F(A) in its target category

An important object in this work is the functor R C rarr C2plusmn also defined in Sect 24We show that R is left and right adjoint to T As a consequence R is automatically alax and colax tensor functor but it is in general not a tensor functor However we willshow that it is still a Frobenius functor and so takes Frobenius algebras in C to Frobeniusalgebras in C2plusmn In fact it also preserves the properties simple special and symmetricof a Frobenius algebra In the case C = CV the functor R CV rarr CV otimesV was firstconstructed in [Li1Li2] using techniques from vertex operator algebras This motivatedthe present construction and notation The functor R was also considered in a slightlydifferent context in [ENO2]

The above results imply that R and T form an ambidextrous adjunction and we willuse this adjunction to transport algebraic structures between C and C2plusmn For examplethe algebra homomorphism ιcl-op T (Acl) rarr Aop in C is transported to an algebrahomomorphism ιcl-op Acl rarr R(Aop) in C2plusmn This gives rise to an alternative definitionof a Cardy C|C2plusmn-algebra as a triple (Aop|Acl ιcl-op)

To prepare the definition of a Cardy algebra in Sect 31 we discuss the so-calledmodular invariance condition for algebras in C2plusmn (Definition 31 below) We show thatwhen Acl is simple the modular invariance condition can be replaced by an easier con-dition on the quantum dimension of Acl (namely the dimension of Acl has to be that ofthe modular tensor category C) see Theorem 34

In Sect 32 we give the two definitions of a Cardy algebra and prove their equiva-lence Section 33 contains our main results We first show that for each special symmet-ric Frobenius algebra A in C (see Sect 22 for the definition of special) one obtains aCardy algebra (A|Z(A) e) where Z(A) is the full centre of A (Theorem 318) Thefull centre [Fj Def 49] is a subobject of R(A) and e Z(A) rarr R(A) is the canonicalembedding Next we prove a uniqueness theorem (Theorem 321) which states that if(Aop|Acl ιcl-op) is a Cardy algebra such that dim Aop = 0 and Acl is simple then Aopis special and (Aop|Acl ιcl-op) is isomorphic to (Aop|Z(Aop) e) When combined withpart II of this work this result amounts to [Fj Thm 426] and provides an alternative

Cardy Algebras and Sewing Constraints I 875

(and shorter) proof Finally we show that for every simple modular invariant commu-tative symmetric Frobenius algebra Acl in C2plusmn there exists a simple special symmetricFrobenius algebra Aop and an algebra homomorphism ιcl-op Acl rarr R(Aop) such that(Aop Acl ιcl-op) is a Cardy algebra (Theorem 322) This theorem is closely related toa result announced in [Muuml2] and provides an independent proof in the framework ofCardy algebras

In physical terms these two theorems mean that a rational open-closed CFT with aunique closed vacuum state can be uniquely reconstructed from its correlators involvingonly discs with boundary punctures and that every closed CFT with unique vacuum andleftright rational chiral algebra V otimes V occurs as part of such an open-closed CFT

2 Preliminaries on Tensor Categories

In this section we review some basic facts of tensor categories and fix our conventionsand notations along the way

21 Tensor categories and (co)lax tensor functors In a tensor (or monoidal) categoryC with tensor product bifunctor otimes and unit object 1 for U VW isin C we denote the

associator U otimes (V otimes W )sim=minusrarr (U otimes V ) otimes W by αUVW the left unit isomorphism

1 otimes Usim=minusrarr U by lU and the right unit isomorphism U otimes 1

sim=minusrarr U by rU If C is braidedfor U V isin C we write the braiding isomorphism as cUV U otimes V rarr V otimes U

Let C1 and C2 be two tensor categories with units 11 and 12 respectively For simplic-ity we will often write otimes α l r for the data of both C1 and C2 Lax and colax tensorfunctors are defined as follows see eg [Y Ch I3] or [Ln Ch I12]

Definition 21 A lax tensor functor G C1 rarr C2 is a functor equipped with a mor-phism φG

0 12 rarr G(11) in C2 and a natural transformation φG2 otimes(G times G) rarr G otimes

such that the following three diagrams commute

G(A)otimes (G(B)otimes G(C))α

idG(A)otimesφG2

(G(A)otimes G(B))otimes G(C)

φG2 otimesidG(C)

G(A)otimes G(B otimes C)

φG2

G(A otimes B)otimes G(C)

φG2

G(A otimes (B otimes C))

G(α) G((A otimes B)otimes C)

(21)

12 otimes G(A)

φG0 otimesidG(A)

lG(A) G(A)

G(lminus1A )

G(11)otimes G(A)

φG2 G(11 otimes A)

G(A)otimes 12

idG(A)otimesφG0

rG(A) G(A)

G(rminus1A )

G(A)otimes G(11)

φG2 G(A otimes 11)

(22)

Definition 22 A colax tensor functor is a functor F C1 rarr C2 equipped with a mor-phismψ F

0 F(11) rarr 12 in C2 and a natural transformationψ F2 Fotimes rarr otimes(F times F)

such that the following three diagrams commute

876 L Kong I Runkel

F(A)otimes (F(B)otimes F(C))α (F(A)otimes F(B))otimes F(C)

F(A)otimes F(B otimes C)

idF(A)otimesψ F2

F(A otimes B)otimes F(C)

ψF2 otimesidF(C)

F(A otimes (B otimes C))F(α)

ψF2

F((A otimes B)otimes C)

ψF2

(23)

12 otimes F(A) F(A)lminus1F(A)

F(11)otimes F(A)

ψF0 otimesidF(A)

F(11 otimes A)ψF

2

F(lA)

F(A)otimes 12 F(A)rminus1

F(A)

F(A)otimes F(11)

idF(A)otimesψ F0

F(A otimes 11)

F(rA)

ψF2

(24)

We denote a lax tensor functor by (G φG2 φ

G0 ) or just G and a colax tensor functor

by (F ψ F2 ψ

F0 ) or F

Definition 23 A tensor functor T C1 rarr C2 is a lax tensor functor (T φT2 φ

T0 ) such

that φT0 φ

T2 are both isomorphisms

A tensor functor (T φT2 φ

T0 ) is automatically a colax tensor functor (T ψT

2 ψT0 )

with ψT0 = (φT

0 )minus1 and ψT

2 = (φT2 )

minus1In the next section we will discuss algebras in tensor categories The defining proper-

ties (21) and (22) of a lax tensor functor are analogues of the associativity the left-unitand the right-unit properties of an algebra Indeed a lax tensor functor G C1 rarr C2maps a C1-algebra to a C2-algebra Similarly (23) and (24) are analogues of the coasso-ciativity the left-counit and the right-counit properties of a coalgebra and a colax tensorfunctor F C1 rarr C2 maps a C1-coalgebra to a C2-coalgebra We will later make useof functors that take Frobenius algebras to Frobenius algebras This requires a strongercondition than being lax and colax and leads to the notion of a lsquofunctor with Frobeniusstructurersquo or lsquoFrobenius monoidal functorrsquo [SzDPP] which we will simply refer to asFrobenius functor

Definition 24 A Frobenius functor F C1 rarr C2 is a tuple F equiv (F φF2 φ

F0 ψ

F2 ψ

F0 )

such that (F φF2 φ

F0 ) is a lax tensor functor (F ψ F

2 ψF0 ) is a colax tensor functor

and such that the following two diagrams commute

F(A)otimes (F(B)otimes F(C))α (F(A)otimes F(B))otimes F(C)

φF2 otimesidF(C)

F(A)otimes F(B otimes C)

idF(A)otimesψ F2

φF2

F(A otimes B)otimes F(C)

F(A otimes (B otimes C))F(α) F((A otimes B)otimes C)

ψF2

(25)

Cardy Algebras and Sewing Constraints I 877

F(A)otimes (F(B)otimes F(C))

idF(A)otimesφF2

(F(A)otimes F(B))otimes F(C)αminus1

F(A)otimes F(B otimes C) F(A otimes B)otimes F(C)

ψF2 otimesidF(C)

φF2

F(A otimes (B otimes C))

ψF2

F((A otimes B)otimes C)F(αminus1)

(26)

Proposition 25 If (F φF2 φ

F0 ) is a tensor functor then F is a Frobenius functor with

ψ F0 = (φF

0 )minus1 and ψ F

2 = (φF2 )

minus1

Proof Since F is a tensor functor it is lax and colax If we replace ψ F2 by (φF

2 )minus1 in

(25) and (26) both commuting diagrams are equivalent to (21) which holds becauseF is lax Thus F is a Frobenius functor

The converse statement does not hold For example the functor R which we definein Sect 24 is Frobenius but not tensor

Let us recall the notion of adjunctions and adjoint functors [Ma Ch IV1]

Definition 26 An adjunction from C1 to C2 is a triple 〈FG χ〉 where F and G arefunctors

F C1 rarr C2 G C2 rarr C1

and χ is a natural isomorphism which assigns to each pair of objects A1 isin C1 A2 isin C2a bijective map

χA1A2 HomC2(F(A1) A2)sim=minusminusrarr HomC1(A1G(A2))

which is natural in both A1 and A2 F is called a left-adjoint of G and G is called aright-adjoint of F

For simplicity we will often abbreviate χA1A2 as χ Associated to each adjunction

〈FG χ〉 there are two natural transformations idC1

δminusrarr G F and FGρminusrarr idC2 where

idC1 and idC2 are identity functors given by

δA1 = χ(idF(A1)) ρA2 = χminus1(idG(A2)) (27)

for Ai isin Ci i = 1 2 They satisfy the following two identities

GδGminusrarr G FG

Gρminusrarr G = GidGminusminusrarr G F

Fδminusrarr FG FρFminusrarr F = F

idFminusminusrarr F (28)

We have for g F(A1) rarr A2 and f A1 rarr G(A2)

χ(g) = G(g) δA1 χminus1( f ) = ρA2 F( f ) (29)

For simplicity δA1 and ρA2 are often abbreviated as δ and ρ respectivelyLet 〈FG χ〉 be an adjunction from a tensor category C1 to a tensor category C2

and (F ψ F2 ψ

F0 ) a colax tensor functor from C1 to C2 We can define a morphism

878 L Kong I Runkel

φG0 11 rarr G(12) and a natural transformation φG

2 otimes (G times G) rarr G otimes by forA B isin C2

φG0 = χ(ψ F

0 ) = 11δ11minusrarr G F(11)

G(ψ F0 )minusminusminusminusrarr G(12)

φG2 = χ((ρA otimes ρB) ψ F

2 ) = G(A)otimes G(B)δminusrarr G F (G(A)otimes G(B))

G(ψ F2 )minusminusminusminusrarr G (FG(A)otimes FG(B))

G(ρAotimesρB )minusminusminusminusminusminusrarr G(A otimes B) (210)

where we have used the first identity in (29) Notice that φG2 is natural because it is a

composition of natural transformations One can easily show that ψ F0 and ψ F

2 can bere-obtained from φG

0 and φG2 as follows

ψ F0 = χminus1(φG

0 ) = F(11)FφG

0minusminusrarr FG(12)ρminusrarr 12

ψ F2 = χminus1(φG

2 (δ otimes δ)) = F(U otimes V )F(δotimesδ)minusminusminusminusrarr F (G F(U )otimes G F(V ))

FφG2minusminusrarr FG (F(U )otimes F(V ))

ρminusrarr F(U )otimes F(V ) (211)

for U V isin C1 The following result is standard for the sake of completeness we givea proof in Appendix A1

Lemma 27 (F ψ F2 ψ

F0 ) is a colax tensor functor iff (G φG

2 φG0 ) is a lax tensor

functor

22 Algebras in tensor categories An algebra in a tensor category C or a C-algebra isa triple A = (Am η) where A is an object of C m (the multiplication) is a morphismA otimes A rarr A such that m (m otimes idA) αAAA = m (idA otimes m) and η (the unit) is amorphism 1 rarr A such that m (idA otimes η) = idA rA and m (ηotimes idA) = idA lAIf C is braided and m cAA = m then A is called commutative

A left A-module is a pair (MmM ) where M isin C and m M is a morphism AotimesM rarr Msuch that mM(idAotimesm M ) = m M(m AotimesidM )αAAM and m M(ηAotimesidM ) = idMlM Right A-modules and A-bimodules are defined similarly

Definition 28 Let C be a tensor category and let A be an algebra in C

(i) A is called simple iff it is simple as a bimodule over itselfLet C be in addition k-linear for k a field

(ii) A is called absolutely simple iff the space of A-bimodule maps from A to itself isone-dimensional dimk HomA|A(A A) = 1

(iii) A is called haploid iff dimk Hom(1 A) = 1 [FS Def 43]

In the following we will assume that all tensor categories are strict to avoid spelling outassociators and unit constraints

A C-coalgebra A = (A ε) is defined analogously to a C-algebra ie A rarrA otimes A and ε A rarr 1 obey coassociativity and counit conditions

If C is braided and if A and B are C-algebras there are two in general non-isomorphicalgebra structures on A otimes B We choose A otimes B to be the C-algebra with multiplicationm AotimesB = (m Aotimesm B)(idAotimescminus1

ABotimesidB) and unitηAotimesB = ηAotimesηB Similarly if A and Bare C-coalgebras then A otimes B becomes a C-coalgebra if we choose the comultiplicationAotimesB = (idA otimes cAB otimes idB) (A otimesB) and the counit εAotimesB = εA otimes εB

Cardy Algebras and Sewing Constraints I 879

Definition 29 A Frobenius algebra A = (Am η ε) is an algebra and a coalge-bra such that the coproduct is an intertwiner of A-bimodules

(idA otimes m) (otimes idA) = otimes m = (m otimes idA) (idA otimes)

We will use the following graphical representation for the morphisms of a Frobeniusalgebra

m =A A

A

η =A

=A A

A

ε =A

(212)

A Frobenius algebra A in a k-linear tensor category for k a field is called special iffm = ζ idA and ε η = ξ id1 for nonzero constants ζ ξ isin k If ζ = 1 we call A nor-malised-special A Frobenius algebra homomorphism between two Frobenius algebrasis both an algebra homomorphism and a coalgebra homomorphism

A (strictly) sovereign tensor category is a tensor category equipped with a left and aright duality which agree on objects and morphisms (see eg [BiFS] for more details)We will write the dualities as

Uor U

= dU Uor otimes U rarr 1U Uor

= dU U otimes Uor rarr 1

U Uor

= bU 1 rarr U otimes UorUor U

= bU 1 rarr Uor otimes U

(213)

In terms of these we define the left and right dimension of an object U as

diml U = dU bU dimr U = dU bU (214)

both of which are elements of Hom(1 1)Let now C be a sovereign tensor category For a Frobenius algebra A in C we define

two morphisms

A =

A

Aor

primeA =

A

Aor

(215)

Definition 210 A Frobenius algebra A is symmetric iff A = primeA

The following lemma shows that under certain conditions we do not need to distin-guish the various notions of simplicity in Definition 28

Lemma 211 Let A be a commutative symmetric Frobenius algebra in a C-linear semi-simple sovereign braided tensor category C and suppose that diml A = 0 Then thefollowing are equivalent

880 L Kong I Runkel

(i) A is simple(ii) A is absolutely simple

(iii) A is haploid

Proof (ii)hArr(iii) A is haploid iff it is absolutely simple as a left module over itself[FS Eq (417)] Furthermore for a commutative algebra we have HomA(A A) =HomA|A(A A) and so A is haploid iff it is absolutely simple(i)rArr(ii) If A is simple then every nonzero element of HomA|A(A A) is invertibleHence this space forms a division algebra over C and is therefore isomorphic to C(iii)rArr(i) Since C is semi-simple and A is haploid also Hom(A 1) is one-dimensionalThe counit ε is a nonzero element in this space and so gives a basis This implies firstlythat ε η = 0 and secondly that there is a constant β isin C such that

β middot ε = dA (idAor otimes m) (bA otimes idA) (216)

Composing with η from the right yields β εη = diml A The right-hand side is nonzeroand so β = 0 By [FRS Lem 311] A is special We have already proved (ii)hArr(iii)and so A is absolutely simple A special Frobenius algebra in a semi-simple categoryhas a semi-simple category of bimodules (apply [FS Prop 524] to the algebra tensoredwith its opposite algebra) For semi-simple C-linear categories simple and absolutelysimple are equivalent2 Thus A is simple Remark 212 For a Frobenius algebra A the morphisms (215) are invertible and henceA sim= Aor In this case one has diml A = dimr A [FS Rem 363] and so we could havestated the above lemma equivalently with the condition dimr A = 0

Let F C1 rarr C2 be a lax tensor functor between two tensor categories C1 C2 and

let (Am A ηA) be an algebra in C1 Define morphisms F(A)otimes F(A)m F(A)minusminusminusrarr F(A) and

12ηF(A)minusminusminusrarr F(A) as

m F(A) = F(m A) φF2 ηF(A) = F(ηA) φF

0 (217)

Then (F(A)m F(A) ηF(A)) is an algebra in C2 [JS Prop 55] If f A rarr B is analgebra homomorphism between two algebras A B isin C1 then F( f ) F(A) rarr F(B)is also an algebra homomorphism If (MmM ) is a left (or right) A-module in C1 then(F(M) F(mM )φF

2 ) is a left (or right) F(A)-module if M has a A-bimodule structurethen F(M) naturally has a F(A)-bimodule structure

Similarly if (AA εA) is a coalgebra in C1 and F C1 rarr C2 is a colax tensor func-

tor then F(A) with coproduct F(A)F(A)minusminusminusrarr F(A) otimes F(A) and counit F(A)

εF(A)minusminusminusrarr 12given by

F(A) = ψ F2 F(A) εF(A) = ψ F

0 F(εA) (218)

is a coalgebra in C2 If f A rarr B is a coalgebra homomorphism between two coalgebrasA B isin C1 then F( f ) F(A) rarr F(B) is also a coalgebra homomorphism

2 To see this note that if U is simple then the C-vector space Hom(UU ) is a division algebra and henceHom(UU ) = C idU Conversely if U is not simple then U = U1 oplus U2 and Hom(UU ) contains at leasttwo linearly independent elements namely idU1 and idU2

Cardy Algebras and Sewing Constraints I 881

Proposition 2133 If F C1 rarr C2 is a Frobenius functor and (Am A ηAA εA) aFrobenius algebra in C1 then (F(A)m F(A) ηF(A)F(A) εF(A)) is a Frobenius alge-bra in C2

Proof One Frobenius property (m F(A)otimes idF(A)) (idF(A)otimesF(A)) = F(A) m F(A)follows from the commutativity of the following diagram (we spell out the associativityisomorphisms)

F(A)otimes F(A)idF(A)otimesF(A)

φF2

F(A)otimes F(A otimes A)idF(A)otimesψF

2

φF2

F(A)otimes (F(A)otimes F(A))

αF(A)F(A)F(A)

F(A otimes A)

F(idAotimesA)

F(m A)

F(A otimes (A otimes A))

F(αAAA)

(F(A)otimes F(A))otimes F(A)

φF2 otimesidF(A)

F((A otimes A)otimes A)

F(m AotimesidA)

ψF2 F(A otimes A)otimes F(A)

F(m A)otimesidF(A)

F(A)

F(A) F(A otimes A)ψF

2 F(A)otimes F(A)

(219)

The commutativity of the upper-left subdiagram follows from the naturalness of φF2 that

of the upper-right subdiagram follows from (25) that of the lower-left subdiagram fol-lows from the Frobenius properties of A and that of the lower-right subdiagram followsfrom the naturalness of ψ F

2 The proof of the other Frobenius property is similar Proposition 214 If F C1 rarr C2 is a tensor functor and A a Frobenius algebra in C1then

(i) F(A) has a natural structure of Frobenius algebra as given in Proposition 213(ii) If A is (normalised-)special so is F(A)

Proof Part (i) follows from Propositions 25 and 213 Part (ii) is a straightforwardverification of the definition using ψ F

2 = (φF2 )

minus1 and ψ F0 = (φF

0 )minus1

Let C1 C2 be sovereign tensor categories and F C1 rarr C2 a Frobenius functor Wedefine two morphisms IF(Aor) I prime

F(Aor) F(Aor) rarr F(A)or for a Frobenius algebra A inC1 as follows

IF(Aor) = ((ψ F0 F(dA) φF

2 )otimes idF(A)or) (idF(Aor) otimes bF(A))

I primeF(Aor) = (idF(A)or otimes (ψ F

0 F(dA) φF2 )) (bF(A) otimes idF(Aor))

(220)

It is easy to see that these are isomorphisms

Lemma 215 If F C1 rarr C2 is a Frobenius functor and A a Frobenius algebra in C1then

F(A) = IF(Aor) F(A) primeF(A) = I prime

F(Aor) F(primeA) (221)

3 After the preprint of the present paper appeared we noticed that this proposition is also proved in[DP Cor 5]

882 L Kong I Runkel

Proof We only prove the first equality the second one can be seen in the same way Bydefinition we have

IF(Aor) F(A)

= ((ψ F0 F(dA) φF

2 )otimes idF(A)or) (idF(Aor) otimes bF(A)) F(A)

=[(ψ F

0 F(dA) φF2 ) (F(A)otimes idF(A))

]otimes idF(A)or

(idF(A) otimes bF(A))

For the term inside the square brackets we find

ψ F0 F(dA) φF

2 (F(A)otimes idF(A)) = ψ F0 F(dA) F(A otimes idA) φF

2

= ψ F0 F(dA (A otimes idA)) φF

2 = ψ F0 F(εA m A) φF

2 (222)

On the other hand by definitionF(A) = [((ψ F0 F(εA) (F(m A)φF

2 ))otimes idF(A)or] (idF(A) otimes bF(A)) This demonstrates the first equality in (221) Proposition 216 Let F C1 rarr C2 be a tensor functor G C2 rarr C1 a functor〈FG χ〉 an adjunction A a C1-algebra and B a C2-algebra Then f A rarr G(B)is an algebra homomorphism if and only if f = χminus1( f ) F(A) rarr B is an algebrahomomorphism

Proof We need to show that

mG(B) ( f otimes f ) = f m A and f ηA = ηG(B) (223)

is equivalent to

m B ( f otimes f ) = f m F(A) and f ηF(A) = ηBs (224)

We first prove that the first identity in (223) is equivalent to the first identity in (224)For the left-hand side of the first identity in (223) we have the following equalities

mG(B) ( f otimes f )(1)= G(m B) φG

2 ( f otimes f )(2)= G(m B) G(ρ otimes ρ) G(ψ F

2 ) δ ( f otimes f )(3)= G(m B) G(ρ otimes ρ) G(ψ F

2 ) G F( f otimes f ) δ(4)= G(m B) G(ρ otimes ρ) G(F( f )otimes F( f )) G(ψ F

2 ) δ(5)= G

(m B (ρ otimes ρ) (F( f )otimes F( f )) ψ F

2

) δ

(6)= χ(

m B (ρ otimes ρ) (F( f )otimes F( f )) ψ F2

) (225)

where (1) is the definition of mG(B) in (217) (2) is the second identity in (210) (3)and (4) are naturality of δ and ψ F

2 respectively step (5) is functoriality of G and finallystep (6) is (29) For the right hand side of the first identity in (223) we get

f m A(1)= Gρ δG ( f m A)

(2)= Gρ G F( f m A) δ(3)= G(ρ F( f m A)) δ(4)= χ(ρ F( f ) F(m A)) (226)

Cardy Algebras and Sewing Constraints I 883

where (1) is the adjunction property (28) (2) is naturality of δ (3) functoriality of Gand (4) amounts to (29) and functoriality of F

On the other hand we see that the first equality in (224) is equivalent to

m B (ρ otimes ρ) (F( f )otimes F( f )) = ρ F( f ) F(m A) φF2 (227)

Using that φF2 is invertible with inverse (φF

2 )minus1 = ψ F

2 and that χ is an isomorphismit follows that the statement that (225) is equal to (226) is equivalent to the identity(227)

Now we prove that the second identity in (223) is equivalent to the second identity in(224) Using (217) and (210) we can write ηG(B) = G(ηB)φF

0 = G(ηB)G(ψ F0 )δ1

Together with (29) this shows that the second identity in (223) is equivalent to

f ηA = χ(ηB ψ F0 ) (228)

On the other hand the second identity in (224) is equivalent to

ρ F( f ) F(ηA) φF0 = ηB (229)

which by φF0 = (ψ F

0 )minus1 and (29) is further equivalent to (228)

Definition 217 Let (Am A ηAA εA) and (Bm B ηBB εB) be two Frobeniusalgebras in a tensor category C For f A rarr B we define f lowast B rarr A by

f lowast = ((εB m B)otimes idA) (idB otimes f otimes idA) (idB otimes (A ηA)) (230)

The following lemma is immediate from the definition of (middot)lowast and the properties ofFrobenius algebras We omit the proof

Lemma 218 Let C be a tensor category let A BC be Frobenius algebras in C andlet f A rarr B and g B rarr C be morphisms

(i) (g f )lowast = f lowast glowast(ii) f is a monomorphism iff f lowast is an epimorphism

(iii) f is an algebra map iff f lowast is a coalgebra map(iv) If f is a homomorphism of Frobenius algebras then f lowast f = idA and f f lowast =

idB(v) If C is sovereign and if A and B are symmetric then f lowastlowast = f

Let C and D be tensor categories and let F C rarr D be a Frobenius functor GivenFrobenius algebras A B in C and a morphism f A rarr B the next lemma shows how(middot)lowast behaves under F

Lemma 219 F( f lowast) = F( f )lowast

Proof The definition of the structure morphisms of the Frobenius algebra F(A) is givenin (217) and (218) Substituting these definitions gives

F( f )lowast =[(ψ F

0 F(εB) F(m B) φF2

)otimes idF(A)

] [

idF(B) otimes F( f )otimes idF(A)]

[idF(B) otimes

(ψ F

2 F(A) F(ηA) φF0

)]

= (ψ F0 otimes idF(A)) [

F (εB m B (idB otimes f ))otimes idF(A)]

(φF2 otimes idF(A)) (idF(B) otimes ψ F

2 )

[idF(B) otimes F(A ηA)

] (idF(B) otimes φF0 ) (231)

884 L Kong I Runkel

In the middle line of the last expression we can use the defining property (25) of F namely we substitute (φF

2 otimes idF(A)) (idF(B) otimes ψ F2 ) = ψ F

2 φF2 Then ψ F

2 can bemoved to the left and φF

2 to the right until they can be omitted against ψ F0 and φF

0 respectively using (22) and (24) This results in

F( f )lowast = F ((εB m B (idB otimes f ))otimes idA) F (idB otimes (A ηA)) (232)

which is nothing but F( f lowast)

23 Modular tensor categories Let C be a modular tensor category [TBK] ie anabelian semi-simple finite C-linear ribbon category with simple tensor unit 1 and a non-degeneracy condition on the braiding (to be stated in a moment) We denote the set ofequivalence classes of simple objects in C by I elements in I by i j k isin I and theirrepresentatives by Ui U j Uk We also set U0 = 1 and for an index k isin I we define kby Uk

sim= Uork

Since the tensor unit is simple we shall for modular tensor categories identifyHom(1 1) sim= C (cf footnote 2) Define numbers si j isin C by4

si j = UiU j (233)

They obey si j = s ji and s0i = dim Ui see eg [BK Sect 31] (In a ribbon cate-gory the left and right dimension (214) of Ui coincide and are denoted by dim Ui )The non-degeneracy condition on the braiding of a modular tensor category is that the|I|times|I|-matrix s should be invertible In fact [BK Thm 317]

sumkisinI

sik sk j = Dim C δij (234)

where Dim C = sumiisinI(dim Ui )

2 One can show (even in the weaker context of fusioncategories over C) that Dim C ge 1 [ENO1 Thm 23] In particular Dim C = 0 We fixonce and for all a square root

radicDim C of Dim C

Let us fix a basis λα(i j)kN k

i jα=1 in HomC(Ui otimesU j Uk) and the dual basis ϒ(i j)k

α N ki j

α=1

in HomC(UkUi otimesU j ) The duality of the bases means that λα(i j)k ϒ(i j)kβ = δαβ idUk

We also fix λ(0i)i = λ(i0)i = idUi We denote the basis vectors graphically as follows

λα(i j)k = α

Uk

Ui U j

ϒ(i j)kα = α

Uk

Ui U j

(235)

4 In the graphical notation used below we have given an orientation to the ribbons indicated by the arrowsFor example it is understood that this orientation determines which of the duality morphisms in (213) to use

Cardy Algebras and Sewing Constraints I 885

For V isin C we also choose a basis b(iα)V of HomC(VUi ) and the dual basis bV(iβ)

of HomC(Ui V ) for i isin I such that b(iα)V bV(iβ) = δαβ idUi We use the graphical

notation

b(iα)V = α

Ui

V

bV(iα) = α

Ui

V

(236)

Given two modular tensor categories C and D by C D we mean the tensor productof additive categories over C [BK Def 1115] ie the category whose objects are directsums of pairs V times W of objects V isin C and W isin D and whose morphism spaces are

HomCD(V times W V prime times W prime) = HomC(V V prime)otimesC HomC(WW prime) (237)

for pairs and direct sums of these if the objects are direct sums of pairsIf we replace the braiding and the twist in C by the antibraiding cminus1 and the antitwist

θminus1 respectively we obtain another ribbon category structure on C In order to distin-guish these two distinct structures we denote (C c θ) and (C cminus1 θminus1) by C+ and Cminusrespectively As in the Introduction we will abbreviate

C2plusmn = C+ Cminus (238)

Note that a set of representatives of the simple objects in C2plusmn is given by Ui times U j fori j isin I

For the remainder of Sect 2 we fix a modular tensor category C

24 The functors T and R The tensor product bifunctor otimes can be naturally extendedto a functor T C2plusmn rarr C Namely T (oplusN

i=1Vi times Wi ) = oplusNi=1Vi otimes Wi for all Vi Wi isin C

and N isin N The functor T becomes a tensor functor as follows For φT0 1 rarr T (1 times 1)

take φT0 = id1 (or lminus1

1 in the non-strict case) Next notice that for U VW X isin C

T (U times V )otimes T (W times X)= (U otimes V )otimes (W otimes X)

T ((U times V )otimes (W times X))= (U otimes W )otimes (V otimes X)(239)

We define φT2 T (U times V )otimes T (W times X) rarr T ((U times V )otimes (W times X)) by

φT2 = idU otimes cminus1

W V otimes idX (240)

(In the non-strict case the appropriate associators have to be added) The above definitionof φT

2 can be naturally extended to a morphism φT2 T (M1 otimes M2) rarr T (M1)otimes T (M2)

for any pair of objects M1M2 in C2plusmn The following result can be checked by directcalculation [JS Prop 52]

Lemma 220 The triple (T φT2 φ

T0 ) gives a tensor functor

886 L Kong I Runkel

In particular (T φT2 φ

T0 ψ

T2 ψ

T0 ) where ψT

2 = (φT2 )

minus1 and ψT0 = (φT

0 )minus1 gives

a Frobenius functorDefine the functor R C rarr C2plusmn as follows for A isin C and f isin HomC(A B)

R(A) =oplusiisinI

(A otimes Uori )times Ui R( f ) =

oplusiisinI

( f otimes idUori)times idUi (241)

This functor was also considered in a slightly different context in [ENO2 Prop 23]The family of isomorphisms γ R

A = oplusiisinI DimCdim Ui

id(AotimesUori )timesUi

isin Aut(R(A)) defines a

natural isomorphism γ R R rarr ROur next aim is to show that R is left and right adjoint to T in other words R

and T form an ambidextrous adjunction (see eg [Ld] for a discussion of ambidextrousadjunctions) To this end we introduce two linear isomorphisms for A isin C and M isin C2plusmn

χ HomC(T (M) A) minusrarr HomC2plusmn(M R(A))

χ HomC(A T (M)) minusrarr HomC2plusmn(R(A)M)(242)

If we decompose M as M = oplusNn=1 Ml

n times Mrn then χ and χ are given by

χ Noplus

n=1

fn

Mln Mr

n

A

rarrNoplus

n=1

oplusiisinI

sumα

fn

Mln

Mrn

A

α

Uori

times α

Mrn

Ui

(243)

and

χ Noplus

n=1 gn

Mln Mr

n

A

rarrNoplus

n=1

oplusiisinI

sumα gn

Mln

Mrn

A

α

Uori

times α

Ui

Mrn

DimCdim Ui

(244)

Notice that χ and χ are independent of the choice of basis

Theorem 221 〈T R χ〉 and 〈R T χminus1〉 are adjunctions ie R is both left and rightadjoint of T

Proof Write M as M = oplusNn=1 Ml

n times Mrn The isomorphism χ amounts to the following

composition of natural isomorphisms

HomC(T (M) A) = oplusnHomC(Mln otimes Mr

n A)sim= oplusni HomC(Ml

n otimes Ui A)otimes HomC(Mrn Ui )

sim= oplusni HomC(Mln A otimes Uor

i )otimes HomC(Mrn Ui ) = HomC2plusmn(M R(A)) (245)

Cardy Algebras and Sewing Constraints I 887

Thus χ is natural Let (γ RA )

lowast HomC2plusmn(R(A)M) rarr HomC2plusmn(R(A)M) denote the

pull-back of γ RA The isomorphism χ is equal to the composition of (γ R

A )lowast and the

following sequence of natural isomorphisms

HomC(A T (M)) = oplusn HomC(AMln otimes Mr

n)

sim= oplusni HomC(AMln otimes Ui )otimes HomC(Ui Mr

n)

sim= oplusni HomC(A otimes Uori Ml

n)otimes HomC(Ui Mrn) = HomC2plusmn(R(A)M) (246)

We have proved that both χ and χ are natural isomorphisms There are four natural transformations associated to χ and χ namely

idC2plusmnδminusrarr RT

ρminusrarr idC2plusmn and idCδminusrarr T R

ρminusrarr idC (247)

defined by for A isin C M isin C2plusmn

δM = χ(idT (M)) ρA = χminus1(idR(A))

ρM = χ (idT (M)) δA = χminus1(idR(A))(248)

They can be expressed graphically as follows with M = oplusNn=1 Ml

n times Mrn

δM =oplusni

sumα

Mln

Mln Mr

n Uori

αtimes

Ui

Mrn

α

ρA =oplusiisinI

A

A

Uori Ui

(249)

ρM =oplusni

sumα

Mln

Mln

Mrn Uor

i

α times

Ui

Mrn

α DimCdim Ui

δA =oplusiisinI

A

A Uori Ui

dim UiDimC

Note that

ρM δM = Dim C middot idM and ρA δA = idA (250)

Lemma 222 The functors T and R as maps on the sets of morphisms have left inversesand thus are injective

Proof Let f A rarr B be a morphism in C We define a map Q R HomC2plusmn(R(A)

R(B)) rarr HomC(A B) by f prime rarr ρB T ( f prime) δA Then we have

Q R R( f ) = ρB T R( f ) δA = ρB δB f = f (251)

888 L Kong I Runkel

where we used naturality of δ and (250) in the second and third equalities respectivelySo Q R is a left inverse of R on morphisms Thus R is injective on morphisms Similarlylet g M rarr N be a morphism in C2plusmn We define a map QT HomC(T (M) T (N )) rarrHomC2plusmn(M N ) by gprime rarr (Dim C)minus1 middot ρN R(gprime) δM Then we have

QT T (g) = (Dim C)minus1 middot ρN RT (g) δM = (Dim C)minus1 middot ρN δN g = g (252)

So QT is a left inverse of T on morphisms Thus T is injective on morphisms Using (29) and (250) one can express the two inverse maps χminus1 χminus1 as follows

for f isin HomC2plusmn(M R(A)) and g isin HomC2plusmn(R(A)M)

χminus1( f ) = ρ T ( f ) χminus1(g) = T (g) δ (253)

By Proposition 25 and Lemma 27 R is both a lax and colax tensor functor In particularφR

0 1 times 1 rarr R(1) is given by

φR0 = χ (ψT

0 ) = R(ψT0 ) δ1 times 1 = id1times1 (254)

and φR2 R(A)otimes R(B) rarr R(A otimes B) by φR

2 = R(ρA otimes ρB) R(ψT2 ) δ which can

be expressed graphically as

φR2 =

oplusi jkisinI

sumα

A

A

Uori B

B

Uorj

Uork

α

times

Ui U j

α

Uk

(255)

Similarly ψ R0 R(1) rarr 1 times 1 is given by

ψ R0 = ρ1 R(φT

0 ) = Dim C id1times1 (256)

and ψ R2 R(A otimes B) rarr R(A) otimes R(B) by ψ R

2 = ρ R(φT2 ) R(δA otimes δB) which in

graphical notation reads

ψ R2 =

oplusi jkisinI

sumα

A

A Uori

B

B Uorj

Uork

α

times

Ui U j

α

Uk

dim Ui dim U j

dim Uk Dim C

(257)

If C has more than one simple object then R does not take the tensor unit of C to thetensor unit of C2plusmn and so is clearly not a tensor functor However we will show that R isstill a Frobenius functor This will imply that if A is a Frobenius algebra in C then

R(A) = (R(A)m R(A) ηR(A)R(A) εR(A)) (258)

Cardy Algebras and Sewing Constraints I 889

is a Frobenius algebra in C2plusmn where the structure morphisms were given in (217) and(218) In the case A = 1 it was proved in [Muuml1 Prop 41] (see also [Fr Lem 619]and [K1 Thm 52]) that (258) is a commutative simple symmetric normalised-specialFrobenius algebra in C2plusmn In fact given a Frobenius algebra A in C it is straightforwardto verify that the structure morphisms in (258) are precisely those of (A times 1)otimes R(1)cf Sect 22

Proposition 223 (R φR2 φ

R0 ψ

R2 ψ

R0 ) is a Frobenius functor

Proof Using the explicit graphical expression of φR2 φ

R0 ψ

R2 ψ

R0 it is easy to see that

the commutativity of the diagrams (25) and (26) are equivalent to the statement thatR(1) with structure morphisms as in (258) is a Frobenius algebra in C2plusmn The latterstatement is true by [Muuml1 Prop 41]

From Lemma 220 and Proposition 223 we see that T and R take Frobenius alge-bras to Frobenius algebras The following two propositions show how the properties ofFrobenius algebras are transported

Proposition 224 Let A be a Frobenius algebra in C2plusmn Then T (A) is a Frobenius algebrain C and

(i) A is symmetric iff T (A) is symmetric(ii) A is (normalised-)special iff T (A) is (normalised-)special

Proof For part (i) write A as a direct sum oplusNn=1 Al

n times Arn Then the maps IT (A) I prime

T (A) T (Aor) rarr T (A)or defined in (220) are given by

IT (A) = I primeT (A) = oplusN

n=1c(Aln)

or(Arn)

or (259)

Therefore by (221) T (A) = primeT (A) is equivalent to T (A) = T (prime

A) Since byLemma 222 T is injective on morphisms this proves part (i) Part (ii) can be checkedin the same way for example the condition mT (A) T (A) = ζ idT (A) is easily checkedto be equivalent to T (m A A) = ζ T (idA) Proposition 225 Let A be a Frobenius algebra A in C Then R(A) is a Frobeniusalgebra in C2plusmn and

(i) A is symmetric iff R(A) is symmetric(ii) A is (normalised-)special iff R(A) is (normalised-)special

Proof Recall that the structure morphisms of the Frobenius algebra R(A) are equal tothose of (A times 1) otimes R(1) Using this equality part (i) and (ii) follow because R(1) issymmetric and normalised-special For example

m R(A) R(A) = [(m A A)times id1] otimes idR(1) = R(m A A) (260)

so that m R(A) R(A) = ζ idR(A) is equivalent to R(m A A) = ζ R(idA) which byLemma 222 is equivalent to m A A = ζ idA

The functor R has one additional property not shared by T namely R takes absolutelysimple algebras to absolutely simple algebras We will see explicitly in Sect 33 thatthis is not true for T

890 L Kong I Runkel

Lemma 226 For a C-algebra A the map

R HomA|A(A A) rarr HomR(A)|R(A)(R(A) R(A)) (261)

given by f rarr R( f ) is well-defined and an isomorphism

Proof Since R is a lax tensor functor R(A) is naturally a R(A)-bimodule It is easy tosee that R in (261) is a well-defined map R(A) is also naturally a R(1)-bimodule whichcan be identified with the induced R(1)-bimodule structure on (A times 1)otimes R(1) wherethe left R(1) action on (Atimes1)otimes R(1) is given by (idAtimes1 otimesm R(1))(cminus1

Atimes1R(1)otimes idR(1))We have the following natural isomorphisms

HomR(1)|R(1)(R(A) R(A))sim=minusrarr HomC2plusmn(A times 1 R(A))

χminus1

minusminusrarr HomC(A A) (262)

which by (253) are given by for f isin HomR(1)|R(1)(R(A) R(A))

f rarr f prime = f (idAtimes1 otimes ηR(1)) rarr f primeprime = ρ T ( f prime) (263)

and its inverse is given by for g isin HomC(A A)

g rarr gprime = R(g) δ rarr gprimeprime = (idAtimes1 otimes m R(1)) (gprime otimes idR(1)) (264)

where gprimeprime is indeed a R(1)-bimodule map due to the commutativity of R(1) It is easy tocheck that gprimeprime = R(g) in (264) Therefore R gives an isomorphism from HomC(A A)to HomR(1)|R(1)(R(A) R(A)) Moreover one verifies that R(g) is an R(A)-bimodulemap iff g is an A-bimodule map In other words R g rarr R(g) gives an isomorphism

HomA|A(A A)sim=minusrarr HomR(A)|R(A)(R(A) R(A))

Corollary 227 Let A be a C-algebra

(i) A is absolutely simple iff R(A) is absolutely simple(ii) Let A be in addition Frobenius Then A is simple and special iff R(A) is simple and

special

Proof Part (i) immediately follows from Lemma 226 The statement of part (ii) withoutthe qualifier lsquosimplersquo is proved in Proposition 225 But as in the proof of (iii)rArr(i) inLemma 211 a special Frobenius algebra in a semi-simple category has a semi-simplecategory of bimodules and for a semi-simple C-linear category simple and absolutelysimple are equivalent Part (ii) then follows from part (i)

The following lemma will be needed in Sect 32 below to discuss the properties ofCardy algebras

Lemma 228 Let A be a Frobenius algebra in C Then (δA)lowast = ρA

Proof Recall from (250) that δA is a morphism A rarr T R(A) Since T and R are bothFrobenius functors T R(A) is a Frobenius algebra in C Substituting the definitions aftera short calculation one finds

Cardy Algebras and Sewing Constraints I 891

εT R(A) mT R(A) = Dim(C)oplusiisinI

A Uori Ui A U or

ı Uı

(265)

Substituting this in the definition of (δA)lowast gives again after a short calculation the mor-

phism ρA At an intermediate step one uses that the part of the morphism (δA)lowast which

is made up of Ui and Uı ribbons their duals and the basis morphisms λ(iı)0 andϒ(iı)0can be replaced by 1

dim Uimiddot dUi

3 Cardy Algebras

In this section we start by investigating the properties of Frobenius algebras which sat-isfy the so-called modular invariance condition We then give two definitions of a Cardyalgebra and prove their equivalence Finally in Sect 33 we study the properties of thesealgebras and state our main results

We fix a modular tensor category C Recall that C2plusmn is an abbreviation for C+ Cminus

31 Modular invariance In C2plusmn we define the object K and the morphism ω K rarr Kas

K =oplus

i jisinIUi times U j ω =

sumi jisinI

dim Ui dim U j

Dim C idUi timesU j (31)

They have the property (see eg [BK Cor 3111])

ω

K

(Uk times Ul )or

(Uk times Ul )or

Ui times U j

Ui times U j

= δik δ jlDim C

dim Ui dim U jbUi timesU j dUi timesU j (32)

892 L Kong I Runkel

Definition 31

(i) Let A B be objects of C2plusmn A morphism f A otimes B rarr B is called S-invariant iff

A

B

W

W

f =

A

B

W

W

f

K

ω(33)

holds for all for W isin C2plusmn(ii) A C2plusmn-algebra (Aclmcl ηcl) is called modular invariant iff θAcl = idAcl and mcl

is S-invariant

Lemma 32 The morphism f A otimes B rarr B is S-invariant if and only if

A

B

Ui times U j

Ui times U j

f = Dim Cdim Ui dim U j

sumα

A

B

B

f

Ui times U j

Ui times U j

α

α

(34)

holds for all i j isin I

Proof Condition (33) holds for all W iff it holds for all W = Ui times U j i j isin I so itis enough to show that the right-hand side of (33) with W = Ui times U j is equal to theright-hand side of (34) Recall the notation for basis morphisms in (236) Starting from(33) write

idBor otimes idUi timesU j =⎛⎝sum

klα

(b(ktimeslα)

B

)or (

bB(ktimeslα)

)or⎞⎠ otimes idUi timesU j (35)

and then apply (32) The graphical representation of the resulting morphism can bedeformed to give (34) Remark 33 As shown in [K3 Sect 61] the modular invariance condition of a C2plusmn-alge-bra exactly coincides with the modular invariance condition for torus 1-point correlationfunctions of a genus-01 closed CFT In particular the condition θAcl = idAcl is equiv-alent to invariance under the modular transformation T τ rarr τ + 1 and the condition(34) with f = mcl is equivalent to invariance under S τ rarr minus 1

τ Combining the

modular invariance condition with the genus zero properties of a genus-01 closed CFTresults in a modular invariant commutative symmetric Frobenius algebra in C2plusmn

Cardy Algebras and Sewing Constraints I 893

Let Acl be a modular invariant C2plusmn-algebra Evaluating (34) for f = mcl composingit with ηcl otimes idUi timesU j and taking the trace implies the following identity

Zi j = 1

DimCAclUi timesU j where

(36)

Zi j = dimC HomC2plusmn(UitimesU j Acl)

Decomposing Acl into simple objects this gives

Zi j =sum

klisinISik Zkl Sminus1

l j where Si j = si jradic

DimC (37)

which in CFT terms is of course nothing but the invariance of the torus partition functionunder the modular S-transformation

The following theorem gives a simple criterion for modular invariance

Theorem 34 Let Acl be a haploid commutative symmetric Frobenius algebra in C2plusmn

(i) If Acl is modular invariant then dim Acl = Dim C(ii) If dim Acl = Dim C then Acl is special and modular invariant

Proof Part (i) Since Acl is haploid for i = j = 0 Equation (36) reduces to 1 =dim AclDimCPart (ii) By the same reasoning as in the proof of (iii)rArr(i) in Lemma 211 one showsthat Acl is special Thus mcl cl = ζclidAcl for some ζcl = 0

By [KO Thm 45] the category (C2plusmn)locAcl

of local Acl-modules is again a mod-

ular tensor category and Dim (C2plusmn)locAcl

= (Dim C dim Acl)2 (see [Fr Prop 321 amp

Rem 323] for the same statement in the notation used here) Thus by assumption wehave Dim(C2plusmn)loc

Acl= 1 It then follows from [ENO1 Thm 23] that up to isomorphism

(C2plusmn)locAcl

has a unique simple object (namely the tensor unit) In other words every simplelocal Acl-module is isomorphic to Acl (seen as a left-module over itself)

We have the following isomorphisms between morphism spaces [FS Prop 47 amp 411]

HomAcl(Acl otimes (Ui times U j ) Acl) sim= HomC2plusmn(Ui times U j Acl)

HomAcl(Acl Acl otimes (Ui times U j )) sim= HomC2plusmn(AclUi times U j )

Using these to transport the bases (236) from the right to the left we obtain bases bα(i j)αof HomAcl(Acl otimes (Ui times U j ) Acl) and b(i j)

β β of HomAcl(Acl Acl otimes (Ui times U j )) Thesecan be expressed graphically as

894 L Kong I Runkel

bα(i j) =

Acl

Acl

Acl

Ui times U j

α

b(i j)β = DimC

dim Ui dim U j

1

ζcl

Acl

Ui times U jAcl

Acl

β

(38)

where the nonzero factor in b(i j)β is included for convenience Notice that bα(i j) b(i j)

β isa left Acl-module map Since Acl is simple as a left module over itself we have

bα(i j) b(i j)β = λαβ idAcl (39)

for some λαβ isin C By computing tr(bα(i j) b(i j)β ) it is easy to verify that λαβ = δαβ

We will now prove the following identity

1

ζcl

Acl

Acl

Ui times U j

Ui times U j

Acl

=sumα

DimCdim Ui dim U j

1

ζcl

Acl

Ui times U j

Ui times U j

Acl

Acl

Acl

α

α

(310)

One checks that the left-hand side of this equation is an idempotent which we denote byζminus1

cl PlAcl(Ui timesU j ) cf [Fr Sect 31] By [Fr Prop 41] the image Im(ζminus1

cl PlAcl(Ui timesU j ))

is a local Acl-module and hence isomorphic to AoplusNcl for some N isin Zge0

All left-module morphisms from Acl to Aclotimes(Ui timesU j ) are linear combinations of the

b(i j)β Furthermore one verifies that ζminus1

cl PlAcl(Ui timesU j )b(i j)

β = b(i j)β Therefore the b(i j)

β

describe precisely the image of the idempotent ie ζminus1cl Pl

Acl(Ui timesU j ) = sum

α b(i j)α bα(i j)

which is nothing but (310) Composing (310) with ζcl middot εcl otimes idUi timesU j from the left (iefrom the top) produces (34)

In addition since Acl is commutative symmetric Frobenius it satisfies θAcl = idAcl

[Fr Prop 225] Altogether this shows that Acl is modular invariant Remark 35 As we were writing this paper we heard that the results in Theorem 34were obtained independently by Kitaev and Muumlger [Ki]

Remark 36 Setting i = j = 0 in (36) gives the identity dim Acl = Z00 Dim C [KRProp 23] Combining this with Theorem 34 (ii) one may wonder if a general modularinvariant commutative symmetric Frobenius algebra Acl in C2plusmn is isomorphic to a directsum of simple such algebras However this is not so For example one can take the com-mutative symmetric Frobenius algebra Acl = C[x]x2 in the category of vector spacesequipped with the non-degenerate trace ε(ax + b) = a In this case the modular invari-ance condition holds automatically but Acl is clearly not a direct sum of two algebras

Cardy Algebras and Sewing Constraints I 895

For a general modular tensor category C the algebra C[x]x2 R(1) understood as analgebra in C2plusmn via the braided monoidal isomorphism Vect f (C) C2plusmn rarr C2plusmn providesanother counter-example

32 Two definitions Define a morphism PlA A rarr A for a Frobenius algebra A in C

or C2plusmn as follows [Fr Sect 24]

PlA =

A

A

A

m

(311)

If A is also commutative and obeys m A A = ζA idA we have PlA = ζA idA In partic-

ular this holds if A is commutative and special Using the fact that the Frobenius algebraR(1) is commutative and normalised-special one can check that Pl

R(A) R(A) rarr R(A)takes the following form

PlR(A) =

oplusiisinI

A

A

Uori

Uori

A

m

times

Ui

Ui

(312)

With these ingredients we can now give the first definition of a Cardy C|C2plusmn-algebrawhich was introduced in [K3 Def 514] cf Remark 315 below

Definition 37 (CardyC|C2plusmn-algebra I) A CardyC|C2plusmn-algebra is a triple (Aop|Acl ιcl-op)where (Aclmcl ηclcl εcl) is a modular invariant commutative symmetric Frobe-nius C2plusmn-algebra (Aopmop ηopop εop) is a symmetric Frobenius C-algebra andιcl-op Acl rarr R(Aop) an algebra homomorphism such that the following conditionsare satisfied

(i) Centre condition

Acl R(Aop)

R(Aop)

R(Aop)

ιcl-op

m R(Aop)

=

Acl

ιcl-op

R(Aop)

R(Aop)

R(Aop)

m R(Aop)

(313)

896 L Kong I Runkel

(ii) Cardy condition

ιcl-op ιlowastcl-op =

R(Aop)

R(Aop)

R(Aop)

(314)

Remark 38

(i) The name ldquoCardy C|C2plusmn-algebrardquo in Definition 37 was chosen because many ofthe important ingredients were first studied by Cardy the modular invariance ofthe closed theory [C1] the consistency of the annulus amplitude [C2] and thebulk-boundary OPE [CL] On the other hand the boundary-boundary OPE andthe OPE analogue of the centre condition were first considered in [Lw]

(ii) One can easily see that in the special case that C is the category Vect f (C) of finite-dimensional C-vector spaces a Cardy C|C2plusmn-algebra gives exactly the algebraicformulation of two-dimensional open-closed topological field theory over C (cfRemark 614 in [K3]) see [Lz Sect 48] [Mo Thm 11] [AN Thm 45] [LPCor 43] [MS Sect 22] When passing to a general modular tensor category Cthere are two important differences to the two-dimensional topological field the-ory Firstly the algebras Acl and Aop now live in different categories which inparticular affects the formulation of the centre condition and the Cardy conditionSecondly the modular invariance condition has to be imposed on Acl In the caseC = Vect f (C) modular invariance holds automatically

Definition 39 A homomorphism of Cardy C|C2plusmn-algebras (A(1)op |A(1)cl ι(1)cl-op) rarr (A(2)op

|A(2)cl ι(2)cl-op) is a pair ( fop fcl) of Frobenius algebra homomorphisms fop A(1)op rarr A(2)op

and fcl A(1)cl rarr A(2)cl such that the diagram

A(1)cl

fcl

ι(1)cl-op

A(2)cl

ι(2)cl-op

R(A(1)op )

R( fop) R(A(2)op )

(315)

commutes

Remark 310 Since a homomorphism of Frobenius algebras is invertible (cf Lemma218 (iv)) a homomorphism of Cardy algebras is always an isomorphism

For a homomorphism ( fop fcl) of Cardy C|C2plusmn-algebras using the commutativity of(315) and the fact that fcl and fop are both algebra and coalgebra homomorphisms itis easy to show that (315) commutes iff

R(A(1)op )R( fop)

(ι(1)cl-op)

lowast

R(A(2)op )

(ι(2)cl-op)

lowast

A(1)cl

fcl A(2)cl

(316)

Cardy Algebras and Sewing Constraints I 897

commutesLet (Aop|Acl ιcl-op) be a Cardy C|C2plusmn-algebra Define the morphism

ιcl-op = χminus1(ιcl-op) T (Acl) minusrarr Aop (317)

Decompose Acl as Acl = oplusNn=1Cl

n times Crn such that Cl

1 times Cr1 = ηcl(1 times 1) We use ι(n)cl-op

to denote the restriction of ιcl-op to Cln times Cr

n and ι(n)cl-op to denote the restriction of ιcl-op

to Cln otimes Cr

n We introduce the following graphical notation

ι(n)cl-op =

Cln Cr

n

Aop

(318)

By (243) ιcl-op can be expressed in terms of ιcl-op as follows

ιcl-op =Noplus

n=1

oplusiisinI

sumα

Cln

Crn

Aop

α

Uori

times α

Crn

Ui

(319)

Lemma 311 The centre condition (313) is equivalent to the following condition in C

Cln Cr

n

Aop

Aop

Aop

=

Cln Cr

n

Aop

Aop

Aop

for n = 1 N (320)

Proof First insert (319) and the definition (217) of m R(Aop) into (313) Then apply thecommutativity of R(1) to the left hand side of (313) The equivalence between (313)and (320) follows immediately Remark 312 The centre condition (320) is very natural from the open-closed confor-mal field theory point of view Correlators on the upper half plane are expressed in termsof conformal blocks on the full complex plane The objects Cl

n and Crn are associated to

the field insertion at a point z in the upper half plane and at the complex conjugate pointz in the lower half plane respectively The object Aop corresponds to a field inserted ata point r on the real axis The centre condition (320) simply says that the correlationfunctions in the disjoint domains |z| gt r gt 0 and r gt |z| gt 0 are analytic continuationsof each other see [K2 Prop 118]

898 L Kong I Runkel

Recall that we define ιlowastcl-op Aop rarr T (Acl) as in (230) We introduce the graphicalnotation

ιlowastcl-op =Noplus

n=1

Cln Cr

n

Aop

=Noplus

n=1

Cln Cr

n

Aop

Aop T (minus1Acl) (321)

where the second equality follows from (215) (221) and (259)

Lemma 313 The Cardy condition (314) is equivalent to the following identity in C

Noplusn=1

DimCdim Ui

sumα

Cln

Crn

Crn

α

α

Aop

Aop Uori

Uori

=Aop

Aop

Aop

Uori

Uori

op

mop for all i isin I (322)

Proof By (244) (312) and (319) it is easy to see that (322) is equivalent to thefollowing identity

ιcl-op χ(ιlowastcl-op) = PlR(Aop)

(323)

Therefore it is enough to show that

χ(ιlowastcl-op) = ιlowastcl-op (324)

We have

χminus1(ιlowastcl-op)(1)= T (ιlowastcl-op) δ (2)= T (ιcl-op)

lowast δlowastlowast (3)= (ρ T (ιcl-op)

)lowast (4)= (ιcl-op

)lowast (325)

In step (1) we use the expression (253) for χminus1 step (2) follows from Lemma 218 (v)and Lemma 219 Step (3) is Lemma 218 (i) and Lemma 228 and finally step (4)amounts to substituting (253) and (317) Acting with χ on both sides of the aboveequality produces (324)

Combining Lemmas 311 and 313 and Proposition 216 we obtain the followingequivalent definition of Cardy C|C2plusmn-algebra (recall the graphical notation (318) forιcl-op and (321) for ιlowastcl-op)

Cardy Algebras and Sewing Constraints I 899

Definition 314 (CardyC|C2plusmn-algebra II) A CardyC-algebra is a triple (Aop|Acl ιcl-op)where Acl is a commutative symmetric Frobenius C2plusmn-algebra satisfying property (34)with f = mcl Aop is a symmetric Frobenius C-algebra and ιcl-op T (Acl) rarr Aop isan algebra homomorphism satisfying the conditions (320) and (322)

Remark 315 Up to a choice of normalisation Definition 314 is the same as the orig-inal one in [K3 Def 613] The difference between the two definitions is the factorDim C dim Ui on the left hand side of (322) which in [K3 Def 613] is given byradic

Dim C dim Ui The two definitions are related by rescaling the coproduct cl andcounit εcl of Acl by 1

radicDim C and

radicDim C respectively We chose the convention in

(322) to remove all dimension factors from the expression (314) for the Cardy condition

33 Uniqueness and existence theorems In this subsection we investigate the structureof Cardy algebras We start with the following proposition which when combined withthe results of part II provides an alternative proof of [Fj Prop 422]

Proposition 316 Let (Aop|Acl ιcl-op) be a Cardy C|C2plusmn-algebra If Acl is simple anddim Aop = 0 then Aop is simple and special

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis therefore haploid Restricting the Cardy condition (322) to the case Ui = 1 andcomposing both sides with εop from the left we see that εop kills all terms associated toU j times 1 isin Acl in the sum except for a single 1 times 1 term Thus we obtain the followingidentity

β εop = dAop (mop otimes idAorop) (idAop otimes bAop) (326)

where β isin C Composing with ηop from the right in turn implies that βεop ηop =dim Aop which is nonzero by assumption Thus also β = 0 and εop is a nonzero multipleof the morphism on the right hand side of (326) By [FRS Lem 311] Aop is special

Since Aop is a special Frobenius algebra Aop is semi-simple as an Aop-bimodule(apply [FS Prop 524] to Aop tensored with its opposite algebra) Suppose Aop is not

simple so that we can write Aop = A(1)op oplus A(2)op for nonzero Aop-bimodules A(1)op and A(2)op We denote the canonical embeddings and projections associated to this decompositionas ι12 and π12 We have the identities

mop (ι1 otimes ι2) = 0 εop ηop =2sum

i=1

εop ιi πi ηop (327)

The first identity follows since π1 mop (ι1 otimes ι2) = 0 (as mop gives the left action of

Aop on Aop and hence it preserves A(2)op ) and similarly π2 mop (ι1 otimes ι2) = 0 Thesecond identity is just the completeness of ι12 π12

Since εop ηop = 0 without losing generality we can assume εop ι1 π1 ηop = 0Using that π2 is a bimodule map we compute

π2 [LHS of (322)]Ui =1 ι1 π1 ηop = π2 [RHS of (322)]Ui =1 ι1 π1 ηop

= PlA(2)op

π2 ι1 π1 ηop = 0 (328)

900 L Kong I Runkel

On the other hand using that Acl is haploid that ιcl-op is an algebra map and thatι lowastcl-op is a coalgebra map one can check that the left-hand side of (328) is equal toλ(εop ι1 π1 ηop) π2 ηop for some λ = 0 This implies that π2 ηop = 0 Thusηop = sum

i ιi πi ηop = ι1 π1 ηop Hence we have

0 = π2 ι2 = π2 mop (ηop otimes ι2) = π2 mop ((ι1 π1 ηop)otimes ι2) (329)

However the right-hand side is zero by (327) This is a contradiction and hence Aopmust be simple

To formulate the next theorem we need the notion of the full centre of an algebra [FjDef 49] Recall that an algebra A in a braided tensor category has a left centre and aright centre [VZO] both of which are sub-algebras of A Of these two we will onlyneed the left centre The following definition is [Fr Def 231] which in our setting isequivalent to that of [VZO]

Definition 317 Let A be a symmetric special Frobenius algebra such that m A A =ζA idA

(i) The left centre Cl(A) of A is the image of the idempotent ζminus1A Pl

A(ii) The full centre Z(A) is Cl(R(A))

That ζminus1A Pl

A is an idempotent follows from [FRS Lem 52] when keeping track ofthe factors ζA ([FRS] assumes normalised-special ie ζA = 1) Note that Cl(A) is againan object of C while Z(A) is an object of C2plusmn Let el

A Cl(A) rarr A be the embeddingof Cl(A) into A The left centre is in fact the maximal subobject of A such that

m A cAA (elA otimes idA) = m A (330)

see [Fr Lem 232] This observation explains the name left centre and also makes theconnection to [O Def 15]

The full centre is by definition the image of the idempotent ζminus1A Pl

R(A) R(A) rarrR(A) Since C2plusmn is abelian the idempotent splits and we obtain the embedding andrestriction morphisms

e Z(A) rarr R(A) and r R(A) Z(A) (331)

which obey r e = idZ(A) and e r = ζminus1A Pl

R(A) It follows from Proposition 225 and

[Fr Prop 237] that Z(A) is a commutative symmetric Frobenius algebra in C2plusmn withstructure morphisms5

m Z(A) = r m R(A) (e otimes e) ηZ(A) = r ηR(A)

Z(A) = ζA middot (r otimes r) R(A) e εZ(A) = ζminus1A middot εR(A) e

(332)

Moreover if A is simple then Z(A) is simple and if A is simple and dim A = 0 thenZ(A) is simple and special The normalisation of the counit is such that

εZ(A) ηZ(A) = ζminus2A dim A DimC (333)

5 The normalisation of product and unit is the standard one The factors in the coproduct and counit haveto be included in order for (A|Z(A) e) to be a Cardy algebra see Theorem 318 below The normalisation ofthe counit enters the Cardy condition (314) through the definition of ( middot )lowast

Cardy Algebras and Sewing Constraints I 901

Theorem 318 Let A be a special symmetric Frobenius C-algebra Then (A|Z(A) e)is a Cardy C|C2plusmn-algebra

The proof of this theorem makes use the following two lemmas

Lemma 319 e Z(A) rarr R(A) is an algebra map and elowast = ζA middot r

Proof It follows from [Fr Lem 229] (or by direct calculation using in particularm R(A) R(A) = ζAidR(A)) that

m R(A) (e otimes e) = ζminus1A middot Pl

R(A) m R(A) (e otimes e) (334)

Substituting e r = ζminus1A Pl

R(A) shows that e is compatible with multiplication For theunit one finds

e ηZ(A) = e r ηR(A) = ζminus1A Pl

R(A) ηR(A) = ηR(A) (335)

Thus e is an algebra map For the second statement one computes

elowast (1)= ζA

R(A)

r

R(A)

R(A)

Z(A)

R(A)

Z(A)

r

e

e

(2)=

R(A)

r

R(A)

R(A)

R(A)

Z(A)

(3)=

R(A)

r

R(A)

Z(A)

(4)= ζA middot r (336)

where in (1) the definitions (230) and (332) have been substituted step (2) is e r = ζminus1

A PlR(A) step (3) uses that R(A) is symmetric Frobenius and step (4) is again

e r = ζminus1A Pl

R(A) Lemma 320 Let A be a symmetric Frobenius algebra in C The morphism

PlR(A) m R(A)

(Pl

R(A) otimes PlR(A)

) R(A)otimes R(A) minusrarr R(A) (337)

is S-invariant

The proof of this lemma is a slightly lengthy explicit calculation and has been deferredto Appendix A2

902 L Kong I Runkel

Proof of Theorem 318 That e is an algebra map was proved in Lemma 319 The centrecondition (313) holds by property (330) of the left centre The Cardy condition (314)also is an immediate consequence of Lemma 319

ιcl-op ιlowastcl-op = e (ζA r) = PlR(A) (338)

The full centre Z(A) is a commutative symmetric Frobenius algebra It remains to provemodular invariance That θZ(A) = idZ(A) is implied by commutativity and symmetryof Z(A) [Fr Prop 225] The S-invariance condition (33) follows from Lemma 320In (337) substitute Pl

R(A) = ζA e r and then put the resulting morphism into (33)Compose the resulting equation with e otimes idW from the right (ie from the bottom) andsubstitute the definition (332) of m Z(A) This results in the statement that m Z(A) isS-invariant

The following theorem is analogous to [LR Prop 29] and [Fj Thm 426] whichroughly speaking answer the question under which circumstances the restriction of atwo-dimensional conformal field theory to the boundary already determines the entireconformal field theory The first work is set in Minkowski space and uses operator alge-bras and subfactors while the second work is set in Euclidean space and uses modulartensor categories

Theorem 321 Let (A|Acl ιcl-op) be a Cardy C|C2plusmn-algebra such that dim A = 0 andAcl is simple Then A is special and (A|Acl ιcl-op) sim= (A|Z(A) e) as Cardy algebras

Proof By Proposition 316 A is simple and special Since Acl is simple the algebramap ιcl-op Acl rarr R(A) is either zero or a monomorphism But ιcl-op ηcl = ηR(A)and so ιcl-op = 0 Thus ιcl-op is monic By Lemma 218 (ii) ζminus1

A ιlowastcl-op is epi The Cardycondition (314) implies

ιcl-op ζminus1A ιlowastcl-op = ζminus1

A PlR(A) = e r (339)

Composing this with e r from the left yields e r ιcl-op ζminus1A ιlowastcl-op = e r =

ιcl-op ζminus1A ιlowastcl-op Since ζminus1

A ιlowastcl-op is epi we have

e r ιcl-op = ιcl-op (340)

Actually (340) also follows from (313) and specialness of R(A) We will prove that

( fop fcl) (A|Acl ιcl-op) minusrarr (A|Z(A) e) where fop = idA fcl = r ιcl-op

(341)

is an isomorphism of Cardy algebrasfcl is an algebra map Compatibility with the units follows since ιcl-op is an algebramap

fcl ηcl = r ιcl-op ηcl = r ηR(A) = ηZ(A) (342)

Compatibility with the multiplication also follows since ιcl-op is an algebra map

m Z(A) ( fcl otimes fcl) = r m R(A) (e otimes e) (r otimes r) (ιcl-op otimes ιcl-op)

= r m R(A) (ιcl-op otimes ιcl-op) = r ιcl-op mcl = fcl mcl (343)

where in the second step we used (340)

Cardy Algebras and Sewing Constraints I 903

fcl is an isomorphism As above since fcl is an algebra map and since Acl is simple fcl

has to be monic By Lemma 319 rlowast = ζminus1A e Thus f lowast

cl = ιlowastcl-op rlowast = ζminus1A ιlowastcl-op e and

fcl f lowastcl = r ιcl-op ζminus1

A ιlowastcl-op e = r e r e = idZ(A) (344)

and so fcl is also epi and hence isofcl is a coalgebra map Since fcl is an algebra map so is f minus1

cl By (344) f minus1cl = f lowast

cland by Lemma 218 (iii) this implies that fcl is a also coalgebra mapThe diagram (315) commutes Commutativity of (315) is equivalent to e fcl = ιcl-opwhich holds by (340)

Let A be a special symmetric Frobenius algebra So far we have seen that (A|Z(A) e)is a Cardy algebra and that all Cardy algebras with Aop = A and simple Acl are of thisform It is now natural to ask if every simple Acl does occur as part of a Cardy algebraThe following theorem provides an affirmative answer Recall that for an A-left moduleM the object Mor otimesA M is an algebra (see eg [KR Lem 42])

Theorem 322 If Acl is a simple modular invariant commutative symmetric FrobeniusC2plusmn-algebra then there exist a simple special symmetric Frobenius C-algebra A and amorphism ιcl-op Acl rarr R(A) such that

(i) Acl sim= Z(A) as Frobenius algebras(ii) (A|Acl ιcl-op) is a Cardy C|C2plusmn-algebra

(iii) T (Acl) sim= oplusκisinJ Morκ otimesA Mκ as algebras where MκκisinJ is a set of representa-

tives of the isomorphism classes of simple A-left modules

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis haploid It then follows from Theorem 34 that Acl is special By Proposition 224T (Acl) is a special symmetric Frobenius algebra in C Thus T (Acl) = oplusi Ai where theAi are simple symmetric Frobenius algebras We will show that at least one of the Aiis special Since T (Acl) is special we have mT (Acl) T (Acl) = ζ idT (Acl) for someζ isin C

times Restricting this to the summand Ai shows mi i = ζ idAi FurthermoreεT (Acl) ηT (Acl) = ξ id1 for some ξ isin C

times But εT (Acl) ηT (Acl) = sumi εi ηi and so

at least one of the εi ηi has to be nonzero Therefore at least one of the Ai is speciallet A equiv Ai be this summand We denote the embedding A rarr T (Acl) by e0 and therestriction T (Acl) A by r0 Notice that r0 is an algebra homomorphism Define

ιcl-op = χ (r0) Acl minusrarr R(A) (345)

By Proposition 216 ιcl-op is an algebra homomorphism Next we verify the centre con-dition (313) or rather its equivalent form (320) By substituting the definitions onecan convince oneself that the commutativity mcl cAclAcl = mcl of Acl in C2plusmn implies thecondition mT (Acl) = mT (Acl) in C see [K2 Prop 36] Here T (Acl)otimes T (Acl) rarrT (Acl)otimes T (Acl) is given by

=oplusmn

(idCl

notimes cCl

m Crnotimes idCr

m

)

(cCl

m Clnotimes cminus1

Crn C

rm

)

(idCl

motimes cminus1

Cln C

rm

otimes idCrn

)

(346)

and we decomposed Acl as Acl = oplusnCln times Cr

n As a consequence we obtain the identity

r0 mT (Acl) (idAcl otimes e0) = r0 mT (Acl) (idAcl otimes e0) (347)

904 L Kong I Runkel

Using that r0 is an algebra map and that by definition ιcl-op = r0 we obtain (320)In order to show that (A|Acl ιcl-op) is a Cardy algebra it remains to show that the

Cardy condition (314) is satisfied We will demonstrate this via a detour by first provingthat Acl sim= Z(A) as Frobenius algebras

Recall the notations e and r given in (331) Using the centre condition (320) onecan check that Pl

R(A) ιcl-op = m R(A) R(A) ιcl-op By specialness of A we have

m A A = ζA idA and so together with e r = ζminus1A Pl

R(A) we get

e r ιcl-op = ιcl-op (348)

Next consider the morphism

fcl = r ιcl-op Acl minusrarr Z(A) (349)

By the same derivation as in (342) and (343) one sees that fcl is an algebra map Inparticular fcl ηcl = ηZ(A) = 0 and so fcl = 0 Since Acl is simple fcl has to be amonomorphism

By the same argument as used in the proof of Theorem 34 (ii) up to isomorphismAcl is the unique simple local Acl-(left-)module The algebra monomorphism fcl turnsZ(A) into an Acl-module Since Z(A) is commutative it is local as an Acl-module andso Z(A) sim= AoplusN

cl for some N ge 1 By construction A is a simple special symmetricFrobenius algebra Proposition 225 and Corollary 227 show that R(A) inherits all theseproperties and thus Z(A) is simple (see the comment below Eq (332)) By Theorem318 Z(A) is modular invariant and then by Theorem 34 (i) dim Z(A) = Dim C Thisimplies that N = 1 in Z(A) sim= AoplusN

cl and so fcl is in fact an isomorphismSince Acl and Z(A) are both haploid we have εZ(A) fcl = ξ εcl for some ξ isin C

timesThe counit uniquely determines the Frobenius structure on Acl and Z(A) (see eg [FRSLemma 37]) so that fcl is a coalgebra isomorphism iff ξ = 1 To compute ξ we composethe above identity with ηcl from the right Defining ζcl via εcl ηcl = ζminus1

cl Dim C middot id1 andusing (333) gives ξ = dim A ζclζ

2A By rescaling the comultiplication and the counit

of A and consequently changing ζA we can always achieve ξ = 1 This proves part (i)of the theorem

Equation (348) implies that ιcl-op = e fcl Since fcl is an isomorphism of Frobeniusalgebras by Lemmas 218 and 319 we have

ιcl-op ιlowastcl-op = e fcl f lowastcl elowast = ζA e r = Pl

R(A) (350)

Thus (A|Acl ιcl-op) is a Cardy algebra This proves part (ii) of the theoremPart (iii) can be seen as follows By [KR Prop 43] T Z(A) sim= oplusκisinJ Mor

κ otimesA Mκ asalgebras Together with the observation that T ( fcl) T (Acl) rarr T Z(A) is an isomor-phism of algebras this proves part (iii) Remark 323 Part (i) of Theorem 322 was announced by Muumlger [Muuml2] We provide anindependent proof in the setting of Cardy algebras

The above theorem together with Lemma 211 and Theorem 34 shows that a sim-ple commutative symmetric Frobenius C2plusmn-algebra Acl with dim Acl = Dim C is alwayspart of a Cardy algebra (Aop|Acl ιcl-op) for some simple special symmetric Frobeniusalgebra Aop in C However the above proof also illustrates that Aop is not unique Thisraises the question how two Cardy algebras with a given Acl can differ This question isanswered by [KR Thm 11] which in the present framework can be restated as follows

Cardy Algebras and Sewing Constraints I 905

Theorem 324 If (A(i)op |A(i)cl ι(i)cl-op) i = 1 2 are two Cardy C|C2plusmn-algebras such that

A(i)cl is simple and dim A(i)op = 0 for i = 1 2 then A(1)clsim= A(2)cl as algebras if and only

if A(1)op and A(2)op are Morita equivalent

Proof Theorem 11 in [KR] is stated for A(i)op being non-degenerate algebras and A(i)cl =Z(A(i)op ) for i = 1 2 By Proposition 316 A(i)op are simple and special for i = 1 2

Then by [KR Lem 21] A(i)op are non-degenerate algebras By Theorem 321 we have

A(i)clsim= Z(A(i)op ) as Frobenius algebras Finally by [KR Thm 11] Z(A(1)op ) sim= Z(A(2)op )

as algebras iff A(1)op and A(2)op are Morita equivalent Let Cmax(C2plusmn) be the set of equivalence classes [B] of simple modular invariant

commutative symmetric Frobenius algebras B in C2plusmn Two such algebras B and B prime areequivalent if B and B prime are isomorphic as algebras (but not necessarily as Frobenius alge-bras) Let Msimp(C) be the set of Morita classes of simple special symmetric Frobeniusalgebras in C Define the map z Msimp(C) rarr Cmax(C2plusmn) by z A rarr [Z(A)] whereA denotes the Morita class of A From Theorem 322 (i) and [KR Thm 11] we learn

Corollary 325 The map z Msimp(C) rarr Cmax(C2plusmn) is a bijection

A Appendix

A1 Proof of Lemma 27 We will show that if (F ψ F2 ψ

F0 ) is a colax tensor functor

from C1 to C2 then (G φG2 φ

G0 ) is a lax tensor functor from C2 to C1 Applying this

result to the opposed categories then gives the converse statementWe need to show that φG

0 and φG2 make the diagrams (21) and (22) commute We

first prove the commutativity of (21) Consider the following diagram

G(A)otimes (G(B)otimes G(C))

GψF2 δ

idG(A)otimesφG2 G(A)otimes G(B otimes C)

GψF2 δ

G(FG(A)otimes F(G(B)otimes G(C)))

G(F(idG(A))otimesF(φG2 ))

G(idFG(A)otimesψ F2 )

G(FG(A)otimes FG(B otimes C))

G(ρAotimesρBotimesC )

G(FG(A)otimes (FG(B)otimes FG(C)))

G(ρAotimes(ρBotimesρC )) G(A otimes (B otimes C))

(A1)

The top subdiagram is commutative because of the naturality of Gψ F2 δ The commu-

tativity of the bottom subdiagram follows from the following identities

(ρB otimes ρC ) ψ F2 = (ρB otimes ρC ) ψ F

2 ρF Fδ

= ρBotimesC FG(ρB otimes ρC ) FG(ψ F2 ) Fδ

= ρBotimesC F(φG2 ) (A2)

as a map F(G(B) otimes G(C)) rarr B otimes C The commutativity of (A1) implies that thecomposition of maps in the left column in (21) can be replaced by

G(ρA otimes (ρB otimes ρC )) G((idFG(A) otimes ψ F

2 ) ψ F2

) δ (A3)

906 L Kong I Runkel

Similarly we can show that the composition of maps in the right column in (21) can bereplaced by

G((ρA otimes ρB)otimes ρC ) G((ψ F

2 otimes idFG(C)) ψ F2

) δ (A4)

Using the commutativity of (23) it is easy to see that (21) with the left and right col-umns of (21) replaced by (A3) and (A4) respectively is commutative Hence (21) iscommutative

Now we prove the commutativity of the first diagram in (22)

φG2 (φG

0 otimes idG(A))

(1)= G(ρ12 otimes ρA) Gψ F2 δ

[(Gψ F

0 δ11)otimes idG(A)

]

(2)= G(ρ12 otimes ρA) Gψ F2 G F(Gψ0 otimes idG(A)) G F(δ11 otimes idG(A)) δ

(3)= G(ρ12 otimes ρA) G(FG(ψ F0 )otimes idFG(A)) Gψ F

2 G F(δ11 otimes idG(A)) δ(4)= G(id12 otimes ρA) G

([ψ F

0 ρF(11) (Fδ)11] otimes idFG(A)

) Gψ F

2 δ(5)= G(id12 otimes ρA) G(ψ F

0 otimes idFG(A)) Gψ F2 δ

(6)= G(id12 otimes ρA) G(lminus1FG(A)) G F(lG(A)) δ

(7)= G(lminus1A ) GρA δG lG(A)

(8)= G(lminus1A ) lG(A) (A5)

where in step (1) we substituted the definition of φG0 φ

G2 given in (210) in step (2)

we used the naturality of δ in step (3) we used the naturality of Gψ F2 in step (4) we

switched the position between Gρ12 and G FG(ψ F0 ) and the position between Gψ2 and

G F(δ12 otimes idG(A)) using the naturality of ρ and FψG2 respectively in step (5) we applied

the second identity in (28) in step (6) we used (24) in step (7) we used the naturalityof lminus1 and δ in step (8) we used the first identity in (28)

The proof of the commutativity of the second diagram in (22) is similar Thus wehave shown that G is a lax tensor functor

A2 Proof of Lemma 320 To prepare the proof recall that for a given object B isin Cthe modular group P SL(2Z) acts on the space oplusi HomC(B otimes Ui Ui ) see eg [BKSect 31] and [K3 Eq (455)] We will only need the action of S and Sminus1 Let f isinoplusi HomC(B otimes Ui Ui ) Then

S oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A6)

Cardy Algebras and Sewing Constraints I 907

Sminus1 oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A7)

By Lemma 32 to establish that (337) is S-invariant it is enough to prove the identity(34) when f is given by (337) Using (A6) and (A7) we can see that Eq (34) simplysays that oplusi j [RHS of (34)] is invariant under the action of S times S Consider the elementg of oplus jkisinIHomC2plusmn(R(A)otimes (Uor

j times Uk)Uorj times Uk) given by

g =oplusjkisinI

sumα

R(A)

R(A)

R(A)

R(A)

Uorj times Uk

Uorj times Uk

α

α

(A8)

By the above arguments proving S-invariance of (337) is equivalent to proving invari-ance of g under the action of S times S

For i isin I we denote by gi the component of g in(oplus jisinIHomC(A otimes Uor

i otimes Uorj U

orj )

)otimes (opluskisinIHomC(Ui otimes UkUk))

We view the second Hom-space in the above tensor product as a Hom-space in C+ insteadof Cminus It is enough to show that gi is invariant under the action of S times Sminus1 Note thatthe action of Sminus1 in C+ is equivalent to that of S in Cminus

The morphism gi can be canonically identified with a bilinear pairing

( middot middot )i ⎛⎝oplus

jisinIHomC(Uor

j A otimes Uori otimes Uor

j )

⎞⎠

times(oplus

kisinIHomC(UkUi otimes Uk)

)minusrarr C (A9)

as follows For h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) we set

(h1 h2)i = (dim U j dim Uk)minus1 trUor

j timesUk[gi (h1 times h2)] (A10)

908 L Kong I Runkel

When substituting the explicit form of the product m R(A) of R(A) = (A times 1) otimes R(1)after a short calculation one finds

(h1 h2)i =sumα

1

dim U j

A

A

A

A

A A

A α

α

h1

h2

U jUk

Uorj

Ui

Uk (A11)

Here the top morphism PlR(A) has been simplified with the help of the identity

PR(A) m R(A) (PR(A) otimes PR(A))

=(oplus

iisinI((m A A)otimes idUor

i)times idUi

) m R(A) (PR(A) otimes PR(A)) (A12)

which can be checked by direct calculation along the same lines as in the proof of [FrLem 310]

The action of the modular transformation S on oplusiisinI(B otimes Ui Ui ) for B isin C nat-urally induces an action on oplusiisinI(Ui B otimes Ui ) [K3 Prop 514] which we denote bySlowast In the present case we get an action of Slowast on oplus jisinIHomC(Uor

j A otimes Uori otimes Uor

j ) and

opluskisinIHomC(UkUi otimes Uk) Then to show gi is invariant under the action of S times Sminus1

amounts to showing that

(h1 h2)i = ((Sminus1)lowast h1 Slowast h2)i (A13)

Cardy Algebras and Sewing Constraints I 909

for all h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) We have

((Sminus1)lowast h1 Slowast h2)i =sum

mnα

dim Un

DimC

A

A

A

A

A A

A

Uk

α

α

h1

h2

Ui

U j

UmUn

(A14)

Now drag the upper vertex indexed by α in the above graph along its Uorm -leg until it

meets the lower vertex also indexed by α then sum over α and m This gives

((Sminus1)lowast h1 Slowast h2)i =sum

n

dim Un

DimC

A

A

A

A

A A

A

Uk

Un

h1

h2

Ui

U j

(A15)

910 L Kong I Runkel

If we just look at the neighbourhood of the Un-loop in the above graph we see thefollowing subgraphs

sumn

dim Un

DimC

A Aor

Uorj U j

Uk Uork

Un=

sumα

1

dim U j

A Aor Uk Uork

Uorj U j

αα

(A16)

where we have applied [BK Cor 3111] Substituting this subgraph back to the originalgraph in (A15) we obtain

((Sminus1)lowast h1 Slowast h2)i =sumα

1

dim U j

A

A

A

A

A A

A

A

A

Uorj

α

α

h1

h2m A

Ui

U jUk

Uk

(A17)

The graph in (A17) is equal to that in (A11) In order to see this we first drag theldquobubblerdquo (m A A) along A lines and through the m A vertex (because m A A is abimodule map) until it reaches the lower-left leg of the upper vertex indexed by α Thendrag the m A vertex along the (red) dotted line in the above graph Finally we applythe associativity of A (A12) and [Fr Lem 311] Then we see that the graph in (A17)exactly matches with the one in (A11)

Acknowledgements We would like to thank the organisers of the Oberwolfach Arbeitsgemeinschaft ldquoAlge-braic structures in conformal field theoriesrdquo (April 2007) where this work was started for an inspiring meetingWe would further like to thank the Hausdorff Institute for Mathematics in Bonn and the organisers of the stim-ulating meeting ldquoGeometry and Physicsrdquo (May 2008) We are indebted to Alexei Davydov Jens FjelstadJuumlrgen Fuchs Yi-Zhi Huang Alexei Kitaev Urs Schreiber Christoph Schweigert Stephan Stolz and PeterTeichner for helpful discussions andor comments on a draft of this paper The research of IR was partiallysupported by the EPSRC First Grant EPE0050471 the PPARC rolling grant PPC5071451 and the MarieCurie network lsquoSuperstring Theoryrsquo (MRTN-CT-2004-512194)

Cardy Algebras and Sewing Constraints I 911

References

[AN] Alexeevski A Natanzon SM Noncommutative two-dimensional topological field theories andhurwitz numbers for real algebraic curves Sel Math New Ser 12 307ndash377 (2006)

[Bi] Bichon J Cosovereign hopf algebras J Pure Appl Alg 157 121ndash133 (2001)[BK] Bakalov B Kirillov AA Lectures on Tensor Categories and Modular Functors Providence

RI Amer Math Soc 2001[C1] Cardy JL Operator content of two-dimensional conformal invariant theories Nucl Phys

B 270 186ndash204 (1986)[C2] Cardy JL Boundary conditions fusion rules and the verlinde formula Nucl Phys B 324

581ndash596 (1989)[CL] Cardy JL Lewellen DC Bulk and boundary operators in conformal field theory Phys Lett

B 259 274ndash278 (1991)[DMZ] Dong C-Y Mason G Zhu Y-C Discrete series of the Virasoro algebra and the moonshine

module In Algebraic Groups and Their Generalizations Quantum and infinite-dimensional Meth-ods Proc Symp Pure Math 56 Part 2 Providence RI Amer MathSoc 1994 pp 295ndash316

[DP] Day B Pastro C Note on frobenius monoidal functors New York J Math 14 733ndash742 (2008)[ENO1] Etingof PI Nikshych D Ostrik V On fusion categories Ann Math 162 581ndash642 (2005)[ENO2] Etingof PI Nikshych D Ostrik V An analogue of radfordrsquos s4 formula for finite tensor

categories Int Math Research Notices 54 2915ndash2933 (2004)[Fe] Felder G Froumlhlich J Fuchs J Schweigert C Correlation functions and boundary conditions in

rational conformal field theory and three-dimensional topology Comp Math 131 189ndash237 (2002)[FHL] Frenkel IB Huang Y-Z Lepowsky J On axiomatic approaches to vertex operator algebras

and modules Mem Amer Math Soc 104 (1993)[Fj] Fjelstad J Fuchs J Runkel I Schweigert C Uniqueness of openclosed rational cft with

given algebra of open states Adv Theor Math Phys 12 1283ndash1375 (2008)[FK] Froumlhlich J King C The chern-simons theory and knot polynomials Commun Math

Phys 126 167ndash199 (1989)[Fr] Froumlhlich J Fuchs J Runkel I Schweigert C Correspondences of ribbon categories Adv

Math 199 192ndash329 (2006)[FRS] Fuchs J Runkel I Schweigert C Tft construction of rcft correlators I partition functions Nucl

Phys B 646 353ndash497 (2002)[FS] Fuchs J Schweigert C Category theory for conformal boundary conditions Fields Inst Com-

mun 39 25ndash71 (2003)[H1] Huang Y-Z Two-dimensional conformal geometry and vertex operator algebras Progress in

Mathematics Vol 148 Boston Birkhaumluser 1997[H2] Huang Y-Z Rigidity and modularity of vertex tensor categories Comm Contemp Math 10

871ndash911 (2008)[HK1] Huang Y-Z Kong L Open-string vertex algebra category and operads Commun Math

Phys 250 433ndash471 (2004)[HK2] Huang Y-Z Kong L Full field algebras Commun Math Phys 272 345ndash396 (2007)[HK3] Huang Y-Z Kong L Modular invariance for conformal full field algebras httparxivorgabs

math0609570v2[mathQA] 2006[HL] Huang Y-Z Lepowsky J Tensor products of modules for a vertex operator algebra and vertex

tensor categories In Lie Theory and Geometry in honor of Bertram Kostant ed R BrylinskiJ-L Brylinski V Guillemin V Kac Boston Birkhaumluser 1994 pp 349ndash383

[JS] Joyal A Street R Braided tensor categories Adv Math 102 20ndash78 (1993)[K1] Kong L Full field algebras operads and tensor categories Adv Math 213 271ndash340 (2007)[K2] Kong L Open-closed field algebras Commun Math Phys 280 207ndash261 (2008)[K3] Kong L Cardy condition for open-closed field algebras Commun Math Phys 283 25ndash92 (2008)[Ki] Kitaev A Private communication[KO] Kirillov AA Ostrik V On q-analog of mckay correspondence and ade classification of sl(2)

conformal field theories Adv Math 171 183ndash227 (2002)[KR] Kong L Runkel I Morita classes of algebras in modular tensor categories Adv Math 219

1548ndash1576 (2008)[Ld] Lauda AD Frobenius algebras and ambidextrous adjunctions Theo Appl Cat 16 84ndash

122 (2006)[Li1] Li H-S Regular representations of vertex operator algebras Commun Contemp Math 4

639ndash683 (2002)[Li2] Li H-S Regular representations and huang-lepowsky tensor functors for vertex operator

algebras J Alge 255 423ndash462 (2002)

Cardy Algebras and Sewing Constraints I 875

(and shorter) proof Finally we show that for every simple modular invariant commu-tative symmetric Frobenius algebra Acl in C2plusmn there exists a simple special symmetricFrobenius algebra Aop and an algebra homomorphism ιcl-op Acl rarr R(Aop) such that(Aop Acl ιcl-op) is a Cardy algebra (Theorem 322) This theorem is closely related toa result announced in [Muuml2] and provides an independent proof in the framework ofCardy algebras

In physical terms these two theorems mean that a rational open-closed CFT with aunique closed vacuum state can be uniquely reconstructed from its correlators involvingonly discs with boundary punctures and that every closed CFT with unique vacuum andleftright rational chiral algebra V otimes V occurs as part of such an open-closed CFT

2 Preliminaries on Tensor Categories

In this section we review some basic facts of tensor categories and fix our conventionsand notations along the way

21 Tensor categories and (co)lax tensor functors In a tensor (or monoidal) categoryC with tensor product bifunctor otimes and unit object 1 for U VW isin C we denote the

associator U otimes (V otimes W )sim=minusrarr (U otimes V ) otimes W by αUVW the left unit isomorphism

1 otimes Usim=minusrarr U by lU and the right unit isomorphism U otimes 1

sim=minusrarr U by rU If C is braidedfor U V isin C we write the braiding isomorphism as cUV U otimes V rarr V otimes U

Let C1 and C2 be two tensor categories with units 11 and 12 respectively For simplic-ity we will often write otimes α l r for the data of both C1 and C2 Lax and colax tensorfunctors are defined as follows see eg [Y Ch I3] or [Ln Ch I12]

Definition 21 A lax tensor functor G C1 rarr C2 is a functor equipped with a mor-phism φG

0 12 rarr G(11) in C2 and a natural transformation φG2 otimes(G times G) rarr G otimes

such that the following three diagrams commute

G(A)otimes (G(B)otimes G(C))α

idG(A)otimesφG2

(G(A)otimes G(B))otimes G(C)

φG2 otimesidG(C)

G(A)otimes G(B otimes C)

φG2

G(A otimes B)otimes G(C)

φG2

G(A otimes (B otimes C))

G(α) G((A otimes B)otimes C)

(21)

12 otimes G(A)

φG0 otimesidG(A)

lG(A) G(A)

G(lminus1A )

G(11)otimes G(A)

φG2 G(11 otimes A)

G(A)otimes 12

idG(A)otimesφG0

rG(A) G(A)

G(rminus1A )

G(A)otimes G(11)

φG2 G(A otimes 11)

(22)

Definition 22 A colax tensor functor is a functor F C1 rarr C2 equipped with a mor-phismψ F

0 F(11) rarr 12 in C2 and a natural transformationψ F2 Fotimes rarr otimes(F times F)

such that the following three diagrams commute

876 L Kong I Runkel

F(A)otimes (F(B)otimes F(C))α (F(A)otimes F(B))otimes F(C)

F(A)otimes F(B otimes C)

idF(A)otimesψ F2

F(A otimes B)otimes F(C)

ψF2 otimesidF(C)

F(A otimes (B otimes C))F(α)

ψF2

F((A otimes B)otimes C)

ψF2

(23)

12 otimes F(A) F(A)lminus1F(A)

F(11)otimes F(A)

ψF0 otimesidF(A)

F(11 otimes A)ψF

2

F(lA)

F(A)otimes 12 F(A)rminus1

F(A)

F(A)otimes F(11)

idF(A)otimesψ F0

F(A otimes 11)

F(rA)

ψF2

(24)

We denote a lax tensor functor by (G φG2 φ

G0 ) or just G and a colax tensor functor

by (F ψ F2 ψ

F0 ) or F

Definition 23 A tensor functor T C1 rarr C2 is a lax tensor functor (T φT2 φ

T0 ) such

that φT0 φ

T2 are both isomorphisms

A tensor functor (T φT2 φ

T0 ) is automatically a colax tensor functor (T ψT

2 ψT0 )

with ψT0 = (φT

0 )minus1 and ψT

2 = (φT2 )

minus1In the next section we will discuss algebras in tensor categories The defining proper-

ties (21) and (22) of a lax tensor functor are analogues of the associativity the left-unitand the right-unit properties of an algebra Indeed a lax tensor functor G C1 rarr C2maps a C1-algebra to a C2-algebra Similarly (23) and (24) are analogues of the coasso-ciativity the left-counit and the right-counit properties of a coalgebra and a colax tensorfunctor F C1 rarr C2 maps a C1-coalgebra to a C2-coalgebra We will later make useof functors that take Frobenius algebras to Frobenius algebras This requires a strongercondition than being lax and colax and leads to the notion of a lsquofunctor with Frobeniusstructurersquo or lsquoFrobenius monoidal functorrsquo [SzDPP] which we will simply refer to asFrobenius functor

Definition 24 A Frobenius functor F C1 rarr C2 is a tuple F equiv (F φF2 φ

F0 ψ

F2 ψ

F0 )

such that (F φF2 φ

F0 ) is a lax tensor functor (F ψ F

2 ψF0 ) is a colax tensor functor

and such that the following two diagrams commute

F(A)otimes (F(B)otimes F(C))α (F(A)otimes F(B))otimes F(C)

φF2 otimesidF(C)

F(A)otimes F(B otimes C)

idF(A)otimesψ F2

φF2

F(A otimes B)otimes F(C)

F(A otimes (B otimes C))F(α) F((A otimes B)otimes C)

ψF2

(25)

Cardy Algebras and Sewing Constraints I 877

F(A)otimes (F(B)otimes F(C))

idF(A)otimesφF2

(F(A)otimes F(B))otimes F(C)αminus1

F(A)otimes F(B otimes C) F(A otimes B)otimes F(C)

ψF2 otimesidF(C)

φF2

F(A otimes (B otimes C))

ψF2

F((A otimes B)otimes C)F(αminus1)

(26)

Proposition 25 If (F φF2 φ

F0 ) is a tensor functor then F is a Frobenius functor with

ψ F0 = (φF

0 )minus1 and ψ F

2 = (φF2 )

minus1

Proof Since F is a tensor functor it is lax and colax If we replace ψ F2 by (φF

2 )minus1 in

(25) and (26) both commuting diagrams are equivalent to (21) which holds becauseF is lax Thus F is a Frobenius functor

The converse statement does not hold For example the functor R which we definein Sect 24 is Frobenius but not tensor

Let us recall the notion of adjunctions and adjoint functors [Ma Ch IV1]

Definition 26 An adjunction from C1 to C2 is a triple 〈FG χ〉 where F and G arefunctors

F C1 rarr C2 G C2 rarr C1

and χ is a natural isomorphism which assigns to each pair of objects A1 isin C1 A2 isin C2a bijective map

χA1A2 HomC2(F(A1) A2)sim=minusminusrarr HomC1(A1G(A2))

which is natural in both A1 and A2 F is called a left-adjoint of G and G is called aright-adjoint of F

For simplicity we will often abbreviate χA1A2 as χ Associated to each adjunction

〈FG χ〉 there are two natural transformations idC1

δminusrarr G F and FGρminusrarr idC2 where

idC1 and idC2 are identity functors given by

δA1 = χ(idF(A1)) ρA2 = χminus1(idG(A2)) (27)

for Ai isin Ci i = 1 2 They satisfy the following two identities

GδGminusrarr G FG

Gρminusrarr G = GidGminusminusrarr G F

Fδminusrarr FG FρFminusrarr F = F

idFminusminusrarr F (28)

We have for g F(A1) rarr A2 and f A1 rarr G(A2)

χ(g) = G(g) δA1 χminus1( f ) = ρA2 F( f ) (29)

For simplicity δA1 and ρA2 are often abbreviated as δ and ρ respectivelyLet 〈FG χ〉 be an adjunction from a tensor category C1 to a tensor category C2

and (F ψ F2 ψ

F0 ) a colax tensor functor from C1 to C2 We can define a morphism

878 L Kong I Runkel

φG0 11 rarr G(12) and a natural transformation φG

2 otimes (G times G) rarr G otimes by forA B isin C2

φG0 = χ(ψ F

0 ) = 11δ11minusrarr G F(11)

G(ψ F0 )minusminusminusminusrarr G(12)

φG2 = χ((ρA otimes ρB) ψ F

2 ) = G(A)otimes G(B)δminusrarr G F (G(A)otimes G(B))

G(ψ F2 )minusminusminusminusrarr G (FG(A)otimes FG(B))

G(ρAotimesρB )minusminusminusminusminusminusrarr G(A otimes B) (210)

where we have used the first identity in (29) Notice that φG2 is natural because it is a

composition of natural transformations One can easily show that ψ F0 and ψ F

2 can bere-obtained from φG

0 and φG2 as follows

ψ F0 = χminus1(φG

0 ) = F(11)FφG

0minusminusrarr FG(12)ρminusrarr 12

ψ F2 = χminus1(φG

2 (δ otimes δ)) = F(U otimes V )F(δotimesδ)minusminusminusminusrarr F (G F(U )otimes G F(V ))

FφG2minusminusrarr FG (F(U )otimes F(V ))

ρminusrarr F(U )otimes F(V ) (211)

for U V isin C1 The following result is standard for the sake of completeness we givea proof in Appendix A1

Lemma 27 (F ψ F2 ψ

F0 ) is a colax tensor functor iff (G φG

2 φG0 ) is a lax tensor

functor

22 Algebras in tensor categories An algebra in a tensor category C or a C-algebra isa triple A = (Am η) where A is an object of C m (the multiplication) is a morphismA otimes A rarr A such that m (m otimes idA) αAAA = m (idA otimes m) and η (the unit) is amorphism 1 rarr A such that m (idA otimes η) = idA rA and m (ηotimes idA) = idA lAIf C is braided and m cAA = m then A is called commutative

A left A-module is a pair (MmM ) where M isin C and m M is a morphism AotimesM rarr Msuch that mM(idAotimesm M ) = m M(m AotimesidM )αAAM and m M(ηAotimesidM ) = idMlM Right A-modules and A-bimodules are defined similarly

Definition 28 Let C be a tensor category and let A be an algebra in C

(i) A is called simple iff it is simple as a bimodule over itselfLet C be in addition k-linear for k a field

(ii) A is called absolutely simple iff the space of A-bimodule maps from A to itself isone-dimensional dimk HomA|A(A A) = 1

(iii) A is called haploid iff dimk Hom(1 A) = 1 [FS Def 43]

In the following we will assume that all tensor categories are strict to avoid spelling outassociators and unit constraints

A C-coalgebra A = (A ε) is defined analogously to a C-algebra ie A rarrA otimes A and ε A rarr 1 obey coassociativity and counit conditions

If C is braided and if A and B are C-algebras there are two in general non-isomorphicalgebra structures on A otimes B We choose A otimes B to be the C-algebra with multiplicationm AotimesB = (m Aotimesm B)(idAotimescminus1

ABotimesidB) and unitηAotimesB = ηAotimesηB Similarly if A and Bare C-coalgebras then A otimes B becomes a C-coalgebra if we choose the comultiplicationAotimesB = (idA otimes cAB otimes idB) (A otimesB) and the counit εAotimesB = εA otimes εB

Cardy Algebras and Sewing Constraints I 879

Definition 29 A Frobenius algebra A = (Am η ε) is an algebra and a coalge-bra such that the coproduct is an intertwiner of A-bimodules

(idA otimes m) (otimes idA) = otimes m = (m otimes idA) (idA otimes)

We will use the following graphical representation for the morphisms of a Frobeniusalgebra

m =A A

A

η =A

=A A

A

ε =A

(212)

A Frobenius algebra A in a k-linear tensor category for k a field is called special iffm = ζ idA and ε η = ξ id1 for nonzero constants ζ ξ isin k If ζ = 1 we call A nor-malised-special A Frobenius algebra homomorphism between two Frobenius algebrasis both an algebra homomorphism and a coalgebra homomorphism

A (strictly) sovereign tensor category is a tensor category equipped with a left and aright duality which agree on objects and morphisms (see eg [BiFS] for more details)We will write the dualities as

Uor U

= dU Uor otimes U rarr 1U Uor

= dU U otimes Uor rarr 1

U Uor

= bU 1 rarr U otimes UorUor U

= bU 1 rarr Uor otimes U

(213)

In terms of these we define the left and right dimension of an object U as

diml U = dU bU dimr U = dU bU (214)

both of which are elements of Hom(1 1)Let now C be a sovereign tensor category For a Frobenius algebra A in C we define

two morphisms

A =

A

Aor

primeA =

A

Aor

(215)

Definition 210 A Frobenius algebra A is symmetric iff A = primeA

The following lemma shows that under certain conditions we do not need to distin-guish the various notions of simplicity in Definition 28

Lemma 211 Let A be a commutative symmetric Frobenius algebra in a C-linear semi-simple sovereign braided tensor category C and suppose that diml A = 0 Then thefollowing are equivalent

880 L Kong I Runkel

(i) A is simple(ii) A is absolutely simple

(iii) A is haploid

Proof (ii)hArr(iii) A is haploid iff it is absolutely simple as a left module over itself[FS Eq (417)] Furthermore for a commutative algebra we have HomA(A A) =HomA|A(A A) and so A is haploid iff it is absolutely simple(i)rArr(ii) If A is simple then every nonzero element of HomA|A(A A) is invertibleHence this space forms a division algebra over C and is therefore isomorphic to C(iii)rArr(i) Since C is semi-simple and A is haploid also Hom(A 1) is one-dimensionalThe counit ε is a nonzero element in this space and so gives a basis This implies firstlythat ε η = 0 and secondly that there is a constant β isin C such that

β middot ε = dA (idAor otimes m) (bA otimes idA) (216)

Composing with η from the right yields β εη = diml A The right-hand side is nonzeroand so β = 0 By [FRS Lem 311] A is special We have already proved (ii)hArr(iii)and so A is absolutely simple A special Frobenius algebra in a semi-simple categoryhas a semi-simple category of bimodules (apply [FS Prop 524] to the algebra tensoredwith its opposite algebra) For semi-simple C-linear categories simple and absolutelysimple are equivalent2 Thus A is simple Remark 212 For a Frobenius algebra A the morphisms (215) are invertible and henceA sim= Aor In this case one has diml A = dimr A [FS Rem 363] and so we could havestated the above lemma equivalently with the condition dimr A = 0

Let F C1 rarr C2 be a lax tensor functor between two tensor categories C1 C2 and

let (Am A ηA) be an algebra in C1 Define morphisms F(A)otimes F(A)m F(A)minusminusminusrarr F(A) and

12ηF(A)minusminusminusrarr F(A) as

m F(A) = F(m A) φF2 ηF(A) = F(ηA) φF

0 (217)

Then (F(A)m F(A) ηF(A)) is an algebra in C2 [JS Prop 55] If f A rarr B is analgebra homomorphism between two algebras A B isin C1 then F( f ) F(A) rarr F(B)is also an algebra homomorphism If (MmM ) is a left (or right) A-module in C1 then(F(M) F(mM )φF

2 ) is a left (or right) F(A)-module if M has a A-bimodule structurethen F(M) naturally has a F(A)-bimodule structure

Similarly if (AA εA) is a coalgebra in C1 and F C1 rarr C2 is a colax tensor func-

tor then F(A) with coproduct F(A)F(A)minusminusminusrarr F(A) otimes F(A) and counit F(A)

εF(A)minusminusminusrarr 12given by

F(A) = ψ F2 F(A) εF(A) = ψ F

0 F(εA) (218)

is a coalgebra in C2 If f A rarr B is a coalgebra homomorphism between two coalgebrasA B isin C1 then F( f ) F(A) rarr F(B) is also a coalgebra homomorphism

2 To see this note that if U is simple then the C-vector space Hom(UU ) is a division algebra and henceHom(UU ) = C idU Conversely if U is not simple then U = U1 oplus U2 and Hom(UU ) contains at leasttwo linearly independent elements namely idU1 and idU2

Cardy Algebras and Sewing Constraints I 881

Proposition 2133 If F C1 rarr C2 is a Frobenius functor and (Am A ηAA εA) aFrobenius algebra in C1 then (F(A)m F(A) ηF(A)F(A) εF(A)) is a Frobenius alge-bra in C2

Proof One Frobenius property (m F(A)otimes idF(A)) (idF(A)otimesF(A)) = F(A) m F(A)follows from the commutativity of the following diagram (we spell out the associativityisomorphisms)

F(A)otimes F(A)idF(A)otimesF(A)

φF2

F(A)otimes F(A otimes A)idF(A)otimesψF

2

φF2

F(A)otimes (F(A)otimes F(A))

αF(A)F(A)F(A)

F(A otimes A)

F(idAotimesA)

F(m A)

F(A otimes (A otimes A))

F(αAAA)

(F(A)otimes F(A))otimes F(A)

φF2 otimesidF(A)

F((A otimes A)otimes A)

F(m AotimesidA)

ψF2 F(A otimes A)otimes F(A)

F(m A)otimesidF(A)

F(A)

F(A) F(A otimes A)ψF

2 F(A)otimes F(A)

(219)

The commutativity of the upper-left subdiagram follows from the naturalness of φF2 that

of the upper-right subdiagram follows from (25) that of the lower-left subdiagram fol-lows from the Frobenius properties of A and that of the lower-right subdiagram followsfrom the naturalness of ψ F

2 The proof of the other Frobenius property is similar Proposition 214 If F C1 rarr C2 is a tensor functor and A a Frobenius algebra in C1then

(i) F(A) has a natural structure of Frobenius algebra as given in Proposition 213(ii) If A is (normalised-)special so is F(A)

Proof Part (i) follows from Propositions 25 and 213 Part (ii) is a straightforwardverification of the definition using ψ F

2 = (φF2 )

minus1 and ψ F0 = (φF

0 )minus1

Let C1 C2 be sovereign tensor categories and F C1 rarr C2 a Frobenius functor Wedefine two morphisms IF(Aor) I prime

F(Aor) F(Aor) rarr F(A)or for a Frobenius algebra A inC1 as follows

IF(Aor) = ((ψ F0 F(dA) φF

2 )otimes idF(A)or) (idF(Aor) otimes bF(A))

I primeF(Aor) = (idF(A)or otimes (ψ F

0 F(dA) φF2 )) (bF(A) otimes idF(Aor))

(220)

It is easy to see that these are isomorphisms

Lemma 215 If F C1 rarr C2 is a Frobenius functor and A a Frobenius algebra in C1then

F(A) = IF(Aor) F(A) primeF(A) = I prime

F(Aor) F(primeA) (221)

3 After the preprint of the present paper appeared we noticed that this proposition is also proved in[DP Cor 5]

882 L Kong I Runkel

Proof We only prove the first equality the second one can be seen in the same way Bydefinition we have

IF(Aor) F(A)

= ((ψ F0 F(dA) φF

2 )otimes idF(A)or) (idF(Aor) otimes bF(A)) F(A)

=[(ψ F

0 F(dA) φF2 ) (F(A)otimes idF(A))

]otimes idF(A)or

(idF(A) otimes bF(A))

For the term inside the square brackets we find

ψ F0 F(dA) φF

2 (F(A)otimes idF(A)) = ψ F0 F(dA) F(A otimes idA) φF

2

= ψ F0 F(dA (A otimes idA)) φF

2 = ψ F0 F(εA m A) φF

2 (222)

On the other hand by definitionF(A) = [((ψ F0 F(εA) (F(m A)φF

2 ))otimes idF(A)or] (idF(A) otimes bF(A)) This demonstrates the first equality in (221) Proposition 216 Let F C1 rarr C2 be a tensor functor G C2 rarr C1 a functor〈FG χ〉 an adjunction A a C1-algebra and B a C2-algebra Then f A rarr G(B)is an algebra homomorphism if and only if f = χminus1( f ) F(A) rarr B is an algebrahomomorphism

Proof We need to show that

mG(B) ( f otimes f ) = f m A and f ηA = ηG(B) (223)

is equivalent to

m B ( f otimes f ) = f m F(A) and f ηF(A) = ηBs (224)

We first prove that the first identity in (223) is equivalent to the first identity in (224)For the left-hand side of the first identity in (223) we have the following equalities

mG(B) ( f otimes f )(1)= G(m B) φG

2 ( f otimes f )(2)= G(m B) G(ρ otimes ρ) G(ψ F

2 ) δ ( f otimes f )(3)= G(m B) G(ρ otimes ρ) G(ψ F

2 ) G F( f otimes f ) δ(4)= G(m B) G(ρ otimes ρ) G(F( f )otimes F( f )) G(ψ F

2 ) δ(5)= G

(m B (ρ otimes ρ) (F( f )otimes F( f )) ψ F

2

) δ

(6)= χ(

m B (ρ otimes ρ) (F( f )otimes F( f )) ψ F2

) (225)

where (1) is the definition of mG(B) in (217) (2) is the second identity in (210) (3)and (4) are naturality of δ and ψ F

2 respectively step (5) is functoriality of G and finallystep (6) is (29) For the right hand side of the first identity in (223) we get

f m A(1)= Gρ δG ( f m A)

(2)= Gρ G F( f m A) δ(3)= G(ρ F( f m A)) δ(4)= χ(ρ F( f ) F(m A)) (226)

Cardy Algebras and Sewing Constraints I 883

where (1) is the adjunction property (28) (2) is naturality of δ (3) functoriality of Gand (4) amounts to (29) and functoriality of F

On the other hand we see that the first equality in (224) is equivalent to

m B (ρ otimes ρ) (F( f )otimes F( f )) = ρ F( f ) F(m A) φF2 (227)

Using that φF2 is invertible with inverse (φF

2 )minus1 = ψ F

2 and that χ is an isomorphismit follows that the statement that (225) is equal to (226) is equivalent to the identity(227)

Now we prove that the second identity in (223) is equivalent to the second identity in(224) Using (217) and (210) we can write ηG(B) = G(ηB)φF

0 = G(ηB)G(ψ F0 )δ1

Together with (29) this shows that the second identity in (223) is equivalent to

f ηA = χ(ηB ψ F0 ) (228)

On the other hand the second identity in (224) is equivalent to

ρ F( f ) F(ηA) φF0 = ηB (229)

which by φF0 = (ψ F

0 )minus1 and (29) is further equivalent to (228)

Definition 217 Let (Am A ηAA εA) and (Bm B ηBB εB) be two Frobeniusalgebras in a tensor category C For f A rarr B we define f lowast B rarr A by

f lowast = ((εB m B)otimes idA) (idB otimes f otimes idA) (idB otimes (A ηA)) (230)

The following lemma is immediate from the definition of (middot)lowast and the properties ofFrobenius algebras We omit the proof

Lemma 218 Let C be a tensor category let A BC be Frobenius algebras in C andlet f A rarr B and g B rarr C be morphisms

(i) (g f )lowast = f lowast glowast(ii) f is a monomorphism iff f lowast is an epimorphism

(iii) f is an algebra map iff f lowast is a coalgebra map(iv) If f is a homomorphism of Frobenius algebras then f lowast f = idA and f f lowast =

idB(v) If C is sovereign and if A and B are symmetric then f lowastlowast = f

Let C and D be tensor categories and let F C rarr D be a Frobenius functor GivenFrobenius algebras A B in C and a morphism f A rarr B the next lemma shows how(middot)lowast behaves under F

Lemma 219 F( f lowast) = F( f )lowast

Proof The definition of the structure morphisms of the Frobenius algebra F(A) is givenin (217) and (218) Substituting these definitions gives

F( f )lowast =[(ψ F

0 F(εB) F(m B) φF2

)otimes idF(A)

] [

idF(B) otimes F( f )otimes idF(A)]

[idF(B) otimes

(ψ F

2 F(A) F(ηA) φF0

)]

= (ψ F0 otimes idF(A)) [

F (εB m B (idB otimes f ))otimes idF(A)]

(φF2 otimes idF(A)) (idF(B) otimes ψ F

2 )

[idF(B) otimes F(A ηA)

] (idF(B) otimes φF0 ) (231)

884 L Kong I Runkel

In the middle line of the last expression we can use the defining property (25) of F namely we substitute (φF

2 otimes idF(A)) (idF(B) otimes ψ F2 ) = ψ F

2 φF2 Then ψ F

2 can bemoved to the left and φF

2 to the right until they can be omitted against ψ F0 and φF

0 respectively using (22) and (24) This results in

F( f )lowast = F ((εB m B (idB otimes f ))otimes idA) F (idB otimes (A ηA)) (232)

which is nothing but F( f lowast)

23 Modular tensor categories Let C be a modular tensor category [TBK] ie anabelian semi-simple finite C-linear ribbon category with simple tensor unit 1 and a non-degeneracy condition on the braiding (to be stated in a moment) We denote the set ofequivalence classes of simple objects in C by I elements in I by i j k isin I and theirrepresentatives by Ui U j Uk We also set U0 = 1 and for an index k isin I we define kby Uk

sim= Uork

Since the tensor unit is simple we shall for modular tensor categories identifyHom(1 1) sim= C (cf footnote 2) Define numbers si j isin C by4

si j = UiU j (233)

They obey si j = s ji and s0i = dim Ui see eg [BK Sect 31] (In a ribbon cate-gory the left and right dimension (214) of Ui coincide and are denoted by dim Ui )The non-degeneracy condition on the braiding of a modular tensor category is that the|I|times|I|-matrix s should be invertible In fact [BK Thm 317]

sumkisinI

sik sk j = Dim C δij (234)

where Dim C = sumiisinI(dim Ui )

2 One can show (even in the weaker context of fusioncategories over C) that Dim C ge 1 [ENO1 Thm 23] In particular Dim C = 0 We fixonce and for all a square root

radicDim C of Dim C

Let us fix a basis λα(i j)kN k

i jα=1 in HomC(Ui otimesU j Uk) and the dual basis ϒ(i j)k

α N ki j

α=1

in HomC(UkUi otimesU j ) The duality of the bases means that λα(i j)k ϒ(i j)kβ = δαβ idUk

We also fix λ(0i)i = λ(i0)i = idUi We denote the basis vectors graphically as follows

λα(i j)k = α

Uk

Ui U j

ϒ(i j)kα = α

Uk

Ui U j

(235)

4 In the graphical notation used below we have given an orientation to the ribbons indicated by the arrowsFor example it is understood that this orientation determines which of the duality morphisms in (213) to use

Cardy Algebras and Sewing Constraints I 885

For V isin C we also choose a basis b(iα)V of HomC(VUi ) and the dual basis bV(iβ)

of HomC(Ui V ) for i isin I such that b(iα)V bV(iβ) = δαβ idUi We use the graphical

notation

b(iα)V = α

Ui

V

bV(iα) = α

Ui

V

(236)

Given two modular tensor categories C and D by C D we mean the tensor productof additive categories over C [BK Def 1115] ie the category whose objects are directsums of pairs V times W of objects V isin C and W isin D and whose morphism spaces are

HomCD(V times W V prime times W prime) = HomC(V V prime)otimesC HomC(WW prime) (237)

for pairs and direct sums of these if the objects are direct sums of pairsIf we replace the braiding and the twist in C by the antibraiding cminus1 and the antitwist

θminus1 respectively we obtain another ribbon category structure on C In order to distin-guish these two distinct structures we denote (C c θ) and (C cminus1 θminus1) by C+ and Cminusrespectively As in the Introduction we will abbreviate

C2plusmn = C+ Cminus (238)

Note that a set of representatives of the simple objects in C2plusmn is given by Ui times U j fori j isin I

For the remainder of Sect 2 we fix a modular tensor category C

24 The functors T and R The tensor product bifunctor otimes can be naturally extendedto a functor T C2plusmn rarr C Namely T (oplusN

i=1Vi times Wi ) = oplusNi=1Vi otimes Wi for all Vi Wi isin C

and N isin N The functor T becomes a tensor functor as follows For φT0 1 rarr T (1 times 1)

take φT0 = id1 (or lminus1

1 in the non-strict case) Next notice that for U VW X isin C

T (U times V )otimes T (W times X)= (U otimes V )otimes (W otimes X)

T ((U times V )otimes (W times X))= (U otimes W )otimes (V otimes X)(239)

We define φT2 T (U times V )otimes T (W times X) rarr T ((U times V )otimes (W times X)) by

φT2 = idU otimes cminus1

W V otimes idX (240)

(In the non-strict case the appropriate associators have to be added) The above definitionof φT

2 can be naturally extended to a morphism φT2 T (M1 otimes M2) rarr T (M1)otimes T (M2)

for any pair of objects M1M2 in C2plusmn The following result can be checked by directcalculation [JS Prop 52]

Lemma 220 The triple (T φT2 φ

T0 ) gives a tensor functor

886 L Kong I Runkel

In particular (T φT2 φ

T0 ψ

T2 ψ

T0 ) where ψT

2 = (φT2 )

minus1 and ψT0 = (φT

0 )minus1 gives

a Frobenius functorDefine the functor R C rarr C2plusmn as follows for A isin C and f isin HomC(A B)

R(A) =oplusiisinI

(A otimes Uori )times Ui R( f ) =

oplusiisinI

( f otimes idUori)times idUi (241)

This functor was also considered in a slightly different context in [ENO2 Prop 23]The family of isomorphisms γ R

A = oplusiisinI DimCdim Ui

id(AotimesUori )timesUi

isin Aut(R(A)) defines a

natural isomorphism γ R R rarr ROur next aim is to show that R is left and right adjoint to T in other words R

and T form an ambidextrous adjunction (see eg [Ld] for a discussion of ambidextrousadjunctions) To this end we introduce two linear isomorphisms for A isin C and M isin C2plusmn

χ HomC(T (M) A) minusrarr HomC2plusmn(M R(A))

χ HomC(A T (M)) minusrarr HomC2plusmn(R(A)M)(242)

If we decompose M as M = oplusNn=1 Ml

n times Mrn then χ and χ are given by

χ Noplus

n=1

fn

Mln Mr

n

A

rarrNoplus

n=1

oplusiisinI

sumα

fn

Mln

Mrn

A

α

Uori

times α

Mrn

Ui

(243)

and

χ Noplus

n=1 gn

Mln Mr

n

A

rarrNoplus

n=1

oplusiisinI

sumα gn

Mln

Mrn

A

α

Uori

times α

Ui

Mrn

DimCdim Ui

(244)

Notice that χ and χ are independent of the choice of basis

Theorem 221 〈T R χ〉 and 〈R T χminus1〉 are adjunctions ie R is both left and rightadjoint of T

Proof Write M as M = oplusNn=1 Ml

n times Mrn The isomorphism χ amounts to the following

composition of natural isomorphisms

HomC(T (M) A) = oplusnHomC(Mln otimes Mr

n A)sim= oplusni HomC(Ml

n otimes Ui A)otimes HomC(Mrn Ui )

sim= oplusni HomC(Mln A otimes Uor

i )otimes HomC(Mrn Ui ) = HomC2plusmn(M R(A)) (245)

Cardy Algebras and Sewing Constraints I 887

Thus χ is natural Let (γ RA )

lowast HomC2plusmn(R(A)M) rarr HomC2plusmn(R(A)M) denote the

pull-back of γ RA The isomorphism χ is equal to the composition of (γ R

A )lowast and the

following sequence of natural isomorphisms

HomC(A T (M)) = oplusn HomC(AMln otimes Mr

n)

sim= oplusni HomC(AMln otimes Ui )otimes HomC(Ui Mr

n)

sim= oplusni HomC(A otimes Uori Ml

n)otimes HomC(Ui Mrn) = HomC2plusmn(R(A)M) (246)

We have proved that both χ and χ are natural isomorphisms There are four natural transformations associated to χ and χ namely

idC2plusmnδminusrarr RT

ρminusrarr idC2plusmn and idCδminusrarr T R

ρminusrarr idC (247)

defined by for A isin C M isin C2plusmn

δM = χ(idT (M)) ρA = χminus1(idR(A))

ρM = χ (idT (M)) δA = χminus1(idR(A))(248)

They can be expressed graphically as follows with M = oplusNn=1 Ml

n times Mrn

δM =oplusni

sumα

Mln

Mln Mr

n Uori

αtimes

Ui

Mrn

α

ρA =oplusiisinI

A

A

Uori Ui

(249)

ρM =oplusni

sumα

Mln

Mln

Mrn Uor

i

α times

Ui

Mrn

α DimCdim Ui

δA =oplusiisinI

A

A Uori Ui

dim UiDimC

Note that

ρM δM = Dim C middot idM and ρA δA = idA (250)

Lemma 222 The functors T and R as maps on the sets of morphisms have left inversesand thus are injective

Proof Let f A rarr B be a morphism in C We define a map Q R HomC2plusmn(R(A)

R(B)) rarr HomC(A B) by f prime rarr ρB T ( f prime) δA Then we have

Q R R( f ) = ρB T R( f ) δA = ρB δB f = f (251)

888 L Kong I Runkel

where we used naturality of δ and (250) in the second and third equalities respectivelySo Q R is a left inverse of R on morphisms Thus R is injective on morphisms Similarlylet g M rarr N be a morphism in C2plusmn We define a map QT HomC(T (M) T (N )) rarrHomC2plusmn(M N ) by gprime rarr (Dim C)minus1 middot ρN R(gprime) δM Then we have

QT T (g) = (Dim C)minus1 middot ρN RT (g) δM = (Dim C)minus1 middot ρN δN g = g (252)

So QT is a left inverse of T on morphisms Thus T is injective on morphisms Using (29) and (250) one can express the two inverse maps χminus1 χminus1 as follows

for f isin HomC2plusmn(M R(A)) and g isin HomC2plusmn(R(A)M)

χminus1( f ) = ρ T ( f ) χminus1(g) = T (g) δ (253)

By Proposition 25 and Lemma 27 R is both a lax and colax tensor functor In particularφR

0 1 times 1 rarr R(1) is given by

φR0 = χ (ψT

0 ) = R(ψT0 ) δ1 times 1 = id1times1 (254)

and φR2 R(A)otimes R(B) rarr R(A otimes B) by φR

2 = R(ρA otimes ρB) R(ψT2 ) δ which can

be expressed graphically as

φR2 =

oplusi jkisinI

sumα

A

A

Uori B

B

Uorj

Uork

α

times

Ui U j

α

Uk

(255)

Similarly ψ R0 R(1) rarr 1 times 1 is given by

ψ R0 = ρ1 R(φT

0 ) = Dim C id1times1 (256)

and ψ R2 R(A otimes B) rarr R(A) otimes R(B) by ψ R

2 = ρ R(φT2 ) R(δA otimes δB) which in

graphical notation reads

ψ R2 =

oplusi jkisinI

sumα

A

A Uori

B

B Uorj

Uork

α

times

Ui U j

α

Uk

dim Ui dim U j

dim Uk Dim C

(257)

If C has more than one simple object then R does not take the tensor unit of C to thetensor unit of C2plusmn and so is clearly not a tensor functor However we will show that R isstill a Frobenius functor This will imply that if A is a Frobenius algebra in C then

R(A) = (R(A)m R(A) ηR(A)R(A) εR(A)) (258)

Cardy Algebras and Sewing Constraints I 889

is a Frobenius algebra in C2plusmn where the structure morphisms were given in (217) and(218) In the case A = 1 it was proved in [Muuml1 Prop 41] (see also [Fr Lem 619]and [K1 Thm 52]) that (258) is a commutative simple symmetric normalised-specialFrobenius algebra in C2plusmn In fact given a Frobenius algebra A in C it is straightforwardto verify that the structure morphisms in (258) are precisely those of (A times 1)otimes R(1)cf Sect 22

Proposition 223 (R φR2 φ

R0 ψ

R2 ψ

R0 ) is a Frobenius functor

Proof Using the explicit graphical expression of φR2 φ

R0 ψ

R2 ψ

R0 it is easy to see that

the commutativity of the diagrams (25) and (26) are equivalent to the statement thatR(1) with structure morphisms as in (258) is a Frobenius algebra in C2plusmn The latterstatement is true by [Muuml1 Prop 41]

From Lemma 220 and Proposition 223 we see that T and R take Frobenius alge-bras to Frobenius algebras The following two propositions show how the properties ofFrobenius algebras are transported

Proposition 224 Let A be a Frobenius algebra in C2plusmn Then T (A) is a Frobenius algebrain C and

(i) A is symmetric iff T (A) is symmetric(ii) A is (normalised-)special iff T (A) is (normalised-)special

Proof For part (i) write A as a direct sum oplusNn=1 Al

n times Arn Then the maps IT (A) I prime

T (A) T (Aor) rarr T (A)or defined in (220) are given by

IT (A) = I primeT (A) = oplusN

n=1c(Aln)

or(Arn)

or (259)

Therefore by (221) T (A) = primeT (A) is equivalent to T (A) = T (prime

A) Since byLemma 222 T is injective on morphisms this proves part (i) Part (ii) can be checkedin the same way for example the condition mT (A) T (A) = ζ idT (A) is easily checkedto be equivalent to T (m A A) = ζ T (idA) Proposition 225 Let A be a Frobenius algebra A in C Then R(A) is a Frobeniusalgebra in C2plusmn and

(i) A is symmetric iff R(A) is symmetric(ii) A is (normalised-)special iff R(A) is (normalised-)special

Proof Recall that the structure morphisms of the Frobenius algebra R(A) are equal tothose of (A times 1) otimes R(1) Using this equality part (i) and (ii) follow because R(1) issymmetric and normalised-special For example

m R(A) R(A) = [(m A A)times id1] otimes idR(1) = R(m A A) (260)

so that m R(A) R(A) = ζ idR(A) is equivalent to R(m A A) = ζ R(idA) which byLemma 222 is equivalent to m A A = ζ idA

The functor R has one additional property not shared by T namely R takes absolutelysimple algebras to absolutely simple algebras We will see explicitly in Sect 33 thatthis is not true for T

890 L Kong I Runkel

Lemma 226 For a C-algebra A the map

R HomA|A(A A) rarr HomR(A)|R(A)(R(A) R(A)) (261)

given by f rarr R( f ) is well-defined and an isomorphism

Proof Since R is a lax tensor functor R(A) is naturally a R(A)-bimodule It is easy tosee that R in (261) is a well-defined map R(A) is also naturally a R(1)-bimodule whichcan be identified with the induced R(1)-bimodule structure on (A times 1)otimes R(1) wherethe left R(1) action on (Atimes1)otimes R(1) is given by (idAtimes1 otimesm R(1))(cminus1

Atimes1R(1)otimes idR(1))We have the following natural isomorphisms

HomR(1)|R(1)(R(A) R(A))sim=minusrarr HomC2plusmn(A times 1 R(A))

χminus1

minusminusrarr HomC(A A) (262)

which by (253) are given by for f isin HomR(1)|R(1)(R(A) R(A))

f rarr f prime = f (idAtimes1 otimes ηR(1)) rarr f primeprime = ρ T ( f prime) (263)

and its inverse is given by for g isin HomC(A A)

g rarr gprime = R(g) δ rarr gprimeprime = (idAtimes1 otimes m R(1)) (gprime otimes idR(1)) (264)

where gprimeprime is indeed a R(1)-bimodule map due to the commutativity of R(1) It is easy tocheck that gprimeprime = R(g) in (264) Therefore R gives an isomorphism from HomC(A A)to HomR(1)|R(1)(R(A) R(A)) Moreover one verifies that R(g) is an R(A)-bimodulemap iff g is an A-bimodule map In other words R g rarr R(g) gives an isomorphism

HomA|A(A A)sim=minusrarr HomR(A)|R(A)(R(A) R(A))

Corollary 227 Let A be a C-algebra

(i) A is absolutely simple iff R(A) is absolutely simple(ii) Let A be in addition Frobenius Then A is simple and special iff R(A) is simple and

special

Proof Part (i) immediately follows from Lemma 226 The statement of part (ii) withoutthe qualifier lsquosimplersquo is proved in Proposition 225 But as in the proof of (iii)rArr(i) inLemma 211 a special Frobenius algebra in a semi-simple category has a semi-simplecategory of bimodules and for a semi-simple C-linear category simple and absolutelysimple are equivalent Part (ii) then follows from part (i)

The following lemma will be needed in Sect 32 below to discuss the properties ofCardy algebras

Lemma 228 Let A be a Frobenius algebra in C Then (δA)lowast = ρA

Proof Recall from (250) that δA is a morphism A rarr T R(A) Since T and R are bothFrobenius functors T R(A) is a Frobenius algebra in C Substituting the definitions aftera short calculation one finds

Cardy Algebras and Sewing Constraints I 891

εT R(A) mT R(A) = Dim(C)oplusiisinI

A Uori Ui A U or

ı Uı

(265)

Substituting this in the definition of (δA)lowast gives again after a short calculation the mor-

phism ρA At an intermediate step one uses that the part of the morphism (δA)lowast which

is made up of Ui and Uı ribbons their duals and the basis morphisms λ(iı)0 andϒ(iı)0can be replaced by 1

dim Uimiddot dUi

3 Cardy Algebras

In this section we start by investigating the properties of Frobenius algebras which sat-isfy the so-called modular invariance condition We then give two definitions of a Cardyalgebra and prove their equivalence Finally in Sect 33 we study the properties of thesealgebras and state our main results

We fix a modular tensor category C Recall that C2plusmn is an abbreviation for C+ Cminus

31 Modular invariance In C2plusmn we define the object K and the morphism ω K rarr Kas

K =oplus

i jisinIUi times U j ω =

sumi jisinI

dim Ui dim U j

Dim C idUi timesU j (31)

They have the property (see eg [BK Cor 3111])

ω

K

(Uk times Ul )or

(Uk times Ul )or

Ui times U j

Ui times U j

= δik δ jlDim C

dim Ui dim U jbUi timesU j dUi timesU j (32)

892 L Kong I Runkel

Definition 31

(i) Let A B be objects of C2plusmn A morphism f A otimes B rarr B is called S-invariant iff

A

B

W

W

f =

A

B

W

W

f

K

ω(33)

holds for all for W isin C2plusmn(ii) A C2plusmn-algebra (Aclmcl ηcl) is called modular invariant iff θAcl = idAcl and mcl

is S-invariant

Lemma 32 The morphism f A otimes B rarr B is S-invariant if and only if

A

B

Ui times U j

Ui times U j

f = Dim Cdim Ui dim U j

sumα

A

B

B

f

Ui times U j

Ui times U j

α

α

(34)

holds for all i j isin I

Proof Condition (33) holds for all W iff it holds for all W = Ui times U j i j isin I so itis enough to show that the right-hand side of (33) with W = Ui times U j is equal to theright-hand side of (34) Recall the notation for basis morphisms in (236) Starting from(33) write

idBor otimes idUi timesU j =⎛⎝sum

klα

(b(ktimeslα)

B

)or (

bB(ktimeslα)

)or⎞⎠ otimes idUi timesU j (35)

and then apply (32) The graphical representation of the resulting morphism can bedeformed to give (34) Remark 33 As shown in [K3 Sect 61] the modular invariance condition of a C2plusmn-alge-bra exactly coincides with the modular invariance condition for torus 1-point correlationfunctions of a genus-01 closed CFT In particular the condition θAcl = idAcl is equiv-alent to invariance under the modular transformation T τ rarr τ + 1 and the condition(34) with f = mcl is equivalent to invariance under S τ rarr minus 1

τ Combining the

modular invariance condition with the genus zero properties of a genus-01 closed CFTresults in a modular invariant commutative symmetric Frobenius algebra in C2plusmn

Cardy Algebras and Sewing Constraints I 893

Let Acl be a modular invariant C2plusmn-algebra Evaluating (34) for f = mcl composingit with ηcl otimes idUi timesU j and taking the trace implies the following identity

Zi j = 1

DimCAclUi timesU j where

(36)

Zi j = dimC HomC2plusmn(UitimesU j Acl)

Decomposing Acl into simple objects this gives

Zi j =sum

klisinISik Zkl Sminus1

l j where Si j = si jradic

DimC (37)

which in CFT terms is of course nothing but the invariance of the torus partition functionunder the modular S-transformation

The following theorem gives a simple criterion for modular invariance

Theorem 34 Let Acl be a haploid commutative symmetric Frobenius algebra in C2plusmn

(i) If Acl is modular invariant then dim Acl = Dim C(ii) If dim Acl = Dim C then Acl is special and modular invariant

Proof Part (i) Since Acl is haploid for i = j = 0 Equation (36) reduces to 1 =dim AclDimCPart (ii) By the same reasoning as in the proof of (iii)rArr(i) in Lemma 211 one showsthat Acl is special Thus mcl cl = ζclidAcl for some ζcl = 0

By [KO Thm 45] the category (C2plusmn)locAcl

of local Acl-modules is again a mod-

ular tensor category and Dim (C2plusmn)locAcl

= (Dim C dim Acl)2 (see [Fr Prop 321 amp

Rem 323] for the same statement in the notation used here) Thus by assumption wehave Dim(C2plusmn)loc

Acl= 1 It then follows from [ENO1 Thm 23] that up to isomorphism

(C2plusmn)locAcl

has a unique simple object (namely the tensor unit) In other words every simplelocal Acl-module is isomorphic to Acl (seen as a left-module over itself)

We have the following isomorphisms between morphism spaces [FS Prop 47 amp 411]

HomAcl(Acl otimes (Ui times U j ) Acl) sim= HomC2plusmn(Ui times U j Acl)

HomAcl(Acl Acl otimes (Ui times U j )) sim= HomC2plusmn(AclUi times U j )

Using these to transport the bases (236) from the right to the left we obtain bases bα(i j)αof HomAcl(Acl otimes (Ui times U j ) Acl) and b(i j)

β β of HomAcl(Acl Acl otimes (Ui times U j )) Thesecan be expressed graphically as

894 L Kong I Runkel

bα(i j) =

Acl

Acl

Acl

Ui times U j

α

b(i j)β = DimC

dim Ui dim U j

1

ζcl

Acl

Ui times U jAcl

Acl

β

(38)

where the nonzero factor in b(i j)β is included for convenience Notice that bα(i j) b(i j)

β isa left Acl-module map Since Acl is simple as a left module over itself we have

bα(i j) b(i j)β = λαβ idAcl (39)

for some λαβ isin C By computing tr(bα(i j) b(i j)β ) it is easy to verify that λαβ = δαβ

We will now prove the following identity

1

ζcl

Acl

Acl

Ui times U j

Ui times U j

Acl

=sumα

DimCdim Ui dim U j

1

ζcl

Acl

Ui times U j

Ui times U j

Acl

Acl

Acl

α

α

(310)

One checks that the left-hand side of this equation is an idempotent which we denote byζminus1

cl PlAcl(Ui timesU j ) cf [Fr Sect 31] By [Fr Prop 41] the image Im(ζminus1

cl PlAcl(Ui timesU j ))

is a local Acl-module and hence isomorphic to AoplusNcl for some N isin Zge0

All left-module morphisms from Acl to Aclotimes(Ui timesU j ) are linear combinations of the

b(i j)β Furthermore one verifies that ζminus1

cl PlAcl(Ui timesU j )b(i j)

β = b(i j)β Therefore the b(i j)

β

describe precisely the image of the idempotent ie ζminus1cl Pl

Acl(Ui timesU j ) = sum

α b(i j)α bα(i j)

which is nothing but (310) Composing (310) with ζcl middot εcl otimes idUi timesU j from the left (iefrom the top) produces (34)

In addition since Acl is commutative symmetric Frobenius it satisfies θAcl = idAcl

[Fr Prop 225] Altogether this shows that Acl is modular invariant Remark 35 As we were writing this paper we heard that the results in Theorem 34were obtained independently by Kitaev and Muumlger [Ki]

Remark 36 Setting i = j = 0 in (36) gives the identity dim Acl = Z00 Dim C [KRProp 23] Combining this with Theorem 34 (ii) one may wonder if a general modularinvariant commutative symmetric Frobenius algebra Acl in C2plusmn is isomorphic to a directsum of simple such algebras However this is not so For example one can take the com-mutative symmetric Frobenius algebra Acl = C[x]x2 in the category of vector spacesequipped with the non-degenerate trace ε(ax + b) = a In this case the modular invari-ance condition holds automatically but Acl is clearly not a direct sum of two algebras

Cardy Algebras and Sewing Constraints I 895

For a general modular tensor category C the algebra C[x]x2 R(1) understood as analgebra in C2plusmn via the braided monoidal isomorphism Vect f (C) C2plusmn rarr C2plusmn providesanother counter-example

32 Two definitions Define a morphism PlA A rarr A for a Frobenius algebra A in C

or C2plusmn as follows [Fr Sect 24]

PlA =

A

A

A

m

(311)

If A is also commutative and obeys m A A = ζA idA we have PlA = ζA idA In partic-

ular this holds if A is commutative and special Using the fact that the Frobenius algebraR(1) is commutative and normalised-special one can check that Pl

R(A) R(A) rarr R(A)takes the following form

PlR(A) =

oplusiisinI

A

A

Uori

Uori

A

m

times

Ui

Ui

(312)

With these ingredients we can now give the first definition of a Cardy C|C2plusmn-algebrawhich was introduced in [K3 Def 514] cf Remark 315 below

Definition 37 (CardyC|C2plusmn-algebra I) A CardyC|C2plusmn-algebra is a triple (Aop|Acl ιcl-op)where (Aclmcl ηclcl εcl) is a modular invariant commutative symmetric Frobe-nius C2plusmn-algebra (Aopmop ηopop εop) is a symmetric Frobenius C-algebra andιcl-op Acl rarr R(Aop) an algebra homomorphism such that the following conditionsare satisfied

(i) Centre condition

Acl R(Aop)

R(Aop)

R(Aop)

ιcl-op

m R(Aop)

=

Acl

ιcl-op

R(Aop)

R(Aop)

R(Aop)

m R(Aop)

(313)

896 L Kong I Runkel

(ii) Cardy condition

ιcl-op ιlowastcl-op =

R(Aop)

R(Aop)

R(Aop)

(314)

Remark 38

(i) The name ldquoCardy C|C2plusmn-algebrardquo in Definition 37 was chosen because many ofthe important ingredients were first studied by Cardy the modular invariance ofthe closed theory [C1] the consistency of the annulus amplitude [C2] and thebulk-boundary OPE [CL] On the other hand the boundary-boundary OPE andthe OPE analogue of the centre condition were first considered in [Lw]

(ii) One can easily see that in the special case that C is the category Vect f (C) of finite-dimensional C-vector spaces a Cardy C|C2plusmn-algebra gives exactly the algebraicformulation of two-dimensional open-closed topological field theory over C (cfRemark 614 in [K3]) see [Lz Sect 48] [Mo Thm 11] [AN Thm 45] [LPCor 43] [MS Sect 22] When passing to a general modular tensor category Cthere are two important differences to the two-dimensional topological field the-ory Firstly the algebras Acl and Aop now live in different categories which inparticular affects the formulation of the centre condition and the Cardy conditionSecondly the modular invariance condition has to be imposed on Acl In the caseC = Vect f (C) modular invariance holds automatically

Definition 39 A homomorphism of Cardy C|C2plusmn-algebras (A(1)op |A(1)cl ι(1)cl-op) rarr (A(2)op

|A(2)cl ι(2)cl-op) is a pair ( fop fcl) of Frobenius algebra homomorphisms fop A(1)op rarr A(2)op

and fcl A(1)cl rarr A(2)cl such that the diagram

A(1)cl

fcl

ι(1)cl-op

A(2)cl

ι(2)cl-op

R(A(1)op )

R( fop) R(A(2)op )

(315)

commutes

Remark 310 Since a homomorphism of Frobenius algebras is invertible (cf Lemma218 (iv)) a homomorphism of Cardy algebras is always an isomorphism

For a homomorphism ( fop fcl) of Cardy C|C2plusmn-algebras using the commutativity of(315) and the fact that fcl and fop are both algebra and coalgebra homomorphisms itis easy to show that (315) commutes iff

R(A(1)op )R( fop)

(ι(1)cl-op)

lowast

R(A(2)op )

(ι(2)cl-op)

lowast

A(1)cl

fcl A(2)cl

(316)

Cardy Algebras and Sewing Constraints I 897

commutesLet (Aop|Acl ιcl-op) be a Cardy C|C2plusmn-algebra Define the morphism

ιcl-op = χminus1(ιcl-op) T (Acl) minusrarr Aop (317)

Decompose Acl as Acl = oplusNn=1Cl

n times Crn such that Cl

1 times Cr1 = ηcl(1 times 1) We use ι(n)cl-op

to denote the restriction of ιcl-op to Cln times Cr

n and ι(n)cl-op to denote the restriction of ιcl-op

to Cln otimes Cr

n We introduce the following graphical notation

ι(n)cl-op =

Cln Cr

n

Aop

(318)

By (243) ιcl-op can be expressed in terms of ιcl-op as follows

ιcl-op =Noplus

n=1

oplusiisinI

sumα

Cln

Crn

Aop

α

Uori

times α

Crn

Ui

(319)

Lemma 311 The centre condition (313) is equivalent to the following condition in C

Cln Cr

n

Aop

Aop

Aop

=

Cln Cr

n

Aop

Aop

Aop

for n = 1 N (320)

Proof First insert (319) and the definition (217) of m R(Aop) into (313) Then apply thecommutativity of R(1) to the left hand side of (313) The equivalence between (313)and (320) follows immediately Remark 312 The centre condition (320) is very natural from the open-closed confor-mal field theory point of view Correlators on the upper half plane are expressed in termsof conformal blocks on the full complex plane The objects Cl

n and Crn are associated to

the field insertion at a point z in the upper half plane and at the complex conjugate pointz in the lower half plane respectively The object Aop corresponds to a field inserted ata point r on the real axis The centre condition (320) simply says that the correlationfunctions in the disjoint domains |z| gt r gt 0 and r gt |z| gt 0 are analytic continuationsof each other see [K2 Prop 118]

898 L Kong I Runkel

Recall that we define ιlowastcl-op Aop rarr T (Acl) as in (230) We introduce the graphicalnotation

ιlowastcl-op =Noplus

n=1

Cln Cr

n

Aop

=Noplus

n=1

Cln Cr

n

Aop

Aop T (minus1Acl) (321)

where the second equality follows from (215) (221) and (259)

Lemma 313 The Cardy condition (314) is equivalent to the following identity in C

Noplusn=1

DimCdim Ui

sumα

Cln

Crn

Crn

α

α

Aop

Aop Uori

Uori

=Aop

Aop

Aop

Uori

Uori

op

mop for all i isin I (322)

Proof By (244) (312) and (319) it is easy to see that (322) is equivalent to thefollowing identity

ιcl-op χ(ιlowastcl-op) = PlR(Aop)

(323)

Therefore it is enough to show that

χ(ιlowastcl-op) = ιlowastcl-op (324)

We have

χminus1(ιlowastcl-op)(1)= T (ιlowastcl-op) δ (2)= T (ιcl-op)

lowast δlowastlowast (3)= (ρ T (ιcl-op)

)lowast (4)= (ιcl-op

)lowast (325)

In step (1) we use the expression (253) for χminus1 step (2) follows from Lemma 218 (v)and Lemma 219 Step (3) is Lemma 218 (i) and Lemma 228 and finally step (4)amounts to substituting (253) and (317) Acting with χ on both sides of the aboveequality produces (324)

Combining Lemmas 311 and 313 and Proposition 216 we obtain the followingequivalent definition of Cardy C|C2plusmn-algebra (recall the graphical notation (318) forιcl-op and (321) for ιlowastcl-op)

Cardy Algebras and Sewing Constraints I 899

Definition 314 (CardyC|C2plusmn-algebra II) A CardyC-algebra is a triple (Aop|Acl ιcl-op)where Acl is a commutative symmetric Frobenius C2plusmn-algebra satisfying property (34)with f = mcl Aop is a symmetric Frobenius C-algebra and ιcl-op T (Acl) rarr Aop isan algebra homomorphism satisfying the conditions (320) and (322)

Remark 315 Up to a choice of normalisation Definition 314 is the same as the orig-inal one in [K3 Def 613] The difference between the two definitions is the factorDim C dim Ui on the left hand side of (322) which in [K3 Def 613] is given byradic

Dim C dim Ui The two definitions are related by rescaling the coproduct cl andcounit εcl of Acl by 1

radicDim C and

radicDim C respectively We chose the convention in

(322) to remove all dimension factors from the expression (314) for the Cardy condition

33 Uniqueness and existence theorems In this subsection we investigate the structureof Cardy algebras We start with the following proposition which when combined withthe results of part II provides an alternative proof of [Fj Prop 422]

Proposition 316 Let (Aop|Acl ιcl-op) be a Cardy C|C2plusmn-algebra If Acl is simple anddim Aop = 0 then Aop is simple and special

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis therefore haploid Restricting the Cardy condition (322) to the case Ui = 1 andcomposing both sides with εop from the left we see that εop kills all terms associated toU j times 1 isin Acl in the sum except for a single 1 times 1 term Thus we obtain the followingidentity

β εop = dAop (mop otimes idAorop) (idAop otimes bAop) (326)

where β isin C Composing with ηop from the right in turn implies that βεop ηop =dim Aop which is nonzero by assumption Thus also β = 0 and εop is a nonzero multipleof the morphism on the right hand side of (326) By [FRS Lem 311] Aop is special

Since Aop is a special Frobenius algebra Aop is semi-simple as an Aop-bimodule(apply [FS Prop 524] to Aop tensored with its opposite algebra) Suppose Aop is not

simple so that we can write Aop = A(1)op oplus A(2)op for nonzero Aop-bimodules A(1)op and A(2)op We denote the canonical embeddings and projections associated to this decompositionas ι12 and π12 We have the identities

mop (ι1 otimes ι2) = 0 εop ηop =2sum

i=1

εop ιi πi ηop (327)

The first identity follows since π1 mop (ι1 otimes ι2) = 0 (as mop gives the left action of

Aop on Aop and hence it preserves A(2)op ) and similarly π2 mop (ι1 otimes ι2) = 0 Thesecond identity is just the completeness of ι12 π12

Since εop ηop = 0 without losing generality we can assume εop ι1 π1 ηop = 0Using that π2 is a bimodule map we compute

π2 [LHS of (322)]Ui =1 ι1 π1 ηop = π2 [RHS of (322)]Ui =1 ι1 π1 ηop

= PlA(2)op

π2 ι1 π1 ηop = 0 (328)

900 L Kong I Runkel

On the other hand using that Acl is haploid that ιcl-op is an algebra map and thatι lowastcl-op is a coalgebra map one can check that the left-hand side of (328) is equal toλ(εop ι1 π1 ηop) π2 ηop for some λ = 0 This implies that π2 ηop = 0 Thusηop = sum

i ιi πi ηop = ι1 π1 ηop Hence we have

0 = π2 ι2 = π2 mop (ηop otimes ι2) = π2 mop ((ι1 π1 ηop)otimes ι2) (329)

However the right-hand side is zero by (327) This is a contradiction and hence Aopmust be simple

To formulate the next theorem we need the notion of the full centre of an algebra [FjDef 49] Recall that an algebra A in a braided tensor category has a left centre and aright centre [VZO] both of which are sub-algebras of A Of these two we will onlyneed the left centre The following definition is [Fr Def 231] which in our setting isequivalent to that of [VZO]

Definition 317 Let A be a symmetric special Frobenius algebra such that m A A =ζA idA

(i) The left centre Cl(A) of A is the image of the idempotent ζminus1A Pl

A(ii) The full centre Z(A) is Cl(R(A))

That ζminus1A Pl

A is an idempotent follows from [FRS Lem 52] when keeping track ofthe factors ζA ([FRS] assumes normalised-special ie ζA = 1) Note that Cl(A) is againan object of C while Z(A) is an object of C2plusmn Let el

A Cl(A) rarr A be the embeddingof Cl(A) into A The left centre is in fact the maximal subobject of A such that

m A cAA (elA otimes idA) = m A (330)

see [Fr Lem 232] This observation explains the name left centre and also makes theconnection to [O Def 15]

The full centre is by definition the image of the idempotent ζminus1A Pl

R(A) R(A) rarrR(A) Since C2plusmn is abelian the idempotent splits and we obtain the embedding andrestriction morphisms

e Z(A) rarr R(A) and r R(A) Z(A) (331)

which obey r e = idZ(A) and e r = ζminus1A Pl

R(A) It follows from Proposition 225 and

[Fr Prop 237] that Z(A) is a commutative symmetric Frobenius algebra in C2plusmn withstructure morphisms5

m Z(A) = r m R(A) (e otimes e) ηZ(A) = r ηR(A)

Z(A) = ζA middot (r otimes r) R(A) e εZ(A) = ζminus1A middot εR(A) e

(332)

Moreover if A is simple then Z(A) is simple and if A is simple and dim A = 0 thenZ(A) is simple and special The normalisation of the counit is such that

εZ(A) ηZ(A) = ζminus2A dim A DimC (333)

5 The normalisation of product and unit is the standard one The factors in the coproduct and counit haveto be included in order for (A|Z(A) e) to be a Cardy algebra see Theorem 318 below The normalisation ofthe counit enters the Cardy condition (314) through the definition of ( middot )lowast

Cardy Algebras and Sewing Constraints I 901

Theorem 318 Let A be a special symmetric Frobenius C-algebra Then (A|Z(A) e)is a Cardy C|C2plusmn-algebra

The proof of this theorem makes use the following two lemmas

Lemma 319 e Z(A) rarr R(A) is an algebra map and elowast = ζA middot r

Proof It follows from [Fr Lem 229] (or by direct calculation using in particularm R(A) R(A) = ζAidR(A)) that

m R(A) (e otimes e) = ζminus1A middot Pl

R(A) m R(A) (e otimes e) (334)

Substituting e r = ζminus1A Pl

R(A) shows that e is compatible with multiplication For theunit one finds

e ηZ(A) = e r ηR(A) = ζminus1A Pl

R(A) ηR(A) = ηR(A) (335)

Thus e is an algebra map For the second statement one computes

elowast (1)= ζA

R(A)

r

R(A)

R(A)

Z(A)

R(A)

Z(A)

r

e

e

(2)=

R(A)

r

R(A)

R(A)

R(A)

Z(A)

(3)=

R(A)

r

R(A)

Z(A)

(4)= ζA middot r (336)

where in (1) the definitions (230) and (332) have been substituted step (2) is e r = ζminus1

A PlR(A) step (3) uses that R(A) is symmetric Frobenius and step (4) is again

e r = ζminus1A Pl

R(A) Lemma 320 Let A be a symmetric Frobenius algebra in C The morphism

PlR(A) m R(A)

(Pl

R(A) otimes PlR(A)

) R(A)otimes R(A) minusrarr R(A) (337)

is S-invariant

The proof of this lemma is a slightly lengthy explicit calculation and has been deferredto Appendix A2

902 L Kong I Runkel

Proof of Theorem 318 That e is an algebra map was proved in Lemma 319 The centrecondition (313) holds by property (330) of the left centre The Cardy condition (314)also is an immediate consequence of Lemma 319

ιcl-op ιlowastcl-op = e (ζA r) = PlR(A) (338)

The full centre Z(A) is a commutative symmetric Frobenius algebra It remains to provemodular invariance That θZ(A) = idZ(A) is implied by commutativity and symmetryof Z(A) [Fr Prop 225] The S-invariance condition (33) follows from Lemma 320In (337) substitute Pl

R(A) = ζA e r and then put the resulting morphism into (33)Compose the resulting equation with e otimes idW from the right (ie from the bottom) andsubstitute the definition (332) of m Z(A) This results in the statement that m Z(A) isS-invariant

The following theorem is analogous to [LR Prop 29] and [Fj Thm 426] whichroughly speaking answer the question under which circumstances the restriction of atwo-dimensional conformal field theory to the boundary already determines the entireconformal field theory The first work is set in Minkowski space and uses operator alge-bras and subfactors while the second work is set in Euclidean space and uses modulartensor categories

Theorem 321 Let (A|Acl ιcl-op) be a Cardy C|C2plusmn-algebra such that dim A = 0 andAcl is simple Then A is special and (A|Acl ιcl-op) sim= (A|Z(A) e) as Cardy algebras

Proof By Proposition 316 A is simple and special Since Acl is simple the algebramap ιcl-op Acl rarr R(A) is either zero or a monomorphism But ιcl-op ηcl = ηR(A)and so ιcl-op = 0 Thus ιcl-op is monic By Lemma 218 (ii) ζminus1

A ιlowastcl-op is epi The Cardycondition (314) implies

ιcl-op ζminus1A ιlowastcl-op = ζminus1

A PlR(A) = e r (339)

Composing this with e r from the left yields e r ιcl-op ζminus1A ιlowastcl-op = e r =

ιcl-op ζminus1A ιlowastcl-op Since ζminus1

A ιlowastcl-op is epi we have

e r ιcl-op = ιcl-op (340)

Actually (340) also follows from (313) and specialness of R(A) We will prove that

( fop fcl) (A|Acl ιcl-op) minusrarr (A|Z(A) e) where fop = idA fcl = r ιcl-op

(341)

is an isomorphism of Cardy algebrasfcl is an algebra map Compatibility with the units follows since ιcl-op is an algebramap

fcl ηcl = r ιcl-op ηcl = r ηR(A) = ηZ(A) (342)

Compatibility with the multiplication also follows since ιcl-op is an algebra map

m Z(A) ( fcl otimes fcl) = r m R(A) (e otimes e) (r otimes r) (ιcl-op otimes ιcl-op)

= r m R(A) (ιcl-op otimes ιcl-op) = r ιcl-op mcl = fcl mcl (343)

where in the second step we used (340)

Cardy Algebras and Sewing Constraints I 903

fcl is an isomorphism As above since fcl is an algebra map and since Acl is simple fcl

has to be monic By Lemma 319 rlowast = ζminus1A e Thus f lowast

cl = ιlowastcl-op rlowast = ζminus1A ιlowastcl-op e and

fcl f lowastcl = r ιcl-op ζminus1

A ιlowastcl-op e = r e r e = idZ(A) (344)

and so fcl is also epi and hence isofcl is a coalgebra map Since fcl is an algebra map so is f minus1

cl By (344) f minus1cl = f lowast

cland by Lemma 218 (iii) this implies that fcl is a also coalgebra mapThe diagram (315) commutes Commutativity of (315) is equivalent to e fcl = ιcl-opwhich holds by (340)

Let A be a special symmetric Frobenius algebra So far we have seen that (A|Z(A) e)is a Cardy algebra and that all Cardy algebras with Aop = A and simple Acl are of thisform It is now natural to ask if every simple Acl does occur as part of a Cardy algebraThe following theorem provides an affirmative answer Recall that for an A-left moduleM the object Mor otimesA M is an algebra (see eg [KR Lem 42])

Theorem 322 If Acl is a simple modular invariant commutative symmetric FrobeniusC2plusmn-algebra then there exist a simple special symmetric Frobenius C-algebra A and amorphism ιcl-op Acl rarr R(A) such that

(i) Acl sim= Z(A) as Frobenius algebras(ii) (A|Acl ιcl-op) is a Cardy C|C2plusmn-algebra

(iii) T (Acl) sim= oplusκisinJ Morκ otimesA Mκ as algebras where MκκisinJ is a set of representa-

tives of the isomorphism classes of simple A-left modules

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis haploid It then follows from Theorem 34 that Acl is special By Proposition 224T (Acl) is a special symmetric Frobenius algebra in C Thus T (Acl) = oplusi Ai where theAi are simple symmetric Frobenius algebras We will show that at least one of the Aiis special Since T (Acl) is special we have mT (Acl) T (Acl) = ζ idT (Acl) for someζ isin C

times Restricting this to the summand Ai shows mi i = ζ idAi FurthermoreεT (Acl) ηT (Acl) = ξ id1 for some ξ isin C

times But εT (Acl) ηT (Acl) = sumi εi ηi and so

at least one of the εi ηi has to be nonzero Therefore at least one of the Ai is speciallet A equiv Ai be this summand We denote the embedding A rarr T (Acl) by e0 and therestriction T (Acl) A by r0 Notice that r0 is an algebra homomorphism Define

ιcl-op = χ (r0) Acl minusrarr R(A) (345)

By Proposition 216 ιcl-op is an algebra homomorphism Next we verify the centre con-dition (313) or rather its equivalent form (320) By substituting the definitions onecan convince oneself that the commutativity mcl cAclAcl = mcl of Acl in C2plusmn implies thecondition mT (Acl) = mT (Acl) in C see [K2 Prop 36] Here T (Acl)otimes T (Acl) rarrT (Acl)otimes T (Acl) is given by

=oplusmn

(idCl

notimes cCl

m Crnotimes idCr

m

)

(cCl

m Clnotimes cminus1

Crn C

rm

)

(idCl

motimes cminus1

Cln C

rm

otimes idCrn

)

(346)

and we decomposed Acl as Acl = oplusnCln times Cr

n As a consequence we obtain the identity

r0 mT (Acl) (idAcl otimes e0) = r0 mT (Acl) (idAcl otimes e0) (347)

904 L Kong I Runkel

Using that r0 is an algebra map and that by definition ιcl-op = r0 we obtain (320)In order to show that (A|Acl ιcl-op) is a Cardy algebra it remains to show that the

Cardy condition (314) is satisfied We will demonstrate this via a detour by first provingthat Acl sim= Z(A) as Frobenius algebras

Recall the notations e and r given in (331) Using the centre condition (320) onecan check that Pl

R(A) ιcl-op = m R(A) R(A) ιcl-op By specialness of A we have

m A A = ζA idA and so together with e r = ζminus1A Pl

R(A) we get

e r ιcl-op = ιcl-op (348)

Next consider the morphism

fcl = r ιcl-op Acl minusrarr Z(A) (349)

By the same derivation as in (342) and (343) one sees that fcl is an algebra map Inparticular fcl ηcl = ηZ(A) = 0 and so fcl = 0 Since Acl is simple fcl has to be amonomorphism

By the same argument as used in the proof of Theorem 34 (ii) up to isomorphismAcl is the unique simple local Acl-(left-)module The algebra monomorphism fcl turnsZ(A) into an Acl-module Since Z(A) is commutative it is local as an Acl-module andso Z(A) sim= AoplusN

cl for some N ge 1 By construction A is a simple special symmetricFrobenius algebra Proposition 225 and Corollary 227 show that R(A) inherits all theseproperties and thus Z(A) is simple (see the comment below Eq (332)) By Theorem318 Z(A) is modular invariant and then by Theorem 34 (i) dim Z(A) = Dim C Thisimplies that N = 1 in Z(A) sim= AoplusN

cl and so fcl is in fact an isomorphismSince Acl and Z(A) are both haploid we have εZ(A) fcl = ξ εcl for some ξ isin C

timesThe counit uniquely determines the Frobenius structure on Acl and Z(A) (see eg [FRSLemma 37]) so that fcl is a coalgebra isomorphism iff ξ = 1 To compute ξ we composethe above identity with ηcl from the right Defining ζcl via εcl ηcl = ζminus1

cl Dim C middot id1 andusing (333) gives ξ = dim A ζclζ

2A By rescaling the comultiplication and the counit

of A and consequently changing ζA we can always achieve ξ = 1 This proves part (i)of the theorem

Equation (348) implies that ιcl-op = e fcl Since fcl is an isomorphism of Frobeniusalgebras by Lemmas 218 and 319 we have

ιcl-op ιlowastcl-op = e fcl f lowastcl elowast = ζA e r = Pl

R(A) (350)

Thus (A|Acl ιcl-op) is a Cardy algebra This proves part (ii) of the theoremPart (iii) can be seen as follows By [KR Prop 43] T Z(A) sim= oplusκisinJ Mor

κ otimesA Mκ asalgebras Together with the observation that T ( fcl) T (Acl) rarr T Z(A) is an isomor-phism of algebras this proves part (iii) Remark 323 Part (i) of Theorem 322 was announced by Muumlger [Muuml2] We provide anindependent proof in the setting of Cardy algebras

The above theorem together with Lemma 211 and Theorem 34 shows that a sim-ple commutative symmetric Frobenius C2plusmn-algebra Acl with dim Acl = Dim C is alwayspart of a Cardy algebra (Aop|Acl ιcl-op) for some simple special symmetric Frobeniusalgebra Aop in C However the above proof also illustrates that Aop is not unique Thisraises the question how two Cardy algebras with a given Acl can differ This question isanswered by [KR Thm 11] which in the present framework can be restated as follows

Cardy Algebras and Sewing Constraints I 905

Theorem 324 If (A(i)op |A(i)cl ι(i)cl-op) i = 1 2 are two Cardy C|C2plusmn-algebras such that

A(i)cl is simple and dim A(i)op = 0 for i = 1 2 then A(1)clsim= A(2)cl as algebras if and only

if A(1)op and A(2)op are Morita equivalent

Proof Theorem 11 in [KR] is stated for A(i)op being non-degenerate algebras and A(i)cl =Z(A(i)op ) for i = 1 2 By Proposition 316 A(i)op are simple and special for i = 1 2

Then by [KR Lem 21] A(i)op are non-degenerate algebras By Theorem 321 we have

A(i)clsim= Z(A(i)op ) as Frobenius algebras Finally by [KR Thm 11] Z(A(1)op ) sim= Z(A(2)op )

as algebras iff A(1)op and A(2)op are Morita equivalent Let Cmax(C2plusmn) be the set of equivalence classes [B] of simple modular invariant

commutative symmetric Frobenius algebras B in C2plusmn Two such algebras B and B prime areequivalent if B and B prime are isomorphic as algebras (but not necessarily as Frobenius alge-bras) Let Msimp(C) be the set of Morita classes of simple special symmetric Frobeniusalgebras in C Define the map z Msimp(C) rarr Cmax(C2plusmn) by z A rarr [Z(A)] whereA denotes the Morita class of A From Theorem 322 (i) and [KR Thm 11] we learn

Corollary 325 The map z Msimp(C) rarr Cmax(C2plusmn) is a bijection

A Appendix

A1 Proof of Lemma 27 We will show that if (F ψ F2 ψ

F0 ) is a colax tensor functor

from C1 to C2 then (G φG2 φ

G0 ) is a lax tensor functor from C2 to C1 Applying this

result to the opposed categories then gives the converse statementWe need to show that φG

0 and φG2 make the diagrams (21) and (22) commute We

first prove the commutativity of (21) Consider the following diagram

G(A)otimes (G(B)otimes G(C))

GψF2 δ

idG(A)otimesφG2 G(A)otimes G(B otimes C)

GψF2 δ

G(FG(A)otimes F(G(B)otimes G(C)))

G(F(idG(A))otimesF(φG2 ))

G(idFG(A)otimesψ F2 )

G(FG(A)otimes FG(B otimes C))

G(ρAotimesρBotimesC )

G(FG(A)otimes (FG(B)otimes FG(C)))

G(ρAotimes(ρBotimesρC )) G(A otimes (B otimes C))

(A1)

The top subdiagram is commutative because of the naturality of Gψ F2 δ The commu-

tativity of the bottom subdiagram follows from the following identities

(ρB otimes ρC ) ψ F2 = (ρB otimes ρC ) ψ F

2 ρF Fδ

= ρBotimesC FG(ρB otimes ρC ) FG(ψ F2 ) Fδ

= ρBotimesC F(φG2 ) (A2)

as a map F(G(B) otimes G(C)) rarr B otimes C The commutativity of (A1) implies that thecomposition of maps in the left column in (21) can be replaced by

G(ρA otimes (ρB otimes ρC )) G((idFG(A) otimes ψ F

2 ) ψ F2

) δ (A3)

906 L Kong I Runkel

Similarly we can show that the composition of maps in the right column in (21) can bereplaced by

G((ρA otimes ρB)otimes ρC ) G((ψ F

2 otimes idFG(C)) ψ F2

) δ (A4)

Using the commutativity of (23) it is easy to see that (21) with the left and right col-umns of (21) replaced by (A3) and (A4) respectively is commutative Hence (21) iscommutative

Now we prove the commutativity of the first diagram in (22)

φG2 (φG

0 otimes idG(A))

(1)= G(ρ12 otimes ρA) Gψ F2 δ

[(Gψ F

0 δ11)otimes idG(A)

]

(2)= G(ρ12 otimes ρA) Gψ F2 G F(Gψ0 otimes idG(A)) G F(δ11 otimes idG(A)) δ

(3)= G(ρ12 otimes ρA) G(FG(ψ F0 )otimes idFG(A)) Gψ F

2 G F(δ11 otimes idG(A)) δ(4)= G(id12 otimes ρA) G

([ψ F

0 ρF(11) (Fδ)11] otimes idFG(A)

) Gψ F

2 δ(5)= G(id12 otimes ρA) G(ψ F

0 otimes idFG(A)) Gψ F2 δ

(6)= G(id12 otimes ρA) G(lminus1FG(A)) G F(lG(A)) δ

(7)= G(lminus1A ) GρA δG lG(A)

(8)= G(lminus1A ) lG(A) (A5)

where in step (1) we substituted the definition of φG0 φ

G2 given in (210) in step (2)

we used the naturality of δ in step (3) we used the naturality of Gψ F2 in step (4) we

switched the position between Gρ12 and G FG(ψ F0 ) and the position between Gψ2 and

G F(δ12 otimes idG(A)) using the naturality of ρ and FψG2 respectively in step (5) we applied

the second identity in (28) in step (6) we used (24) in step (7) we used the naturalityof lminus1 and δ in step (8) we used the first identity in (28)

The proof of the commutativity of the second diagram in (22) is similar Thus wehave shown that G is a lax tensor functor

A2 Proof of Lemma 320 To prepare the proof recall that for a given object B isin Cthe modular group P SL(2Z) acts on the space oplusi HomC(B otimes Ui Ui ) see eg [BKSect 31] and [K3 Eq (455)] We will only need the action of S and Sminus1 Let f isinoplusi HomC(B otimes Ui Ui ) Then

S oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A6)

Cardy Algebras and Sewing Constraints I 907

Sminus1 oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A7)

By Lemma 32 to establish that (337) is S-invariant it is enough to prove the identity(34) when f is given by (337) Using (A6) and (A7) we can see that Eq (34) simplysays that oplusi j [RHS of (34)] is invariant under the action of S times S Consider the elementg of oplus jkisinIHomC2plusmn(R(A)otimes (Uor

j times Uk)Uorj times Uk) given by

g =oplusjkisinI

sumα

R(A)

R(A)

R(A)

R(A)

Uorj times Uk

Uorj times Uk

α

α

(A8)

By the above arguments proving S-invariance of (337) is equivalent to proving invari-ance of g under the action of S times S

For i isin I we denote by gi the component of g in(oplus jisinIHomC(A otimes Uor

i otimes Uorj U

orj )

)otimes (opluskisinIHomC(Ui otimes UkUk))

We view the second Hom-space in the above tensor product as a Hom-space in C+ insteadof Cminus It is enough to show that gi is invariant under the action of S times Sminus1 Note thatthe action of Sminus1 in C+ is equivalent to that of S in Cminus

The morphism gi can be canonically identified with a bilinear pairing

( middot middot )i ⎛⎝oplus

jisinIHomC(Uor

j A otimes Uori otimes Uor

j )

⎞⎠

times(oplus

kisinIHomC(UkUi otimes Uk)

)minusrarr C (A9)

as follows For h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) we set

(h1 h2)i = (dim U j dim Uk)minus1 trUor

j timesUk[gi (h1 times h2)] (A10)

908 L Kong I Runkel

When substituting the explicit form of the product m R(A) of R(A) = (A times 1) otimes R(1)after a short calculation one finds

(h1 h2)i =sumα

1

dim U j

A

A

A

A

A A

A α

α

h1

h2

U jUk

Uorj

Ui

Uk (A11)

Here the top morphism PlR(A) has been simplified with the help of the identity

PR(A) m R(A) (PR(A) otimes PR(A))

=(oplus

iisinI((m A A)otimes idUor

i)times idUi

) m R(A) (PR(A) otimes PR(A)) (A12)

which can be checked by direct calculation along the same lines as in the proof of [FrLem 310]

The action of the modular transformation S on oplusiisinI(B otimes Ui Ui ) for B isin C nat-urally induces an action on oplusiisinI(Ui B otimes Ui ) [K3 Prop 514] which we denote bySlowast In the present case we get an action of Slowast on oplus jisinIHomC(Uor

j A otimes Uori otimes Uor

j ) and

opluskisinIHomC(UkUi otimes Uk) Then to show gi is invariant under the action of S times Sminus1

amounts to showing that

(h1 h2)i = ((Sminus1)lowast h1 Slowast h2)i (A13)

Cardy Algebras and Sewing Constraints I 909

for all h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) We have

((Sminus1)lowast h1 Slowast h2)i =sum

mnα

dim Un

DimC

A

A

A

A

A A

A

Uk

α

α

h1

h2

Ui

U j

UmUn

(A14)

Now drag the upper vertex indexed by α in the above graph along its Uorm -leg until it

meets the lower vertex also indexed by α then sum over α and m This gives

((Sminus1)lowast h1 Slowast h2)i =sum

n

dim Un

DimC

A

A

A

A

A A

A

Uk

Un

h1

h2

Ui

U j

(A15)

910 L Kong I Runkel

If we just look at the neighbourhood of the Un-loop in the above graph we see thefollowing subgraphs

sumn

dim Un

DimC

A Aor

Uorj U j

Uk Uork

Un=

sumα

1

dim U j

A Aor Uk Uork

Uorj U j

αα

(A16)

where we have applied [BK Cor 3111] Substituting this subgraph back to the originalgraph in (A15) we obtain

((Sminus1)lowast h1 Slowast h2)i =sumα

1

dim U j

A

A

A

A

A A

A

A

A

Uorj

α

α

h1

h2m A

Ui

U jUk

Uk

(A17)

The graph in (A17) is equal to that in (A11) In order to see this we first drag theldquobubblerdquo (m A A) along A lines and through the m A vertex (because m A A is abimodule map) until it reaches the lower-left leg of the upper vertex indexed by α Thendrag the m A vertex along the (red) dotted line in the above graph Finally we applythe associativity of A (A12) and [Fr Lem 311] Then we see that the graph in (A17)exactly matches with the one in (A11)

Acknowledgements We would like to thank the organisers of the Oberwolfach Arbeitsgemeinschaft ldquoAlge-braic structures in conformal field theoriesrdquo (April 2007) where this work was started for an inspiring meetingWe would further like to thank the Hausdorff Institute for Mathematics in Bonn and the organisers of the stim-ulating meeting ldquoGeometry and Physicsrdquo (May 2008) We are indebted to Alexei Davydov Jens FjelstadJuumlrgen Fuchs Yi-Zhi Huang Alexei Kitaev Urs Schreiber Christoph Schweigert Stephan Stolz and PeterTeichner for helpful discussions andor comments on a draft of this paper The research of IR was partiallysupported by the EPSRC First Grant EPE0050471 the PPARC rolling grant PPC5071451 and the MarieCurie network lsquoSuperstring Theoryrsquo (MRTN-CT-2004-512194)

Cardy Algebras and Sewing Constraints I 911

References

[AN] Alexeevski A Natanzon SM Noncommutative two-dimensional topological field theories andhurwitz numbers for real algebraic curves Sel Math New Ser 12 307ndash377 (2006)

[Bi] Bichon J Cosovereign hopf algebras J Pure Appl Alg 157 121ndash133 (2001)[BK] Bakalov B Kirillov AA Lectures on Tensor Categories and Modular Functors Providence

RI Amer Math Soc 2001[C1] Cardy JL Operator content of two-dimensional conformal invariant theories Nucl Phys

B 270 186ndash204 (1986)[C2] Cardy JL Boundary conditions fusion rules and the verlinde formula Nucl Phys B 324

581ndash596 (1989)[CL] Cardy JL Lewellen DC Bulk and boundary operators in conformal field theory Phys Lett

B 259 274ndash278 (1991)[DMZ] Dong C-Y Mason G Zhu Y-C Discrete series of the Virasoro algebra and the moonshine

module In Algebraic Groups and Their Generalizations Quantum and infinite-dimensional Meth-ods Proc Symp Pure Math 56 Part 2 Providence RI Amer MathSoc 1994 pp 295ndash316

[DP] Day B Pastro C Note on frobenius monoidal functors New York J Math 14 733ndash742 (2008)[ENO1] Etingof PI Nikshych D Ostrik V On fusion categories Ann Math 162 581ndash642 (2005)[ENO2] Etingof PI Nikshych D Ostrik V An analogue of radfordrsquos s4 formula for finite tensor

categories Int Math Research Notices 54 2915ndash2933 (2004)[Fe] Felder G Froumlhlich J Fuchs J Schweigert C Correlation functions and boundary conditions in

rational conformal field theory and three-dimensional topology Comp Math 131 189ndash237 (2002)[FHL] Frenkel IB Huang Y-Z Lepowsky J On axiomatic approaches to vertex operator algebras

and modules Mem Amer Math Soc 104 (1993)[Fj] Fjelstad J Fuchs J Runkel I Schweigert C Uniqueness of openclosed rational cft with

given algebra of open states Adv Theor Math Phys 12 1283ndash1375 (2008)[FK] Froumlhlich J King C The chern-simons theory and knot polynomials Commun Math

Phys 126 167ndash199 (1989)[Fr] Froumlhlich J Fuchs J Runkel I Schweigert C Correspondences of ribbon categories Adv

Math 199 192ndash329 (2006)[FRS] Fuchs J Runkel I Schweigert C Tft construction of rcft correlators I partition functions Nucl

Phys B 646 353ndash497 (2002)[FS] Fuchs J Schweigert C Category theory for conformal boundary conditions Fields Inst Com-

mun 39 25ndash71 (2003)[H1] Huang Y-Z Two-dimensional conformal geometry and vertex operator algebras Progress in

Mathematics Vol 148 Boston Birkhaumluser 1997[H2] Huang Y-Z Rigidity and modularity of vertex tensor categories Comm Contemp Math 10

871ndash911 (2008)[HK1] Huang Y-Z Kong L Open-string vertex algebra category and operads Commun Math

Phys 250 433ndash471 (2004)[HK2] Huang Y-Z Kong L Full field algebras Commun Math Phys 272 345ndash396 (2007)[HK3] Huang Y-Z Kong L Modular invariance for conformal full field algebras httparxivorgabs

math0609570v2[mathQA] 2006[HL] Huang Y-Z Lepowsky J Tensor products of modules for a vertex operator algebra and vertex

tensor categories In Lie Theory and Geometry in honor of Bertram Kostant ed R BrylinskiJ-L Brylinski V Guillemin V Kac Boston Birkhaumluser 1994 pp 349ndash383

[JS] Joyal A Street R Braided tensor categories Adv Math 102 20ndash78 (1993)[K1] Kong L Full field algebras operads and tensor categories Adv Math 213 271ndash340 (2007)[K2] Kong L Open-closed field algebras Commun Math Phys 280 207ndash261 (2008)[K3] Kong L Cardy condition for open-closed field algebras Commun Math Phys 283 25ndash92 (2008)[Ki] Kitaev A Private communication[KO] Kirillov AA Ostrik V On q-analog of mckay correspondence and ade classification of sl(2)

conformal field theories Adv Math 171 183ndash227 (2002)[KR] Kong L Runkel I Morita classes of algebras in modular tensor categories Adv Math 219

1548ndash1576 (2008)[Ld] Lauda AD Frobenius algebras and ambidextrous adjunctions Theo Appl Cat 16 84ndash

122 (2006)[Li1] Li H-S Regular representations of vertex operator algebras Commun Contemp Math 4

639ndash683 (2002)[Li2] Li H-S Regular representations and huang-lepowsky tensor functors for vertex operator

algebras J Alge 255 423ndash462 (2002)

876 L Kong I Runkel

F(A)otimes (F(B)otimes F(C))α (F(A)otimes F(B))otimes F(C)

F(A)otimes F(B otimes C)

idF(A)otimesψ F2

F(A otimes B)otimes F(C)

ψF2 otimesidF(C)

F(A otimes (B otimes C))F(α)

ψF2

F((A otimes B)otimes C)

ψF2

(23)

12 otimes F(A) F(A)lminus1F(A)

F(11)otimes F(A)

ψF0 otimesidF(A)

F(11 otimes A)ψF

2

F(lA)

F(A)otimes 12 F(A)rminus1

F(A)

F(A)otimes F(11)

idF(A)otimesψ F0

F(A otimes 11)

F(rA)

ψF2

(24)

We denote a lax tensor functor by (G φG2 φ

G0 ) or just G and a colax tensor functor

by (F ψ F2 ψ

F0 ) or F

Definition 23 A tensor functor T C1 rarr C2 is a lax tensor functor (T φT2 φ

T0 ) such

that φT0 φ

T2 are both isomorphisms

A tensor functor (T φT2 φ

T0 ) is automatically a colax tensor functor (T ψT

2 ψT0 )

with ψT0 = (φT

0 )minus1 and ψT

2 = (φT2 )

minus1In the next section we will discuss algebras in tensor categories The defining proper-

ties (21) and (22) of a lax tensor functor are analogues of the associativity the left-unitand the right-unit properties of an algebra Indeed a lax tensor functor G C1 rarr C2maps a C1-algebra to a C2-algebra Similarly (23) and (24) are analogues of the coasso-ciativity the left-counit and the right-counit properties of a coalgebra and a colax tensorfunctor F C1 rarr C2 maps a C1-coalgebra to a C2-coalgebra We will later make useof functors that take Frobenius algebras to Frobenius algebras This requires a strongercondition than being lax and colax and leads to the notion of a lsquofunctor with Frobeniusstructurersquo or lsquoFrobenius monoidal functorrsquo [SzDPP] which we will simply refer to asFrobenius functor

Definition 24 A Frobenius functor F C1 rarr C2 is a tuple F equiv (F φF2 φ

F0 ψ

F2 ψ

F0 )

such that (F φF2 φ

F0 ) is a lax tensor functor (F ψ F

2 ψF0 ) is a colax tensor functor

and such that the following two diagrams commute

F(A)otimes (F(B)otimes F(C))α (F(A)otimes F(B))otimes F(C)

φF2 otimesidF(C)

F(A)otimes F(B otimes C)

idF(A)otimesψ F2

φF2

F(A otimes B)otimes F(C)

F(A otimes (B otimes C))F(α) F((A otimes B)otimes C)

ψF2

(25)

Cardy Algebras and Sewing Constraints I 877

F(A)otimes (F(B)otimes F(C))

idF(A)otimesφF2

(F(A)otimes F(B))otimes F(C)αminus1

F(A)otimes F(B otimes C) F(A otimes B)otimes F(C)

ψF2 otimesidF(C)

φF2

F(A otimes (B otimes C))

ψF2

F((A otimes B)otimes C)F(αminus1)

(26)

Proposition 25 If (F φF2 φ

F0 ) is a tensor functor then F is a Frobenius functor with

ψ F0 = (φF

0 )minus1 and ψ F

2 = (φF2 )

minus1

Proof Since F is a tensor functor it is lax and colax If we replace ψ F2 by (φF

2 )minus1 in

(25) and (26) both commuting diagrams are equivalent to (21) which holds becauseF is lax Thus F is a Frobenius functor

The converse statement does not hold For example the functor R which we definein Sect 24 is Frobenius but not tensor

Let us recall the notion of adjunctions and adjoint functors [Ma Ch IV1]

Definition 26 An adjunction from C1 to C2 is a triple 〈FG χ〉 where F and G arefunctors

F C1 rarr C2 G C2 rarr C1

and χ is a natural isomorphism which assigns to each pair of objects A1 isin C1 A2 isin C2a bijective map

χA1A2 HomC2(F(A1) A2)sim=minusminusrarr HomC1(A1G(A2))

which is natural in both A1 and A2 F is called a left-adjoint of G and G is called aright-adjoint of F

For simplicity we will often abbreviate χA1A2 as χ Associated to each adjunction

〈FG χ〉 there are two natural transformations idC1

δminusrarr G F and FGρminusrarr idC2 where

idC1 and idC2 are identity functors given by

δA1 = χ(idF(A1)) ρA2 = χminus1(idG(A2)) (27)

for Ai isin Ci i = 1 2 They satisfy the following two identities

GδGminusrarr G FG

Gρminusrarr G = GidGminusminusrarr G F

Fδminusrarr FG FρFminusrarr F = F

idFminusminusrarr F (28)

We have for g F(A1) rarr A2 and f A1 rarr G(A2)

χ(g) = G(g) δA1 χminus1( f ) = ρA2 F( f ) (29)

For simplicity δA1 and ρA2 are often abbreviated as δ and ρ respectivelyLet 〈FG χ〉 be an adjunction from a tensor category C1 to a tensor category C2

and (F ψ F2 ψ

F0 ) a colax tensor functor from C1 to C2 We can define a morphism

878 L Kong I Runkel

φG0 11 rarr G(12) and a natural transformation φG

2 otimes (G times G) rarr G otimes by forA B isin C2

φG0 = χ(ψ F

0 ) = 11δ11minusrarr G F(11)

G(ψ F0 )minusminusminusminusrarr G(12)

φG2 = χ((ρA otimes ρB) ψ F

2 ) = G(A)otimes G(B)δminusrarr G F (G(A)otimes G(B))

G(ψ F2 )minusminusminusminusrarr G (FG(A)otimes FG(B))

G(ρAotimesρB )minusminusminusminusminusminusrarr G(A otimes B) (210)

where we have used the first identity in (29) Notice that φG2 is natural because it is a

composition of natural transformations One can easily show that ψ F0 and ψ F

2 can bere-obtained from φG

0 and φG2 as follows

ψ F0 = χminus1(φG

0 ) = F(11)FφG

0minusminusrarr FG(12)ρminusrarr 12

ψ F2 = χminus1(φG

2 (δ otimes δ)) = F(U otimes V )F(δotimesδ)minusminusminusminusrarr F (G F(U )otimes G F(V ))

FφG2minusminusrarr FG (F(U )otimes F(V ))

ρminusrarr F(U )otimes F(V ) (211)

for U V isin C1 The following result is standard for the sake of completeness we givea proof in Appendix A1

Lemma 27 (F ψ F2 ψ

F0 ) is a colax tensor functor iff (G φG

2 φG0 ) is a lax tensor

functor

22 Algebras in tensor categories An algebra in a tensor category C or a C-algebra isa triple A = (Am η) where A is an object of C m (the multiplication) is a morphismA otimes A rarr A such that m (m otimes idA) αAAA = m (idA otimes m) and η (the unit) is amorphism 1 rarr A such that m (idA otimes η) = idA rA and m (ηotimes idA) = idA lAIf C is braided and m cAA = m then A is called commutative

A left A-module is a pair (MmM ) where M isin C and m M is a morphism AotimesM rarr Msuch that mM(idAotimesm M ) = m M(m AotimesidM )αAAM and m M(ηAotimesidM ) = idMlM Right A-modules and A-bimodules are defined similarly

Definition 28 Let C be a tensor category and let A be an algebra in C

(i) A is called simple iff it is simple as a bimodule over itselfLet C be in addition k-linear for k a field

(ii) A is called absolutely simple iff the space of A-bimodule maps from A to itself isone-dimensional dimk HomA|A(A A) = 1

(iii) A is called haploid iff dimk Hom(1 A) = 1 [FS Def 43]

In the following we will assume that all tensor categories are strict to avoid spelling outassociators and unit constraints

A C-coalgebra A = (A ε) is defined analogously to a C-algebra ie A rarrA otimes A and ε A rarr 1 obey coassociativity and counit conditions

If C is braided and if A and B are C-algebras there are two in general non-isomorphicalgebra structures on A otimes B We choose A otimes B to be the C-algebra with multiplicationm AotimesB = (m Aotimesm B)(idAotimescminus1

ABotimesidB) and unitηAotimesB = ηAotimesηB Similarly if A and Bare C-coalgebras then A otimes B becomes a C-coalgebra if we choose the comultiplicationAotimesB = (idA otimes cAB otimes idB) (A otimesB) and the counit εAotimesB = εA otimes εB

Cardy Algebras and Sewing Constraints I 879

Definition 29 A Frobenius algebra A = (Am η ε) is an algebra and a coalge-bra such that the coproduct is an intertwiner of A-bimodules

(idA otimes m) (otimes idA) = otimes m = (m otimes idA) (idA otimes)

We will use the following graphical representation for the morphisms of a Frobeniusalgebra

m =A A

A

η =A

=A A

A

ε =A

(212)

A Frobenius algebra A in a k-linear tensor category for k a field is called special iffm = ζ idA and ε η = ξ id1 for nonzero constants ζ ξ isin k If ζ = 1 we call A nor-malised-special A Frobenius algebra homomorphism between two Frobenius algebrasis both an algebra homomorphism and a coalgebra homomorphism

A (strictly) sovereign tensor category is a tensor category equipped with a left and aright duality which agree on objects and morphisms (see eg [BiFS] for more details)We will write the dualities as

Uor U

= dU Uor otimes U rarr 1U Uor

= dU U otimes Uor rarr 1

U Uor

= bU 1 rarr U otimes UorUor U

= bU 1 rarr Uor otimes U

(213)

In terms of these we define the left and right dimension of an object U as

diml U = dU bU dimr U = dU bU (214)

both of which are elements of Hom(1 1)Let now C be a sovereign tensor category For a Frobenius algebra A in C we define

two morphisms

A =

A

Aor

primeA =

A

Aor

(215)

Definition 210 A Frobenius algebra A is symmetric iff A = primeA

The following lemma shows that under certain conditions we do not need to distin-guish the various notions of simplicity in Definition 28

Lemma 211 Let A be a commutative symmetric Frobenius algebra in a C-linear semi-simple sovereign braided tensor category C and suppose that diml A = 0 Then thefollowing are equivalent

880 L Kong I Runkel

(i) A is simple(ii) A is absolutely simple

(iii) A is haploid

Proof (ii)hArr(iii) A is haploid iff it is absolutely simple as a left module over itself[FS Eq (417)] Furthermore for a commutative algebra we have HomA(A A) =HomA|A(A A) and so A is haploid iff it is absolutely simple(i)rArr(ii) If A is simple then every nonzero element of HomA|A(A A) is invertibleHence this space forms a division algebra over C and is therefore isomorphic to C(iii)rArr(i) Since C is semi-simple and A is haploid also Hom(A 1) is one-dimensionalThe counit ε is a nonzero element in this space and so gives a basis This implies firstlythat ε η = 0 and secondly that there is a constant β isin C such that

β middot ε = dA (idAor otimes m) (bA otimes idA) (216)

Composing with η from the right yields β εη = diml A The right-hand side is nonzeroand so β = 0 By [FRS Lem 311] A is special We have already proved (ii)hArr(iii)and so A is absolutely simple A special Frobenius algebra in a semi-simple categoryhas a semi-simple category of bimodules (apply [FS Prop 524] to the algebra tensoredwith its opposite algebra) For semi-simple C-linear categories simple and absolutelysimple are equivalent2 Thus A is simple Remark 212 For a Frobenius algebra A the morphisms (215) are invertible and henceA sim= Aor In this case one has diml A = dimr A [FS Rem 363] and so we could havestated the above lemma equivalently with the condition dimr A = 0

Let F C1 rarr C2 be a lax tensor functor between two tensor categories C1 C2 and

let (Am A ηA) be an algebra in C1 Define morphisms F(A)otimes F(A)m F(A)minusminusminusrarr F(A) and

12ηF(A)minusminusminusrarr F(A) as

m F(A) = F(m A) φF2 ηF(A) = F(ηA) φF

0 (217)

Then (F(A)m F(A) ηF(A)) is an algebra in C2 [JS Prop 55] If f A rarr B is analgebra homomorphism between two algebras A B isin C1 then F( f ) F(A) rarr F(B)is also an algebra homomorphism If (MmM ) is a left (or right) A-module in C1 then(F(M) F(mM )φF

2 ) is a left (or right) F(A)-module if M has a A-bimodule structurethen F(M) naturally has a F(A)-bimodule structure

Similarly if (AA εA) is a coalgebra in C1 and F C1 rarr C2 is a colax tensor func-

tor then F(A) with coproduct F(A)F(A)minusminusminusrarr F(A) otimes F(A) and counit F(A)

εF(A)minusminusminusrarr 12given by

F(A) = ψ F2 F(A) εF(A) = ψ F

0 F(εA) (218)

is a coalgebra in C2 If f A rarr B is a coalgebra homomorphism between two coalgebrasA B isin C1 then F( f ) F(A) rarr F(B) is also a coalgebra homomorphism

2 To see this note that if U is simple then the C-vector space Hom(UU ) is a division algebra and henceHom(UU ) = C idU Conversely if U is not simple then U = U1 oplus U2 and Hom(UU ) contains at leasttwo linearly independent elements namely idU1 and idU2

Cardy Algebras and Sewing Constraints I 881

Proposition 2133 If F C1 rarr C2 is a Frobenius functor and (Am A ηAA εA) aFrobenius algebra in C1 then (F(A)m F(A) ηF(A)F(A) εF(A)) is a Frobenius alge-bra in C2

Proof One Frobenius property (m F(A)otimes idF(A)) (idF(A)otimesF(A)) = F(A) m F(A)follows from the commutativity of the following diagram (we spell out the associativityisomorphisms)

F(A)otimes F(A)idF(A)otimesF(A)

φF2

F(A)otimes F(A otimes A)idF(A)otimesψF

2

φF2

F(A)otimes (F(A)otimes F(A))

αF(A)F(A)F(A)

F(A otimes A)

F(idAotimesA)

F(m A)

F(A otimes (A otimes A))

F(αAAA)

(F(A)otimes F(A))otimes F(A)

φF2 otimesidF(A)

F((A otimes A)otimes A)

F(m AotimesidA)

ψF2 F(A otimes A)otimes F(A)

F(m A)otimesidF(A)

F(A)

F(A) F(A otimes A)ψF

2 F(A)otimes F(A)

(219)

The commutativity of the upper-left subdiagram follows from the naturalness of φF2 that

of the upper-right subdiagram follows from (25) that of the lower-left subdiagram fol-lows from the Frobenius properties of A and that of the lower-right subdiagram followsfrom the naturalness of ψ F

2 The proof of the other Frobenius property is similar Proposition 214 If F C1 rarr C2 is a tensor functor and A a Frobenius algebra in C1then

(i) F(A) has a natural structure of Frobenius algebra as given in Proposition 213(ii) If A is (normalised-)special so is F(A)

Proof Part (i) follows from Propositions 25 and 213 Part (ii) is a straightforwardverification of the definition using ψ F

2 = (φF2 )

minus1 and ψ F0 = (φF

0 )minus1

Let C1 C2 be sovereign tensor categories and F C1 rarr C2 a Frobenius functor Wedefine two morphisms IF(Aor) I prime

F(Aor) F(Aor) rarr F(A)or for a Frobenius algebra A inC1 as follows

IF(Aor) = ((ψ F0 F(dA) φF

2 )otimes idF(A)or) (idF(Aor) otimes bF(A))

I primeF(Aor) = (idF(A)or otimes (ψ F

0 F(dA) φF2 )) (bF(A) otimes idF(Aor))

(220)

It is easy to see that these are isomorphisms

Lemma 215 If F C1 rarr C2 is a Frobenius functor and A a Frobenius algebra in C1then

F(A) = IF(Aor) F(A) primeF(A) = I prime

F(Aor) F(primeA) (221)

3 After the preprint of the present paper appeared we noticed that this proposition is also proved in[DP Cor 5]

882 L Kong I Runkel

Proof We only prove the first equality the second one can be seen in the same way Bydefinition we have

IF(Aor) F(A)

= ((ψ F0 F(dA) φF

2 )otimes idF(A)or) (idF(Aor) otimes bF(A)) F(A)

=[(ψ F

0 F(dA) φF2 ) (F(A)otimes idF(A))

]otimes idF(A)or

(idF(A) otimes bF(A))

For the term inside the square brackets we find

ψ F0 F(dA) φF

2 (F(A)otimes idF(A)) = ψ F0 F(dA) F(A otimes idA) φF

2

= ψ F0 F(dA (A otimes idA)) φF

2 = ψ F0 F(εA m A) φF

2 (222)

On the other hand by definitionF(A) = [((ψ F0 F(εA) (F(m A)φF

2 ))otimes idF(A)or] (idF(A) otimes bF(A)) This demonstrates the first equality in (221) Proposition 216 Let F C1 rarr C2 be a tensor functor G C2 rarr C1 a functor〈FG χ〉 an adjunction A a C1-algebra and B a C2-algebra Then f A rarr G(B)is an algebra homomorphism if and only if f = χminus1( f ) F(A) rarr B is an algebrahomomorphism

Proof We need to show that

mG(B) ( f otimes f ) = f m A and f ηA = ηG(B) (223)

is equivalent to

m B ( f otimes f ) = f m F(A) and f ηF(A) = ηBs (224)

We first prove that the first identity in (223) is equivalent to the first identity in (224)For the left-hand side of the first identity in (223) we have the following equalities

mG(B) ( f otimes f )(1)= G(m B) φG

2 ( f otimes f )(2)= G(m B) G(ρ otimes ρ) G(ψ F

2 ) δ ( f otimes f )(3)= G(m B) G(ρ otimes ρ) G(ψ F

2 ) G F( f otimes f ) δ(4)= G(m B) G(ρ otimes ρ) G(F( f )otimes F( f )) G(ψ F

2 ) δ(5)= G

(m B (ρ otimes ρ) (F( f )otimes F( f )) ψ F

2

) δ

(6)= χ(

m B (ρ otimes ρ) (F( f )otimes F( f )) ψ F2

) (225)

where (1) is the definition of mG(B) in (217) (2) is the second identity in (210) (3)and (4) are naturality of δ and ψ F

2 respectively step (5) is functoriality of G and finallystep (6) is (29) For the right hand side of the first identity in (223) we get

f m A(1)= Gρ δG ( f m A)

(2)= Gρ G F( f m A) δ(3)= G(ρ F( f m A)) δ(4)= χ(ρ F( f ) F(m A)) (226)

Cardy Algebras and Sewing Constraints I 883

where (1) is the adjunction property (28) (2) is naturality of δ (3) functoriality of Gand (4) amounts to (29) and functoriality of F

On the other hand we see that the first equality in (224) is equivalent to

m B (ρ otimes ρ) (F( f )otimes F( f )) = ρ F( f ) F(m A) φF2 (227)

Using that φF2 is invertible with inverse (φF

2 )minus1 = ψ F

2 and that χ is an isomorphismit follows that the statement that (225) is equal to (226) is equivalent to the identity(227)

Now we prove that the second identity in (223) is equivalent to the second identity in(224) Using (217) and (210) we can write ηG(B) = G(ηB)φF

0 = G(ηB)G(ψ F0 )δ1

Together with (29) this shows that the second identity in (223) is equivalent to

f ηA = χ(ηB ψ F0 ) (228)

On the other hand the second identity in (224) is equivalent to

ρ F( f ) F(ηA) φF0 = ηB (229)

which by φF0 = (ψ F

0 )minus1 and (29) is further equivalent to (228)

Definition 217 Let (Am A ηAA εA) and (Bm B ηBB εB) be two Frobeniusalgebras in a tensor category C For f A rarr B we define f lowast B rarr A by

f lowast = ((εB m B)otimes idA) (idB otimes f otimes idA) (idB otimes (A ηA)) (230)

The following lemma is immediate from the definition of (middot)lowast and the properties ofFrobenius algebras We omit the proof

Lemma 218 Let C be a tensor category let A BC be Frobenius algebras in C andlet f A rarr B and g B rarr C be morphisms

(i) (g f )lowast = f lowast glowast(ii) f is a monomorphism iff f lowast is an epimorphism

(iii) f is an algebra map iff f lowast is a coalgebra map(iv) If f is a homomorphism of Frobenius algebras then f lowast f = idA and f f lowast =

idB(v) If C is sovereign and if A and B are symmetric then f lowastlowast = f

Let C and D be tensor categories and let F C rarr D be a Frobenius functor GivenFrobenius algebras A B in C and a morphism f A rarr B the next lemma shows how(middot)lowast behaves under F

Lemma 219 F( f lowast) = F( f )lowast

Proof The definition of the structure morphisms of the Frobenius algebra F(A) is givenin (217) and (218) Substituting these definitions gives

F( f )lowast =[(ψ F

0 F(εB) F(m B) φF2

)otimes idF(A)

] [

idF(B) otimes F( f )otimes idF(A)]

[idF(B) otimes

(ψ F

2 F(A) F(ηA) φF0

)]

= (ψ F0 otimes idF(A)) [

F (εB m B (idB otimes f ))otimes idF(A)]

(φF2 otimes idF(A)) (idF(B) otimes ψ F

2 )

[idF(B) otimes F(A ηA)

] (idF(B) otimes φF0 ) (231)

884 L Kong I Runkel

In the middle line of the last expression we can use the defining property (25) of F namely we substitute (φF

2 otimes idF(A)) (idF(B) otimes ψ F2 ) = ψ F

2 φF2 Then ψ F

2 can bemoved to the left and φF

2 to the right until they can be omitted against ψ F0 and φF

0 respectively using (22) and (24) This results in

F( f )lowast = F ((εB m B (idB otimes f ))otimes idA) F (idB otimes (A ηA)) (232)

which is nothing but F( f lowast)

23 Modular tensor categories Let C be a modular tensor category [TBK] ie anabelian semi-simple finite C-linear ribbon category with simple tensor unit 1 and a non-degeneracy condition on the braiding (to be stated in a moment) We denote the set ofequivalence classes of simple objects in C by I elements in I by i j k isin I and theirrepresentatives by Ui U j Uk We also set U0 = 1 and for an index k isin I we define kby Uk

sim= Uork

Since the tensor unit is simple we shall for modular tensor categories identifyHom(1 1) sim= C (cf footnote 2) Define numbers si j isin C by4

si j = UiU j (233)

They obey si j = s ji and s0i = dim Ui see eg [BK Sect 31] (In a ribbon cate-gory the left and right dimension (214) of Ui coincide and are denoted by dim Ui )The non-degeneracy condition on the braiding of a modular tensor category is that the|I|times|I|-matrix s should be invertible In fact [BK Thm 317]

sumkisinI

sik sk j = Dim C δij (234)

where Dim C = sumiisinI(dim Ui )

2 One can show (even in the weaker context of fusioncategories over C) that Dim C ge 1 [ENO1 Thm 23] In particular Dim C = 0 We fixonce and for all a square root

radicDim C of Dim C

Let us fix a basis λα(i j)kN k

i jα=1 in HomC(Ui otimesU j Uk) and the dual basis ϒ(i j)k

α N ki j

α=1

in HomC(UkUi otimesU j ) The duality of the bases means that λα(i j)k ϒ(i j)kβ = δαβ idUk

We also fix λ(0i)i = λ(i0)i = idUi We denote the basis vectors graphically as follows

λα(i j)k = α

Uk

Ui U j

ϒ(i j)kα = α

Uk

Ui U j

(235)

4 In the graphical notation used below we have given an orientation to the ribbons indicated by the arrowsFor example it is understood that this orientation determines which of the duality morphisms in (213) to use

Cardy Algebras and Sewing Constraints I 885

For V isin C we also choose a basis b(iα)V of HomC(VUi ) and the dual basis bV(iβ)

of HomC(Ui V ) for i isin I such that b(iα)V bV(iβ) = δαβ idUi We use the graphical

notation

b(iα)V = α

Ui

V

bV(iα) = α

Ui

V

(236)

Given two modular tensor categories C and D by C D we mean the tensor productof additive categories over C [BK Def 1115] ie the category whose objects are directsums of pairs V times W of objects V isin C and W isin D and whose morphism spaces are

HomCD(V times W V prime times W prime) = HomC(V V prime)otimesC HomC(WW prime) (237)

for pairs and direct sums of these if the objects are direct sums of pairsIf we replace the braiding and the twist in C by the antibraiding cminus1 and the antitwist

θminus1 respectively we obtain another ribbon category structure on C In order to distin-guish these two distinct structures we denote (C c θ) and (C cminus1 θminus1) by C+ and Cminusrespectively As in the Introduction we will abbreviate

C2plusmn = C+ Cminus (238)

Note that a set of representatives of the simple objects in C2plusmn is given by Ui times U j fori j isin I

For the remainder of Sect 2 we fix a modular tensor category C

24 The functors T and R The tensor product bifunctor otimes can be naturally extendedto a functor T C2plusmn rarr C Namely T (oplusN

i=1Vi times Wi ) = oplusNi=1Vi otimes Wi for all Vi Wi isin C

and N isin N The functor T becomes a tensor functor as follows For φT0 1 rarr T (1 times 1)

take φT0 = id1 (or lminus1

1 in the non-strict case) Next notice that for U VW X isin C

T (U times V )otimes T (W times X)= (U otimes V )otimes (W otimes X)

T ((U times V )otimes (W times X))= (U otimes W )otimes (V otimes X)(239)

We define φT2 T (U times V )otimes T (W times X) rarr T ((U times V )otimes (W times X)) by

φT2 = idU otimes cminus1

W V otimes idX (240)

(In the non-strict case the appropriate associators have to be added) The above definitionof φT

2 can be naturally extended to a morphism φT2 T (M1 otimes M2) rarr T (M1)otimes T (M2)

for any pair of objects M1M2 in C2plusmn The following result can be checked by directcalculation [JS Prop 52]

Lemma 220 The triple (T φT2 φ

T0 ) gives a tensor functor

886 L Kong I Runkel

In particular (T φT2 φ

T0 ψ

T2 ψ

T0 ) where ψT

2 = (φT2 )

minus1 and ψT0 = (φT

0 )minus1 gives

a Frobenius functorDefine the functor R C rarr C2plusmn as follows for A isin C and f isin HomC(A B)

R(A) =oplusiisinI

(A otimes Uori )times Ui R( f ) =

oplusiisinI

( f otimes idUori)times idUi (241)

This functor was also considered in a slightly different context in [ENO2 Prop 23]The family of isomorphisms γ R

A = oplusiisinI DimCdim Ui

id(AotimesUori )timesUi

isin Aut(R(A)) defines a

natural isomorphism γ R R rarr ROur next aim is to show that R is left and right adjoint to T in other words R

and T form an ambidextrous adjunction (see eg [Ld] for a discussion of ambidextrousadjunctions) To this end we introduce two linear isomorphisms for A isin C and M isin C2plusmn

χ HomC(T (M) A) minusrarr HomC2plusmn(M R(A))

χ HomC(A T (M)) minusrarr HomC2plusmn(R(A)M)(242)

If we decompose M as M = oplusNn=1 Ml

n times Mrn then χ and χ are given by

χ Noplus

n=1

fn

Mln Mr

n

A

rarrNoplus

n=1

oplusiisinI

sumα

fn

Mln

Mrn

A

α

Uori

times α

Mrn

Ui

(243)

and

χ Noplus

n=1 gn

Mln Mr

n

A

rarrNoplus

n=1

oplusiisinI

sumα gn

Mln

Mrn

A

α

Uori

times α

Ui

Mrn

DimCdim Ui

(244)

Notice that χ and χ are independent of the choice of basis

Theorem 221 〈T R χ〉 and 〈R T χminus1〉 are adjunctions ie R is both left and rightadjoint of T

Proof Write M as M = oplusNn=1 Ml

n times Mrn The isomorphism χ amounts to the following

composition of natural isomorphisms

HomC(T (M) A) = oplusnHomC(Mln otimes Mr

n A)sim= oplusni HomC(Ml

n otimes Ui A)otimes HomC(Mrn Ui )

sim= oplusni HomC(Mln A otimes Uor

i )otimes HomC(Mrn Ui ) = HomC2plusmn(M R(A)) (245)

Cardy Algebras and Sewing Constraints I 887

Thus χ is natural Let (γ RA )

lowast HomC2plusmn(R(A)M) rarr HomC2plusmn(R(A)M) denote the

pull-back of γ RA The isomorphism χ is equal to the composition of (γ R

A )lowast and the

following sequence of natural isomorphisms

HomC(A T (M)) = oplusn HomC(AMln otimes Mr

n)

sim= oplusni HomC(AMln otimes Ui )otimes HomC(Ui Mr

n)

sim= oplusni HomC(A otimes Uori Ml

n)otimes HomC(Ui Mrn) = HomC2plusmn(R(A)M) (246)

We have proved that both χ and χ are natural isomorphisms There are four natural transformations associated to χ and χ namely

idC2plusmnδminusrarr RT

ρminusrarr idC2plusmn and idCδminusrarr T R

ρminusrarr idC (247)

defined by for A isin C M isin C2plusmn

δM = χ(idT (M)) ρA = χminus1(idR(A))

ρM = χ (idT (M)) δA = χminus1(idR(A))(248)

They can be expressed graphically as follows with M = oplusNn=1 Ml

n times Mrn

δM =oplusni

sumα

Mln

Mln Mr

n Uori

αtimes

Ui

Mrn

α

ρA =oplusiisinI

A

A

Uori Ui

(249)

ρM =oplusni

sumα

Mln

Mln

Mrn Uor

i

α times

Ui

Mrn

α DimCdim Ui

δA =oplusiisinI

A

A Uori Ui

dim UiDimC

Note that

ρM δM = Dim C middot idM and ρA δA = idA (250)

Lemma 222 The functors T and R as maps on the sets of morphisms have left inversesand thus are injective

Proof Let f A rarr B be a morphism in C We define a map Q R HomC2plusmn(R(A)

R(B)) rarr HomC(A B) by f prime rarr ρB T ( f prime) δA Then we have

Q R R( f ) = ρB T R( f ) δA = ρB δB f = f (251)

888 L Kong I Runkel

where we used naturality of δ and (250) in the second and third equalities respectivelySo Q R is a left inverse of R on morphisms Thus R is injective on morphisms Similarlylet g M rarr N be a morphism in C2plusmn We define a map QT HomC(T (M) T (N )) rarrHomC2plusmn(M N ) by gprime rarr (Dim C)minus1 middot ρN R(gprime) δM Then we have

QT T (g) = (Dim C)minus1 middot ρN RT (g) δM = (Dim C)minus1 middot ρN δN g = g (252)

So QT is a left inverse of T on morphisms Thus T is injective on morphisms Using (29) and (250) one can express the two inverse maps χminus1 χminus1 as follows

for f isin HomC2plusmn(M R(A)) and g isin HomC2plusmn(R(A)M)

χminus1( f ) = ρ T ( f ) χminus1(g) = T (g) δ (253)

By Proposition 25 and Lemma 27 R is both a lax and colax tensor functor In particularφR

0 1 times 1 rarr R(1) is given by

φR0 = χ (ψT

0 ) = R(ψT0 ) δ1 times 1 = id1times1 (254)

and φR2 R(A)otimes R(B) rarr R(A otimes B) by φR

2 = R(ρA otimes ρB) R(ψT2 ) δ which can

be expressed graphically as

φR2 =

oplusi jkisinI

sumα

A

A

Uori B

B

Uorj

Uork

α

times

Ui U j

α

Uk

(255)

Similarly ψ R0 R(1) rarr 1 times 1 is given by

ψ R0 = ρ1 R(φT

0 ) = Dim C id1times1 (256)

and ψ R2 R(A otimes B) rarr R(A) otimes R(B) by ψ R

2 = ρ R(φT2 ) R(δA otimes δB) which in

graphical notation reads

ψ R2 =

oplusi jkisinI

sumα

A

A Uori

B

B Uorj

Uork

α

times

Ui U j

α

Uk

dim Ui dim U j

dim Uk Dim C

(257)

If C has more than one simple object then R does not take the tensor unit of C to thetensor unit of C2plusmn and so is clearly not a tensor functor However we will show that R isstill a Frobenius functor This will imply that if A is a Frobenius algebra in C then

R(A) = (R(A)m R(A) ηR(A)R(A) εR(A)) (258)

Cardy Algebras and Sewing Constraints I 889

is a Frobenius algebra in C2plusmn where the structure morphisms were given in (217) and(218) In the case A = 1 it was proved in [Muuml1 Prop 41] (see also [Fr Lem 619]and [K1 Thm 52]) that (258) is a commutative simple symmetric normalised-specialFrobenius algebra in C2plusmn In fact given a Frobenius algebra A in C it is straightforwardto verify that the structure morphisms in (258) are precisely those of (A times 1)otimes R(1)cf Sect 22

Proposition 223 (R φR2 φ

R0 ψ

R2 ψ

R0 ) is a Frobenius functor

Proof Using the explicit graphical expression of φR2 φ

R0 ψ

R2 ψ

R0 it is easy to see that

the commutativity of the diagrams (25) and (26) are equivalent to the statement thatR(1) with structure morphisms as in (258) is a Frobenius algebra in C2plusmn The latterstatement is true by [Muuml1 Prop 41]

From Lemma 220 and Proposition 223 we see that T and R take Frobenius alge-bras to Frobenius algebras The following two propositions show how the properties ofFrobenius algebras are transported

Proposition 224 Let A be a Frobenius algebra in C2plusmn Then T (A) is a Frobenius algebrain C and

(i) A is symmetric iff T (A) is symmetric(ii) A is (normalised-)special iff T (A) is (normalised-)special

Proof For part (i) write A as a direct sum oplusNn=1 Al

n times Arn Then the maps IT (A) I prime

T (A) T (Aor) rarr T (A)or defined in (220) are given by

IT (A) = I primeT (A) = oplusN

n=1c(Aln)

or(Arn)

or (259)

Therefore by (221) T (A) = primeT (A) is equivalent to T (A) = T (prime

A) Since byLemma 222 T is injective on morphisms this proves part (i) Part (ii) can be checkedin the same way for example the condition mT (A) T (A) = ζ idT (A) is easily checkedto be equivalent to T (m A A) = ζ T (idA) Proposition 225 Let A be a Frobenius algebra A in C Then R(A) is a Frobeniusalgebra in C2plusmn and

(i) A is symmetric iff R(A) is symmetric(ii) A is (normalised-)special iff R(A) is (normalised-)special

Proof Recall that the structure morphisms of the Frobenius algebra R(A) are equal tothose of (A times 1) otimes R(1) Using this equality part (i) and (ii) follow because R(1) issymmetric and normalised-special For example

m R(A) R(A) = [(m A A)times id1] otimes idR(1) = R(m A A) (260)

so that m R(A) R(A) = ζ idR(A) is equivalent to R(m A A) = ζ R(idA) which byLemma 222 is equivalent to m A A = ζ idA

The functor R has one additional property not shared by T namely R takes absolutelysimple algebras to absolutely simple algebras We will see explicitly in Sect 33 thatthis is not true for T

890 L Kong I Runkel

Lemma 226 For a C-algebra A the map

R HomA|A(A A) rarr HomR(A)|R(A)(R(A) R(A)) (261)

given by f rarr R( f ) is well-defined and an isomorphism

Proof Since R is a lax tensor functor R(A) is naturally a R(A)-bimodule It is easy tosee that R in (261) is a well-defined map R(A) is also naturally a R(1)-bimodule whichcan be identified with the induced R(1)-bimodule structure on (A times 1)otimes R(1) wherethe left R(1) action on (Atimes1)otimes R(1) is given by (idAtimes1 otimesm R(1))(cminus1

Atimes1R(1)otimes idR(1))We have the following natural isomorphisms

HomR(1)|R(1)(R(A) R(A))sim=minusrarr HomC2plusmn(A times 1 R(A))

χminus1

minusminusrarr HomC(A A) (262)

which by (253) are given by for f isin HomR(1)|R(1)(R(A) R(A))

f rarr f prime = f (idAtimes1 otimes ηR(1)) rarr f primeprime = ρ T ( f prime) (263)

and its inverse is given by for g isin HomC(A A)

g rarr gprime = R(g) δ rarr gprimeprime = (idAtimes1 otimes m R(1)) (gprime otimes idR(1)) (264)

where gprimeprime is indeed a R(1)-bimodule map due to the commutativity of R(1) It is easy tocheck that gprimeprime = R(g) in (264) Therefore R gives an isomorphism from HomC(A A)to HomR(1)|R(1)(R(A) R(A)) Moreover one verifies that R(g) is an R(A)-bimodulemap iff g is an A-bimodule map In other words R g rarr R(g) gives an isomorphism

HomA|A(A A)sim=minusrarr HomR(A)|R(A)(R(A) R(A))

Corollary 227 Let A be a C-algebra

(i) A is absolutely simple iff R(A) is absolutely simple(ii) Let A be in addition Frobenius Then A is simple and special iff R(A) is simple and

special

Proof Part (i) immediately follows from Lemma 226 The statement of part (ii) withoutthe qualifier lsquosimplersquo is proved in Proposition 225 But as in the proof of (iii)rArr(i) inLemma 211 a special Frobenius algebra in a semi-simple category has a semi-simplecategory of bimodules and for a semi-simple C-linear category simple and absolutelysimple are equivalent Part (ii) then follows from part (i)

The following lemma will be needed in Sect 32 below to discuss the properties ofCardy algebras

Lemma 228 Let A be a Frobenius algebra in C Then (δA)lowast = ρA

Proof Recall from (250) that δA is a morphism A rarr T R(A) Since T and R are bothFrobenius functors T R(A) is a Frobenius algebra in C Substituting the definitions aftera short calculation one finds

Cardy Algebras and Sewing Constraints I 891

εT R(A) mT R(A) = Dim(C)oplusiisinI

A Uori Ui A U or

ı Uı

(265)

Substituting this in the definition of (δA)lowast gives again after a short calculation the mor-

phism ρA At an intermediate step one uses that the part of the morphism (δA)lowast which

is made up of Ui and Uı ribbons their duals and the basis morphisms λ(iı)0 andϒ(iı)0can be replaced by 1

dim Uimiddot dUi

3 Cardy Algebras

In this section we start by investigating the properties of Frobenius algebras which sat-isfy the so-called modular invariance condition We then give two definitions of a Cardyalgebra and prove their equivalence Finally in Sect 33 we study the properties of thesealgebras and state our main results

We fix a modular tensor category C Recall that C2plusmn is an abbreviation for C+ Cminus

31 Modular invariance In C2plusmn we define the object K and the morphism ω K rarr Kas

K =oplus

i jisinIUi times U j ω =

sumi jisinI

dim Ui dim U j

Dim C idUi timesU j (31)

They have the property (see eg [BK Cor 3111])

ω

K

(Uk times Ul )or

(Uk times Ul )or

Ui times U j

Ui times U j

= δik δ jlDim C

dim Ui dim U jbUi timesU j dUi timesU j (32)

892 L Kong I Runkel

Definition 31

(i) Let A B be objects of C2plusmn A morphism f A otimes B rarr B is called S-invariant iff

A

B

W

W

f =

A

B

W

W

f

K

ω(33)

holds for all for W isin C2plusmn(ii) A C2plusmn-algebra (Aclmcl ηcl) is called modular invariant iff θAcl = idAcl and mcl

is S-invariant

Lemma 32 The morphism f A otimes B rarr B is S-invariant if and only if

A

B

Ui times U j

Ui times U j

f = Dim Cdim Ui dim U j

sumα

A

B

B

f

Ui times U j

Ui times U j

α

α

(34)

holds for all i j isin I

Proof Condition (33) holds for all W iff it holds for all W = Ui times U j i j isin I so itis enough to show that the right-hand side of (33) with W = Ui times U j is equal to theright-hand side of (34) Recall the notation for basis morphisms in (236) Starting from(33) write

idBor otimes idUi timesU j =⎛⎝sum

klα

(b(ktimeslα)

B

)or (

bB(ktimeslα)

)or⎞⎠ otimes idUi timesU j (35)

and then apply (32) The graphical representation of the resulting morphism can bedeformed to give (34) Remark 33 As shown in [K3 Sect 61] the modular invariance condition of a C2plusmn-alge-bra exactly coincides with the modular invariance condition for torus 1-point correlationfunctions of a genus-01 closed CFT In particular the condition θAcl = idAcl is equiv-alent to invariance under the modular transformation T τ rarr τ + 1 and the condition(34) with f = mcl is equivalent to invariance under S τ rarr minus 1

τ Combining the

modular invariance condition with the genus zero properties of a genus-01 closed CFTresults in a modular invariant commutative symmetric Frobenius algebra in C2plusmn

Cardy Algebras and Sewing Constraints I 893

Let Acl be a modular invariant C2plusmn-algebra Evaluating (34) for f = mcl composingit with ηcl otimes idUi timesU j and taking the trace implies the following identity

Zi j = 1

DimCAclUi timesU j where

(36)

Zi j = dimC HomC2plusmn(UitimesU j Acl)

Decomposing Acl into simple objects this gives

Zi j =sum

klisinISik Zkl Sminus1

l j where Si j = si jradic

DimC (37)

which in CFT terms is of course nothing but the invariance of the torus partition functionunder the modular S-transformation

The following theorem gives a simple criterion for modular invariance

Theorem 34 Let Acl be a haploid commutative symmetric Frobenius algebra in C2plusmn

(i) If Acl is modular invariant then dim Acl = Dim C(ii) If dim Acl = Dim C then Acl is special and modular invariant

Proof Part (i) Since Acl is haploid for i = j = 0 Equation (36) reduces to 1 =dim AclDimCPart (ii) By the same reasoning as in the proof of (iii)rArr(i) in Lemma 211 one showsthat Acl is special Thus mcl cl = ζclidAcl for some ζcl = 0

By [KO Thm 45] the category (C2plusmn)locAcl

of local Acl-modules is again a mod-

ular tensor category and Dim (C2plusmn)locAcl

= (Dim C dim Acl)2 (see [Fr Prop 321 amp

Rem 323] for the same statement in the notation used here) Thus by assumption wehave Dim(C2plusmn)loc

Acl= 1 It then follows from [ENO1 Thm 23] that up to isomorphism

(C2plusmn)locAcl

has a unique simple object (namely the tensor unit) In other words every simplelocal Acl-module is isomorphic to Acl (seen as a left-module over itself)

We have the following isomorphisms between morphism spaces [FS Prop 47 amp 411]

HomAcl(Acl otimes (Ui times U j ) Acl) sim= HomC2plusmn(Ui times U j Acl)

HomAcl(Acl Acl otimes (Ui times U j )) sim= HomC2plusmn(AclUi times U j )

Using these to transport the bases (236) from the right to the left we obtain bases bα(i j)αof HomAcl(Acl otimes (Ui times U j ) Acl) and b(i j)

β β of HomAcl(Acl Acl otimes (Ui times U j )) Thesecan be expressed graphically as

894 L Kong I Runkel

bα(i j) =

Acl

Acl

Acl

Ui times U j

α

b(i j)β = DimC

dim Ui dim U j

1

ζcl

Acl

Ui times U jAcl

Acl

β

(38)

where the nonzero factor in b(i j)β is included for convenience Notice that bα(i j) b(i j)

β isa left Acl-module map Since Acl is simple as a left module over itself we have

bα(i j) b(i j)β = λαβ idAcl (39)

for some λαβ isin C By computing tr(bα(i j) b(i j)β ) it is easy to verify that λαβ = δαβ

We will now prove the following identity

1

ζcl

Acl

Acl

Ui times U j

Ui times U j

Acl

=sumα

DimCdim Ui dim U j

1

ζcl

Acl

Ui times U j

Ui times U j

Acl

Acl

Acl

α

α

(310)

One checks that the left-hand side of this equation is an idempotent which we denote byζminus1

cl PlAcl(Ui timesU j ) cf [Fr Sect 31] By [Fr Prop 41] the image Im(ζminus1

cl PlAcl(Ui timesU j ))

is a local Acl-module and hence isomorphic to AoplusNcl for some N isin Zge0

All left-module morphisms from Acl to Aclotimes(Ui timesU j ) are linear combinations of the

b(i j)β Furthermore one verifies that ζminus1

cl PlAcl(Ui timesU j )b(i j)

β = b(i j)β Therefore the b(i j)

β

describe precisely the image of the idempotent ie ζminus1cl Pl

Acl(Ui timesU j ) = sum

α b(i j)α bα(i j)

which is nothing but (310) Composing (310) with ζcl middot εcl otimes idUi timesU j from the left (iefrom the top) produces (34)

In addition since Acl is commutative symmetric Frobenius it satisfies θAcl = idAcl

[Fr Prop 225] Altogether this shows that Acl is modular invariant Remark 35 As we were writing this paper we heard that the results in Theorem 34were obtained independently by Kitaev and Muumlger [Ki]

Remark 36 Setting i = j = 0 in (36) gives the identity dim Acl = Z00 Dim C [KRProp 23] Combining this with Theorem 34 (ii) one may wonder if a general modularinvariant commutative symmetric Frobenius algebra Acl in C2plusmn is isomorphic to a directsum of simple such algebras However this is not so For example one can take the com-mutative symmetric Frobenius algebra Acl = C[x]x2 in the category of vector spacesequipped with the non-degenerate trace ε(ax + b) = a In this case the modular invari-ance condition holds automatically but Acl is clearly not a direct sum of two algebras

Cardy Algebras and Sewing Constraints I 895

For a general modular tensor category C the algebra C[x]x2 R(1) understood as analgebra in C2plusmn via the braided monoidal isomorphism Vect f (C) C2plusmn rarr C2plusmn providesanother counter-example

32 Two definitions Define a morphism PlA A rarr A for a Frobenius algebra A in C

or C2plusmn as follows [Fr Sect 24]

PlA =

A

A

A

m

(311)

If A is also commutative and obeys m A A = ζA idA we have PlA = ζA idA In partic-

ular this holds if A is commutative and special Using the fact that the Frobenius algebraR(1) is commutative and normalised-special one can check that Pl

R(A) R(A) rarr R(A)takes the following form

PlR(A) =

oplusiisinI

A

A

Uori

Uori

A

m

times

Ui

Ui

(312)

With these ingredients we can now give the first definition of a Cardy C|C2plusmn-algebrawhich was introduced in [K3 Def 514] cf Remark 315 below

Definition 37 (CardyC|C2plusmn-algebra I) A CardyC|C2plusmn-algebra is a triple (Aop|Acl ιcl-op)where (Aclmcl ηclcl εcl) is a modular invariant commutative symmetric Frobe-nius C2plusmn-algebra (Aopmop ηopop εop) is a symmetric Frobenius C-algebra andιcl-op Acl rarr R(Aop) an algebra homomorphism such that the following conditionsare satisfied

(i) Centre condition

Acl R(Aop)

R(Aop)

R(Aop)

ιcl-op

m R(Aop)

=

Acl

ιcl-op

R(Aop)

R(Aop)

R(Aop)

m R(Aop)

(313)

896 L Kong I Runkel

(ii) Cardy condition

ιcl-op ιlowastcl-op =

R(Aop)

R(Aop)

R(Aop)

(314)

Remark 38

(i) The name ldquoCardy C|C2plusmn-algebrardquo in Definition 37 was chosen because many ofthe important ingredients were first studied by Cardy the modular invariance ofthe closed theory [C1] the consistency of the annulus amplitude [C2] and thebulk-boundary OPE [CL] On the other hand the boundary-boundary OPE andthe OPE analogue of the centre condition were first considered in [Lw]

(ii) One can easily see that in the special case that C is the category Vect f (C) of finite-dimensional C-vector spaces a Cardy C|C2plusmn-algebra gives exactly the algebraicformulation of two-dimensional open-closed topological field theory over C (cfRemark 614 in [K3]) see [Lz Sect 48] [Mo Thm 11] [AN Thm 45] [LPCor 43] [MS Sect 22] When passing to a general modular tensor category Cthere are two important differences to the two-dimensional topological field the-ory Firstly the algebras Acl and Aop now live in different categories which inparticular affects the formulation of the centre condition and the Cardy conditionSecondly the modular invariance condition has to be imposed on Acl In the caseC = Vect f (C) modular invariance holds automatically

Definition 39 A homomorphism of Cardy C|C2plusmn-algebras (A(1)op |A(1)cl ι(1)cl-op) rarr (A(2)op

|A(2)cl ι(2)cl-op) is a pair ( fop fcl) of Frobenius algebra homomorphisms fop A(1)op rarr A(2)op

and fcl A(1)cl rarr A(2)cl such that the diagram

A(1)cl

fcl

ι(1)cl-op

A(2)cl

ι(2)cl-op

R(A(1)op )

R( fop) R(A(2)op )

(315)

commutes

Remark 310 Since a homomorphism of Frobenius algebras is invertible (cf Lemma218 (iv)) a homomorphism of Cardy algebras is always an isomorphism

For a homomorphism ( fop fcl) of Cardy C|C2plusmn-algebras using the commutativity of(315) and the fact that fcl and fop are both algebra and coalgebra homomorphisms itis easy to show that (315) commutes iff

R(A(1)op )R( fop)

(ι(1)cl-op)

lowast

R(A(2)op )

(ι(2)cl-op)

lowast

A(1)cl

fcl A(2)cl

(316)

Cardy Algebras and Sewing Constraints I 897

commutesLet (Aop|Acl ιcl-op) be a Cardy C|C2plusmn-algebra Define the morphism

ιcl-op = χminus1(ιcl-op) T (Acl) minusrarr Aop (317)

Decompose Acl as Acl = oplusNn=1Cl

n times Crn such that Cl

1 times Cr1 = ηcl(1 times 1) We use ι(n)cl-op

to denote the restriction of ιcl-op to Cln times Cr

n and ι(n)cl-op to denote the restriction of ιcl-op

to Cln otimes Cr

n We introduce the following graphical notation

ι(n)cl-op =

Cln Cr

n

Aop

(318)

By (243) ιcl-op can be expressed in terms of ιcl-op as follows

ιcl-op =Noplus

n=1

oplusiisinI

sumα

Cln

Crn

Aop

α

Uori

times α

Crn

Ui

(319)

Lemma 311 The centre condition (313) is equivalent to the following condition in C

Cln Cr

n

Aop

Aop

Aop

=

Cln Cr

n

Aop

Aop

Aop

for n = 1 N (320)

Proof First insert (319) and the definition (217) of m R(Aop) into (313) Then apply thecommutativity of R(1) to the left hand side of (313) The equivalence between (313)and (320) follows immediately Remark 312 The centre condition (320) is very natural from the open-closed confor-mal field theory point of view Correlators on the upper half plane are expressed in termsof conformal blocks on the full complex plane The objects Cl

n and Crn are associated to

the field insertion at a point z in the upper half plane and at the complex conjugate pointz in the lower half plane respectively The object Aop corresponds to a field inserted ata point r on the real axis The centre condition (320) simply says that the correlationfunctions in the disjoint domains |z| gt r gt 0 and r gt |z| gt 0 are analytic continuationsof each other see [K2 Prop 118]

898 L Kong I Runkel

Recall that we define ιlowastcl-op Aop rarr T (Acl) as in (230) We introduce the graphicalnotation

ιlowastcl-op =Noplus

n=1

Cln Cr

n

Aop

=Noplus

n=1

Cln Cr

n

Aop

Aop T (minus1Acl) (321)

where the second equality follows from (215) (221) and (259)

Lemma 313 The Cardy condition (314) is equivalent to the following identity in C

Noplusn=1

DimCdim Ui

sumα

Cln

Crn

Crn

α

α

Aop

Aop Uori

Uori

=Aop

Aop

Aop

Uori

Uori

op

mop for all i isin I (322)

Proof By (244) (312) and (319) it is easy to see that (322) is equivalent to thefollowing identity

ιcl-op χ(ιlowastcl-op) = PlR(Aop)

(323)

Therefore it is enough to show that

χ(ιlowastcl-op) = ιlowastcl-op (324)

We have

χminus1(ιlowastcl-op)(1)= T (ιlowastcl-op) δ (2)= T (ιcl-op)

lowast δlowastlowast (3)= (ρ T (ιcl-op)

)lowast (4)= (ιcl-op

)lowast (325)

In step (1) we use the expression (253) for χminus1 step (2) follows from Lemma 218 (v)and Lemma 219 Step (3) is Lemma 218 (i) and Lemma 228 and finally step (4)amounts to substituting (253) and (317) Acting with χ on both sides of the aboveequality produces (324)

Combining Lemmas 311 and 313 and Proposition 216 we obtain the followingequivalent definition of Cardy C|C2plusmn-algebra (recall the graphical notation (318) forιcl-op and (321) for ιlowastcl-op)

Cardy Algebras and Sewing Constraints I 899

Definition 314 (CardyC|C2plusmn-algebra II) A CardyC-algebra is a triple (Aop|Acl ιcl-op)where Acl is a commutative symmetric Frobenius C2plusmn-algebra satisfying property (34)with f = mcl Aop is a symmetric Frobenius C-algebra and ιcl-op T (Acl) rarr Aop isan algebra homomorphism satisfying the conditions (320) and (322)

Remark 315 Up to a choice of normalisation Definition 314 is the same as the orig-inal one in [K3 Def 613] The difference between the two definitions is the factorDim C dim Ui on the left hand side of (322) which in [K3 Def 613] is given byradic

Dim C dim Ui The two definitions are related by rescaling the coproduct cl andcounit εcl of Acl by 1

radicDim C and

radicDim C respectively We chose the convention in

(322) to remove all dimension factors from the expression (314) for the Cardy condition

33 Uniqueness and existence theorems In this subsection we investigate the structureof Cardy algebras We start with the following proposition which when combined withthe results of part II provides an alternative proof of [Fj Prop 422]

Proposition 316 Let (Aop|Acl ιcl-op) be a Cardy C|C2plusmn-algebra If Acl is simple anddim Aop = 0 then Aop is simple and special

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis therefore haploid Restricting the Cardy condition (322) to the case Ui = 1 andcomposing both sides with εop from the left we see that εop kills all terms associated toU j times 1 isin Acl in the sum except for a single 1 times 1 term Thus we obtain the followingidentity

β εop = dAop (mop otimes idAorop) (idAop otimes bAop) (326)

where β isin C Composing with ηop from the right in turn implies that βεop ηop =dim Aop which is nonzero by assumption Thus also β = 0 and εop is a nonzero multipleof the morphism on the right hand side of (326) By [FRS Lem 311] Aop is special

Since Aop is a special Frobenius algebra Aop is semi-simple as an Aop-bimodule(apply [FS Prop 524] to Aop tensored with its opposite algebra) Suppose Aop is not

simple so that we can write Aop = A(1)op oplus A(2)op for nonzero Aop-bimodules A(1)op and A(2)op We denote the canonical embeddings and projections associated to this decompositionas ι12 and π12 We have the identities

mop (ι1 otimes ι2) = 0 εop ηop =2sum

i=1

εop ιi πi ηop (327)

The first identity follows since π1 mop (ι1 otimes ι2) = 0 (as mop gives the left action of

Aop on Aop and hence it preserves A(2)op ) and similarly π2 mop (ι1 otimes ι2) = 0 Thesecond identity is just the completeness of ι12 π12

Since εop ηop = 0 without losing generality we can assume εop ι1 π1 ηop = 0Using that π2 is a bimodule map we compute

π2 [LHS of (322)]Ui =1 ι1 π1 ηop = π2 [RHS of (322)]Ui =1 ι1 π1 ηop

= PlA(2)op

π2 ι1 π1 ηop = 0 (328)

900 L Kong I Runkel

On the other hand using that Acl is haploid that ιcl-op is an algebra map and thatι lowastcl-op is a coalgebra map one can check that the left-hand side of (328) is equal toλ(εop ι1 π1 ηop) π2 ηop for some λ = 0 This implies that π2 ηop = 0 Thusηop = sum

i ιi πi ηop = ι1 π1 ηop Hence we have

0 = π2 ι2 = π2 mop (ηop otimes ι2) = π2 mop ((ι1 π1 ηop)otimes ι2) (329)

However the right-hand side is zero by (327) This is a contradiction and hence Aopmust be simple

To formulate the next theorem we need the notion of the full centre of an algebra [FjDef 49] Recall that an algebra A in a braided tensor category has a left centre and aright centre [VZO] both of which are sub-algebras of A Of these two we will onlyneed the left centre The following definition is [Fr Def 231] which in our setting isequivalent to that of [VZO]

Definition 317 Let A be a symmetric special Frobenius algebra such that m A A =ζA idA

(i) The left centre Cl(A) of A is the image of the idempotent ζminus1A Pl

A(ii) The full centre Z(A) is Cl(R(A))

That ζminus1A Pl

A is an idempotent follows from [FRS Lem 52] when keeping track ofthe factors ζA ([FRS] assumes normalised-special ie ζA = 1) Note that Cl(A) is againan object of C while Z(A) is an object of C2plusmn Let el

A Cl(A) rarr A be the embeddingof Cl(A) into A The left centre is in fact the maximal subobject of A such that

m A cAA (elA otimes idA) = m A (330)

see [Fr Lem 232] This observation explains the name left centre and also makes theconnection to [O Def 15]

The full centre is by definition the image of the idempotent ζminus1A Pl

R(A) R(A) rarrR(A) Since C2plusmn is abelian the idempotent splits and we obtain the embedding andrestriction morphisms

e Z(A) rarr R(A) and r R(A) Z(A) (331)

which obey r e = idZ(A) and e r = ζminus1A Pl

R(A) It follows from Proposition 225 and

[Fr Prop 237] that Z(A) is a commutative symmetric Frobenius algebra in C2plusmn withstructure morphisms5

m Z(A) = r m R(A) (e otimes e) ηZ(A) = r ηR(A)

Z(A) = ζA middot (r otimes r) R(A) e εZ(A) = ζminus1A middot εR(A) e

(332)

Moreover if A is simple then Z(A) is simple and if A is simple and dim A = 0 thenZ(A) is simple and special The normalisation of the counit is such that

εZ(A) ηZ(A) = ζminus2A dim A DimC (333)

5 The normalisation of product and unit is the standard one The factors in the coproduct and counit haveto be included in order for (A|Z(A) e) to be a Cardy algebra see Theorem 318 below The normalisation ofthe counit enters the Cardy condition (314) through the definition of ( middot )lowast

Cardy Algebras and Sewing Constraints I 901

Theorem 318 Let A be a special symmetric Frobenius C-algebra Then (A|Z(A) e)is a Cardy C|C2plusmn-algebra

The proof of this theorem makes use the following two lemmas

Lemma 319 e Z(A) rarr R(A) is an algebra map and elowast = ζA middot r

Proof It follows from [Fr Lem 229] (or by direct calculation using in particularm R(A) R(A) = ζAidR(A)) that

m R(A) (e otimes e) = ζminus1A middot Pl

R(A) m R(A) (e otimes e) (334)

Substituting e r = ζminus1A Pl

R(A) shows that e is compatible with multiplication For theunit one finds

e ηZ(A) = e r ηR(A) = ζminus1A Pl

R(A) ηR(A) = ηR(A) (335)

Thus e is an algebra map For the second statement one computes

elowast (1)= ζA

R(A)

r

R(A)

R(A)

Z(A)

R(A)

Z(A)

r

e

e

(2)=

R(A)

r

R(A)

R(A)

R(A)

Z(A)

(3)=

R(A)

r

R(A)

Z(A)

(4)= ζA middot r (336)

where in (1) the definitions (230) and (332) have been substituted step (2) is e r = ζminus1

A PlR(A) step (3) uses that R(A) is symmetric Frobenius and step (4) is again

e r = ζminus1A Pl

R(A) Lemma 320 Let A be a symmetric Frobenius algebra in C The morphism

PlR(A) m R(A)

(Pl

R(A) otimes PlR(A)

) R(A)otimes R(A) minusrarr R(A) (337)

is S-invariant

The proof of this lemma is a slightly lengthy explicit calculation and has been deferredto Appendix A2

902 L Kong I Runkel

Proof of Theorem 318 That e is an algebra map was proved in Lemma 319 The centrecondition (313) holds by property (330) of the left centre The Cardy condition (314)also is an immediate consequence of Lemma 319

ιcl-op ιlowastcl-op = e (ζA r) = PlR(A) (338)

The full centre Z(A) is a commutative symmetric Frobenius algebra It remains to provemodular invariance That θZ(A) = idZ(A) is implied by commutativity and symmetryof Z(A) [Fr Prop 225] The S-invariance condition (33) follows from Lemma 320In (337) substitute Pl

R(A) = ζA e r and then put the resulting morphism into (33)Compose the resulting equation with e otimes idW from the right (ie from the bottom) andsubstitute the definition (332) of m Z(A) This results in the statement that m Z(A) isS-invariant

The following theorem is analogous to [LR Prop 29] and [Fj Thm 426] whichroughly speaking answer the question under which circumstances the restriction of atwo-dimensional conformal field theory to the boundary already determines the entireconformal field theory The first work is set in Minkowski space and uses operator alge-bras and subfactors while the second work is set in Euclidean space and uses modulartensor categories

Theorem 321 Let (A|Acl ιcl-op) be a Cardy C|C2plusmn-algebra such that dim A = 0 andAcl is simple Then A is special and (A|Acl ιcl-op) sim= (A|Z(A) e) as Cardy algebras

Proof By Proposition 316 A is simple and special Since Acl is simple the algebramap ιcl-op Acl rarr R(A) is either zero or a monomorphism But ιcl-op ηcl = ηR(A)and so ιcl-op = 0 Thus ιcl-op is monic By Lemma 218 (ii) ζminus1

A ιlowastcl-op is epi The Cardycondition (314) implies

ιcl-op ζminus1A ιlowastcl-op = ζminus1

A PlR(A) = e r (339)

Composing this with e r from the left yields e r ιcl-op ζminus1A ιlowastcl-op = e r =

ιcl-op ζminus1A ιlowastcl-op Since ζminus1

A ιlowastcl-op is epi we have

e r ιcl-op = ιcl-op (340)

Actually (340) also follows from (313) and specialness of R(A) We will prove that

( fop fcl) (A|Acl ιcl-op) minusrarr (A|Z(A) e) where fop = idA fcl = r ιcl-op

(341)

is an isomorphism of Cardy algebrasfcl is an algebra map Compatibility with the units follows since ιcl-op is an algebramap

fcl ηcl = r ιcl-op ηcl = r ηR(A) = ηZ(A) (342)

Compatibility with the multiplication also follows since ιcl-op is an algebra map

m Z(A) ( fcl otimes fcl) = r m R(A) (e otimes e) (r otimes r) (ιcl-op otimes ιcl-op)

= r m R(A) (ιcl-op otimes ιcl-op) = r ιcl-op mcl = fcl mcl (343)

where in the second step we used (340)

Cardy Algebras and Sewing Constraints I 903

fcl is an isomorphism As above since fcl is an algebra map and since Acl is simple fcl

has to be monic By Lemma 319 rlowast = ζminus1A e Thus f lowast

cl = ιlowastcl-op rlowast = ζminus1A ιlowastcl-op e and

fcl f lowastcl = r ιcl-op ζminus1

A ιlowastcl-op e = r e r e = idZ(A) (344)

and so fcl is also epi and hence isofcl is a coalgebra map Since fcl is an algebra map so is f minus1

cl By (344) f minus1cl = f lowast

cland by Lemma 218 (iii) this implies that fcl is a also coalgebra mapThe diagram (315) commutes Commutativity of (315) is equivalent to e fcl = ιcl-opwhich holds by (340)

Let A be a special symmetric Frobenius algebra So far we have seen that (A|Z(A) e)is a Cardy algebra and that all Cardy algebras with Aop = A and simple Acl are of thisform It is now natural to ask if every simple Acl does occur as part of a Cardy algebraThe following theorem provides an affirmative answer Recall that for an A-left moduleM the object Mor otimesA M is an algebra (see eg [KR Lem 42])

Theorem 322 If Acl is a simple modular invariant commutative symmetric FrobeniusC2plusmn-algebra then there exist a simple special symmetric Frobenius C-algebra A and amorphism ιcl-op Acl rarr R(A) such that

(i) Acl sim= Z(A) as Frobenius algebras(ii) (A|Acl ιcl-op) is a Cardy C|C2plusmn-algebra

(iii) T (Acl) sim= oplusκisinJ Morκ otimesA Mκ as algebras where MκκisinJ is a set of representa-

tives of the isomorphism classes of simple A-left modules

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis haploid It then follows from Theorem 34 that Acl is special By Proposition 224T (Acl) is a special symmetric Frobenius algebra in C Thus T (Acl) = oplusi Ai where theAi are simple symmetric Frobenius algebras We will show that at least one of the Aiis special Since T (Acl) is special we have mT (Acl) T (Acl) = ζ idT (Acl) for someζ isin C

times Restricting this to the summand Ai shows mi i = ζ idAi FurthermoreεT (Acl) ηT (Acl) = ξ id1 for some ξ isin C

times But εT (Acl) ηT (Acl) = sumi εi ηi and so

at least one of the εi ηi has to be nonzero Therefore at least one of the Ai is speciallet A equiv Ai be this summand We denote the embedding A rarr T (Acl) by e0 and therestriction T (Acl) A by r0 Notice that r0 is an algebra homomorphism Define

ιcl-op = χ (r0) Acl minusrarr R(A) (345)

By Proposition 216 ιcl-op is an algebra homomorphism Next we verify the centre con-dition (313) or rather its equivalent form (320) By substituting the definitions onecan convince oneself that the commutativity mcl cAclAcl = mcl of Acl in C2plusmn implies thecondition mT (Acl) = mT (Acl) in C see [K2 Prop 36] Here T (Acl)otimes T (Acl) rarrT (Acl)otimes T (Acl) is given by

=oplusmn

(idCl

notimes cCl

m Crnotimes idCr

m

)

(cCl

m Clnotimes cminus1

Crn C

rm

)

(idCl

motimes cminus1

Cln C

rm

otimes idCrn

)

(346)

and we decomposed Acl as Acl = oplusnCln times Cr

n As a consequence we obtain the identity

r0 mT (Acl) (idAcl otimes e0) = r0 mT (Acl) (idAcl otimes e0) (347)

904 L Kong I Runkel

Using that r0 is an algebra map and that by definition ιcl-op = r0 we obtain (320)In order to show that (A|Acl ιcl-op) is a Cardy algebra it remains to show that the

Cardy condition (314) is satisfied We will demonstrate this via a detour by first provingthat Acl sim= Z(A) as Frobenius algebras

Recall the notations e and r given in (331) Using the centre condition (320) onecan check that Pl

R(A) ιcl-op = m R(A) R(A) ιcl-op By specialness of A we have

m A A = ζA idA and so together with e r = ζminus1A Pl

R(A) we get

e r ιcl-op = ιcl-op (348)

Next consider the morphism

fcl = r ιcl-op Acl minusrarr Z(A) (349)

By the same derivation as in (342) and (343) one sees that fcl is an algebra map Inparticular fcl ηcl = ηZ(A) = 0 and so fcl = 0 Since Acl is simple fcl has to be amonomorphism

By the same argument as used in the proof of Theorem 34 (ii) up to isomorphismAcl is the unique simple local Acl-(left-)module The algebra monomorphism fcl turnsZ(A) into an Acl-module Since Z(A) is commutative it is local as an Acl-module andso Z(A) sim= AoplusN

cl for some N ge 1 By construction A is a simple special symmetricFrobenius algebra Proposition 225 and Corollary 227 show that R(A) inherits all theseproperties and thus Z(A) is simple (see the comment below Eq (332)) By Theorem318 Z(A) is modular invariant and then by Theorem 34 (i) dim Z(A) = Dim C Thisimplies that N = 1 in Z(A) sim= AoplusN

cl and so fcl is in fact an isomorphismSince Acl and Z(A) are both haploid we have εZ(A) fcl = ξ εcl for some ξ isin C

timesThe counit uniquely determines the Frobenius structure on Acl and Z(A) (see eg [FRSLemma 37]) so that fcl is a coalgebra isomorphism iff ξ = 1 To compute ξ we composethe above identity with ηcl from the right Defining ζcl via εcl ηcl = ζminus1

cl Dim C middot id1 andusing (333) gives ξ = dim A ζclζ

2A By rescaling the comultiplication and the counit

of A and consequently changing ζA we can always achieve ξ = 1 This proves part (i)of the theorem

Equation (348) implies that ιcl-op = e fcl Since fcl is an isomorphism of Frobeniusalgebras by Lemmas 218 and 319 we have

ιcl-op ιlowastcl-op = e fcl f lowastcl elowast = ζA e r = Pl

R(A) (350)

Thus (A|Acl ιcl-op) is a Cardy algebra This proves part (ii) of the theoremPart (iii) can be seen as follows By [KR Prop 43] T Z(A) sim= oplusκisinJ Mor

κ otimesA Mκ asalgebras Together with the observation that T ( fcl) T (Acl) rarr T Z(A) is an isomor-phism of algebras this proves part (iii) Remark 323 Part (i) of Theorem 322 was announced by Muumlger [Muuml2] We provide anindependent proof in the setting of Cardy algebras

The above theorem together with Lemma 211 and Theorem 34 shows that a sim-ple commutative symmetric Frobenius C2plusmn-algebra Acl with dim Acl = Dim C is alwayspart of a Cardy algebra (Aop|Acl ιcl-op) for some simple special symmetric Frobeniusalgebra Aop in C However the above proof also illustrates that Aop is not unique Thisraises the question how two Cardy algebras with a given Acl can differ This question isanswered by [KR Thm 11] which in the present framework can be restated as follows

Cardy Algebras and Sewing Constraints I 905

Theorem 324 If (A(i)op |A(i)cl ι(i)cl-op) i = 1 2 are two Cardy C|C2plusmn-algebras such that

A(i)cl is simple and dim A(i)op = 0 for i = 1 2 then A(1)clsim= A(2)cl as algebras if and only

if A(1)op and A(2)op are Morita equivalent

Proof Theorem 11 in [KR] is stated for A(i)op being non-degenerate algebras and A(i)cl =Z(A(i)op ) for i = 1 2 By Proposition 316 A(i)op are simple and special for i = 1 2

Then by [KR Lem 21] A(i)op are non-degenerate algebras By Theorem 321 we have

A(i)clsim= Z(A(i)op ) as Frobenius algebras Finally by [KR Thm 11] Z(A(1)op ) sim= Z(A(2)op )

as algebras iff A(1)op and A(2)op are Morita equivalent Let Cmax(C2plusmn) be the set of equivalence classes [B] of simple modular invariant

commutative symmetric Frobenius algebras B in C2plusmn Two such algebras B and B prime areequivalent if B and B prime are isomorphic as algebras (but not necessarily as Frobenius alge-bras) Let Msimp(C) be the set of Morita classes of simple special symmetric Frobeniusalgebras in C Define the map z Msimp(C) rarr Cmax(C2plusmn) by z A rarr [Z(A)] whereA denotes the Morita class of A From Theorem 322 (i) and [KR Thm 11] we learn

Corollary 325 The map z Msimp(C) rarr Cmax(C2plusmn) is a bijection

A Appendix

A1 Proof of Lemma 27 We will show that if (F ψ F2 ψ

F0 ) is a colax tensor functor

from C1 to C2 then (G φG2 φ

G0 ) is a lax tensor functor from C2 to C1 Applying this

result to the opposed categories then gives the converse statementWe need to show that φG

0 and φG2 make the diagrams (21) and (22) commute We

first prove the commutativity of (21) Consider the following diagram

G(A)otimes (G(B)otimes G(C))

GψF2 δ

idG(A)otimesφG2 G(A)otimes G(B otimes C)

GψF2 δ

G(FG(A)otimes F(G(B)otimes G(C)))

G(F(idG(A))otimesF(φG2 ))

G(idFG(A)otimesψ F2 )

G(FG(A)otimes FG(B otimes C))

G(ρAotimesρBotimesC )

G(FG(A)otimes (FG(B)otimes FG(C)))

G(ρAotimes(ρBotimesρC )) G(A otimes (B otimes C))

(A1)

The top subdiagram is commutative because of the naturality of Gψ F2 δ The commu-

tativity of the bottom subdiagram follows from the following identities

(ρB otimes ρC ) ψ F2 = (ρB otimes ρC ) ψ F

2 ρF Fδ

= ρBotimesC FG(ρB otimes ρC ) FG(ψ F2 ) Fδ

= ρBotimesC F(φG2 ) (A2)

as a map F(G(B) otimes G(C)) rarr B otimes C The commutativity of (A1) implies that thecomposition of maps in the left column in (21) can be replaced by

G(ρA otimes (ρB otimes ρC )) G((idFG(A) otimes ψ F

2 ) ψ F2

) δ (A3)

906 L Kong I Runkel

Similarly we can show that the composition of maps in the right column in (21) can bereplaced by

G((ρA otimes ρB)otimes ρC ) G((ψ F

2 otimes idFG(C)) ψ F2

) δ (A4)

Using the commutativity of (23) it is easy to see that (21) with the left and right col-umns of (21) replaced by (A3) and (A4) respectively is commutative Hence (21) iscommutative

Now we prove the commutativity of the first diagram in (22)

φG2 (φG

0 otimes idG(A))

(1)= G(ρ12 otimes ρA) Gψ F2 δ

[(Gψ F

0 δ11)otimes idG(A)

]

(2)= G(ρ12 otimes ρA) Gψ F2 G F(Gψ0 otimes idG(A)) G F(δ11 otimes idG(A)) δ

(3)= G(ρ12 otimes ρA) G(FG(ψ F0 )otimes idFG(A)) Gψ F

2 G F(δ11 otimes idG(A)) δ(4)= G(id12 otimes ρA) G

([ψ F

0 ρF(11) (Fδ)11] otimes idFG(A)

) Gψ F

2 δ(5)= G(id12 otimes ρA) G(ψ F

0 otimes idFG(A)) Gψ F2 δ

(6)= G(id12 otimes ρA) G(lminus1FG(A)) G F(lG(A)) δ

(7)= G(lminus1A ) GρA δG lG(A)

(8)= G(lminus1A ) lG(A) (A5)

where in step (1) we substituted the definition of φG0 φ

G2 given in (210) in step (2)

we used the naturality of δ in step (3) we used the naturality of Gψ F2 in step (4) we

switched the position between Gρ12 and G FG(ψ F0 ) and the position between Gψ2 and

G F(δ12 otimes idG(A)) using the naturality of ρ and FψG2 respectively in step (5) we applied

the second identity in (28) in step (6) we used (24) in step (7) we used the naturalityof lminus1 and δ in step (8) we used the first identity in (28)

The proof of the commutativity of the second diagram in (22) is similar Thus wehave shown that G is a lax tensor functor

A2 Proof of Lemma 320 To prepare the proof recall that for a given object B isin Cthe modular group P SL(2Z) acts on the space oplusi HomC(B otimes Ui Ui ) see eg [BKSect 31] and [K3 Eq (455)] We will only need the action of S and Sminus1 Let f isinoplusi HomC(B otimes Ui Ui ) Then

S oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A6)

Cardy Algebras and Sewing Constraints I 907

Sminus1 oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A7)

By Lemma 32 to establish that (337) is S-invariant it is enough to prove the identity(34) when f is given by (337) Using (A6) and (A7) we can see that Eq (34) simplysays that oplusi j [RHS of (34)] is invariant under the action of S times S Consider the elementg of oplus jkisinIHomC2plusmn(R(A)otimes (Uor

j times Uk)Uorj times Uk) given by

g =oplusjkisinI

sumα

R(A)

R(A)

R(A)

R(A)

Uorj times Uk

Uorj times Uk

α

α

(A8)

By the above arguments proving S-invariance of (337) is equivalent to proving invari-ance of g under the action of S times S

For i isin I we denote by gi the component of g in(oplus jisinIHomC(A otimes Uor

i otimes Uorj U

orj )

)otimes (opluskisinIHomC(Ui otimes UkUk))

We view the second Hom-space in the above tensor product as a Hom-space in C+ insteadof Cminus It is enough to show that gi is invariant under the action of S times Sminus1 Note thatthe action of Sminus1 in C+ is equivalent to that of S in Cminus

The morphism gi can be canonically identified with a bilinear pairing

( middot middot )i ⎛⎝oplus

jisinIHomC(Uor

j A otimes Uori otimes Uor

j )

⎞⎠

times(oplus

kisinIHomC(UkUi otimes Uk)

)minusrarr C (A9)

as follows For h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) we set

(h1 h2)i = (dim U j dim Uk)minus1 trUor

j timesUk[gi (h1 times h2)] (A10)

908 L Kong I Runkel

When substituting the explicit form of the product m R(A) of R(A) = (A times 1) otimes R(1)after a short calculation one finds

(h1 h2)i =sumα

1

dim U j

A

A

A

A

A A

A α

α

h1

h2

U jUk

Uorj

Ui

Uk (A11)

Here the top morphism PlR(A) has been simplified with the help of the identity

PR(A) m R(A) (PR(A) otimes PR(A))

=(oplus

iisinI((m A A)otimes idUor

i)times idUi

) m R(A) (PR(A) otimes PR(A)) (A12)

which can be checked by direct calculation along the same lines as in the proof of [FrLem 310]

The action of the modular transformation S on oplusiisinI(B otimes Ui Ui ) for B isin C nat-urally induces an action on oplusiisinI(Ui B otimes Ui ) [K3 Prop 514] which we denote bySlowast In the present case we get an action of Slowast on oplus jisinIHomC(Uor

j A otimes Uori otimes Uor

j ) and

opluskisinIHomC(UkUi otimes Uk) Then to show gi is invariant under the action of S times Sminus1

amounts to showing that

(h1 h2)i = ((Sminus1)lowast h1 Slowast h2)i (A13)

Cardy Algebras and Sewing Constraints I 909

for all h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) We have

((Sminus1)lowast h1 Slowast h2)i =sum

mnα

dim Un

DimC

A

A

A

A

A A

A

Uk

α

α

h1

h2

Ui

U j

UmUn

(A14)

Now drag the upper vertex indexed by α in the above graph along its Uorm -leg until it

meets the lower vertex also indexed by α then sum over α and m This gives

((Sminus1)lowast h1 Slowast h2)i =sum

n

dim Un

DimC

A

A

A

A

A A

A

Uk

Un

h1

h2

Ui

U j

(A15)

910 L Kong I Runkel

If we just look at the neighbourhood of the Un-loop in the above graph we see thefollowing subgraphs

sumn

dim Un

DimC

A Aor

Uorj U j

Uk Uork

Un=

sumα

1

dim U j

A Aor Uk Uork

Uorj U j

αα

(A16)

where we have applied [BK Cor 3111] Substituting this subgraph back to the originalgraph in (A15) we obtain

((Sminus1)lowast h1 Slowast h2)i =sumα

1

dim U j

A

A

A

A

A A

A

A

A

Uorj

α

α

h1

h2m A

Ui

U jUk

Uk

(A17)

The graph in (A17) is equal to that in (A11) In order to see this we first drag theldquobubblerdquo (m A A) along A lines and through the m A vertex (because m A A is abimodule map) until it reaches the lower-left leg of the upper vertex indexed by α Thendrag the m A vertex along the (red) dotted line in the above graph Finally we applythe associativity of A (A12) and [Fr Lem 311] Then we see that the graph in (A17)exactly matches with the one in (A11)

Acknowledgements We would like to thank the organisers of the Oberwolfach Arbeitsgemeinschaft ldquoAlge-braic structures in conformal field theoriesrdquo (April 2007) where this work was started for an inspiring meetingWe would further like to thank the Hausdorff Institute for Mathematics in Bonn and the organisers of the stim-ulating meeting ldquoGeometry and Physicsrdquo (May 2008) We are indebted to Alexei Davydov Jens FjelstadJuumlrgen Fuchs Yi-Zhi Huang Alexei Kitaev Urs Schreiber Christoph Schweigert Stephan Stolz and PeterTeichner for helpful discussions andor comments on a draft of this paper The research of IR was partiallysupported by the EPSRC First Grant EPE0050471 the PPARC rolling grant PPC5071451 and the MarieCurie network lsquoSuperstring Theoryrsquo (MRTN-CT-2004-512194)

Cardy Algebras and Sewing Constraints I 911

References

[AN] Alexeevski A Natanzon SM Noncommutative two-dimensional topological field theories andhurwitz numbers for real algebraic curves Sel Math New Ser 12 307ndash377 (2006)

[Bi] Bichon J Cosovereign hopf algebras J Pure Appl Alg 157 121ndash133 (2001)[BK] Bakalov B Kirillov AA Lectures on Tensor Categories and Modular Functors Providence

RI Amer Math Soc 2001[C1] Cardy JL Operator content of two-dimensional conformal invariant theories Nucl Phys

B 270 186ndash204 (1986)[C2] Cardy JL Boundary conditions fusion rules and the verlinde formula Nucl Phys B 324

581ndash596 (1989)[CL] Cardy JL Lewellen DC Bulk and boundary operators in conformal field theory Phys Lett

B 259 274ndash278 (1991)[DMZ] Dong C-Y Mason G Zhu Y-C Discrete series of the Virasoro algebra and the moonshine

module In Algebraic Groups and Their Generalizations Quantum and infinite-dimensional Meth-ods Proc Symp Pure Math 56 Part 2 Providence RI Amer MathSoc 1994 pp 295ndash316

[DP] Day B Pastro C Note on frobenius monoidal functors New York J Math 14 733ndash742 (2008)[ENO1] Etingof PI Nikshych D Ostrik V On fusion categories Ann Math 162 581ndash642 (2005)[ENO2] Etingof PI Nikshych D Ostrik V An analogue of radfordrsquos s4 formula for finite tensor

categories Int Math Research Notices 54 2915ndash2933 (2004)[Fe] Felder G Froumlhlich J Fuchs J Schweigert C Correlation functions and boundary conditions in

rational conformal field theory and three-dimensional topology Comp Math 131 189ndash237 (2002)[FHL] Frenkel IB Huang Y-Z Lepowsky J On axiomatic approaches to vertex operator algebras

and modules Mem Amer Math Soc 104 (1993)[Fj] Fjelstad J Fuchs J Runkel I Schweigert C Uniqueness of openclosed rational cft with

given algebra of open states Adv Theor Math Phys 12 1283ndash1375 (2008)[FK] Froumlhlich J King C The chern-simons theory and knot polynomials Commun Math

Phys 126 167ndash199 (1989)[Fr] Froumlhlich J Fuchs J Runkel I Schweigert C Correspondences of ribbon categories Adv

Math 199 192ndash329 (2006)[FRS] Fuchs J Runkel I Schweigert C Tft construction of rcft correlators I partition functions Nucl

Phys B 646 353ndash497 (2002)[FS] Fuchs J Schweigert C Category theory for conformal boundary conditions Fields Inst Com-

mun 39 25ndash71 (2003)[H1] Huang Y-Z Two-dimensional conformal geometry and vertex operator algebras Progress in

Mathematics Vol 148 Boston Birkhaumluser 1997[H2] Huang Y-Z Rigidity and modularity of vertex tensor categories Comm Contemp Math 10

871ndash911 (2008)[HK1] Huang Y-Z Kong L Open-string vertex algebra category and operads Commun Math

Phys 250 433ndash471 (2004)[HK2] Huang Y-Z Kong L Full field algebras Commun Math Phys 272 345ndash396 (2007)[HK3] Huang Y-Z Kong L Modular invariance for conformal full field algebras httparxivorgabs

math0609570v2[mathQA] 2006[HL] Huang Y-Z Lepowsky J Tensor products of modules for a vertex operator algebra and vertex

tensor categories In Lie Theory and Geometry in honor of Bertram Kostant ed R BrylinskiJ-L Brylinski V Guillemin V Kac Boston Birkhaumluser 1994 pp 349ndash383

[JS] Joyal A Street R Braided tensor categories Adv Math 102 20ndash78 (1993)[K1] Kong L Full field algebras operads and tensor categories Adv Math 213 271ndash340 (2007)[K2] Kong L Open-closed field algebras Commun Math Phys 280 207ndash261 (2008)[K3] Kong L Cardy condition for open-closed field algebras Commun Math Phys 283 25ndash92 (2008)[Ki] Kitaev A Private communication[KO] Kirillov AA Ostrik V On q-analog of mckay correspondence and ade classification of sl(2)

conformal field theories Adv Math 171 183ndash227 (2002)[KR] Kong L Runkel I Morita classes of algebras in modular tensor categories Adv Math 219

1548ndash1576 (2008)[Ld] Lauda AD Frobenius algebras and ambidextrous adjunctions Theo Appl Cat 16 84ndash

122 (2006)[Li1] Li H-S Regular representations of vertex operator algebras Commun Contemp Math 4

639ndash683 (2002)[Li2] Li H-S Regular representations and huang-lepowsky tensor functors for vertex operator

algebras J Alge 255 423ndash462 (2002)

Cardy Algebras and Sewing Constraints I 877

F(A)otimes (F(B)otimes F(C))

idF(A)otimesφF2

(F(A)otimes F(B))otimes F(C)αminus1

F(A)otimes F(B otimes C) F(A otimes B)otimes F(C)

ψF2 otimesidF(C)

φF2

F(A otimes (B otimes C))

ψF2

F((A otimes B)otimes C)F(αminus1)

(26)

Proposition 25 If (F φF2 φ

F0 ) is a tensor functor then F is a Frobenius functor with

ψ F0 = (φF

0 )minus1 and ψ F

2 = (φF2 )

minus1

Proof Since F is a tensor functor it is lax and colax If we replace ψ F2 by (φF

2 )minus1 in

(25) and (26) both commuting diagrams are equivalent to (21) which holds becauseF is lax Thus F is a Frobenius functor

The converse statement does not hold For example the functor R which we definein Sect 24 is Frobenius but not tensor

Let us recall the notion of adjunctions and adjoint functors [Ma Ch IV1]

Definition 26 An adjunction from C1 to C2 is a triple 〈FG χ〉 where F and G arefunctors

F C1 rarr C2 G C2 rarr C1

and χ is a natural isomorphism which assigns to each pair of objects A1 isin C1 A2 isin C2a bijective map

χA1A2 HomC2(F(A1) A2)sim=minusminusrarr HomC1(A1G(A2))

which is natural in both A1 and A2 F is called a left-adjoint of G and G is called aright-adjoint of F

For simplicity we will often abbreviate χA1A2 as χ Associated to each adjunction

〈FG χ〉 there are two natural transformations idC1

δminusrarr G F and FGρminusrarr idC2 where

idC1 and idC2 are identity functors given by

δA1 = χ(idF(A1)) ρA2 = χminus1(idG(A2)) (27)

for Ai isin Ci i = 1 2 They satisfy the following two identities

GδGminusrarr G FG

Gρminusrarr G = GidGminusminusrarr G F

Fδminusrarr FG FρFminusrarr F = F

idFminusminusrarr F (28)

We have for g F(A1) rarr A2 and f A1 rarr G(A2)

χ(g) = G(g) δA1 χminus1( f ) = ρA2 F( f ) (29)

For simplicity δA1 and ρA2 are often abbreviated as δ and ρ respectivelyLet 〈FG χ〉 be an adjunction from a tensor category C1 to a tensor category C2

and (F ψ F2 ψ

F0 ) a colax tensor functor from C1 to C2 We can define a morphism

878 L Kong I Runkel

φG0 11 rarr G(12) and a natural transformation φG

2 otimes (G times G) rarr G otimes by forA B isin C2

φG0 = χ(ψ F

0 ) = 11δ11minusrarr G F(11)

G(ψ F0 )minusminusminusminusrarr G(12)

φG2 = χ((ρA otimes ρB) ψ F

2 ) = G(A)otimes G(B)δminusrarr G F (G(A)otimes G(B))

G(ψ F2 )minusminusminusminusrarr G (FG(A)otimes FG(B))

G(ρAotimesρB )minusminusminusminusminusminusrarr G(A otimes B) (210)

where we have used the first identity in (29) Notice that φG2 is natural because it is a

composition of natural transformations One can easily show that ψ F0 and ψ F

2 can bere-obtained from φG

0 and φG2 as follows

ψ F0 = χminus1(φG

0 ) = F(11)FφG

0minusminusrarr FG(12)ρminusrarr 12

ψ F2 = χminus1(φG

2 (δ otimes δ)) = F(U otimes V )F(δotimesδ)minusminusminusminusrarr F (G F(U )otimes G F(V ))

FφG2minusminusrarr FG (F(U )otimes F(V ))

ρminusrarr F(U )otimes F(V ) (211)

for U V isin C1 The following result is standard for the sake of completeness we givea proof in Appendix A1

Lemma 27 (F ψ F2 ψ

F0 ) is a colax tensor functor iff (G φG

2 φG0 ) is a lax tensor

functor

22 Algebras in tensor categories An algebra in a tensor category C or a C-algebra isa triple A = (Am η) where A is an object of C m (the multiplication) is a morphismA otimes A rarr A such that m (m otimes idA) αAAA = m (idA otimes m) and η (the unit) is amorphism 1 rarr A such that m (idA otimes η) = idA rA and m (ηotimes idA) = idA lAIf C is braided and m cAA = m then A is called commutative

A left A-module is a pair (MmM ) where M isin C and m M is a morphism AotimesM rarr Msuch that mM(idAotimesm M ) = m M(m AotimesidM )αAAM and m M(ηAotimesidM ) = idMlM Right A-modules and A-bimodules are defined similarly

Definition 28 Let C be a tensor category and let A be an algebra in C

(i) A is called simple iff it is simple as a bimodule over itselfLet C be in addition k-linear for k a field

(ii) A is called absolutely simple iff the space of A-bimodule maps from A to itself isone-dimensional dimk HomA|A(A A) = 1

(iii) A is called haploid iff dimk Hom(1 A) = 1 [FS Def 43]

In the following we will assume that all tensor categories are strict to avoid spelling outassociators and unit constraints

A C-coalgebra A = (A ε) is defined analogously to a C-algebra ie A rarrA otimes A and ε A rarr 1 obey coassociativity and counit conditions

If C is braided and if A and B are C-algebras there are two in general non-isomorphicalgebra structures on A otimes B We choose A otimes B to be the C-algebra with multiplicationm AotimesB = (m Aotimesm B)(idAotimescminus1

ABotimesidB) and unitηAotimesB = ηAotimesηB Similarly if A and Bare C-coalgebras then A otimes B becomes a C-coalgebra if we choose the comultiplicationAotimesB = (idA otimes cAB otimes idB) (A otimesB) and the counit εAotimesB = εA otimes εB

Cardy Algebras and Sewing Constraints I 879

Definition 29 A Frobenius algebra A = (Am η ε) is an algebra and a coalge-bra such that the coproduct is an intertwiner of A-bimodules

(idA otimes m) (otimes idA) = otimes m = (m otimes idA) (idA otimes)

We will use the following graphical representation for the morphisms of a Frobeniusalgebra

m =A A

A

η =A

=A A

A

ε =A

(212)

A Frobenius algebra A in a k-linear tensor category for k a field is called special iffm = ζ idA and ε η = ξ id1 for nonzero constants ζ ξ isin k If ζ = 1 we call A nor-malised-special A Frobenius algebra homomorphism between two Frobenius algebrasis both an algebra homomorphism and a coalgebra homomorphism

A (strictly) sovereign tensor category is a tensor category equipped with a left and aright duality which agree on objects and morphisms (see eg [BiFS] for more details)We will write the dualities as

Uor U

= dU Uor otimes U rarr 1U Uor

= dU U otimes Uor rarr 1

U Uor

= bU 1 rarr U otimes UorUor U

= bU 1 rarr Uor otimes U

(213)

In terms of these we define the left and right dimension of an object U as

diml U = dU bU dimr U = dU bU (214)

both of which are elements of Hom(1 1)Let now C be a sovereign tensor category For a Frobenius algebra A in C we define

two morphisms

A =

A

Aor

primeA =

A

Aor

(215)

Definition 210 A Frobenius algebra A is symmetric iff A = primeA

The following lemma shows that under certain conditions we do not need to distin-guish the various notions of simplicity in Definition 28

Lemma 211 Let A be a commutative symmetric Frobenius algebra in a C-linear semi-simple sovereign braided tensor category C and suppose that diml A = 0 Then thefollowing are equivalent

880 L Kong I Runkel

(i) A is simple(ii) A is absolutely simple

(iii) A is haploid

Proof (ii)hArr(iii) A is haploid iff it is absolutely simple as a left module over itself[FS Eq (417)] Furthermore for a commutative algebra we have HomA(A A) =HomA|A(A A) and so A is haploid iff it is absolutely simple(i)rArr(ii) If A is simple then every nonzero element of HomA|A(A A) is invertibleHence this space forms a division algebra over C and is therefore isomorphic to C(iii)rArr(i) Since C is semi-simple and A is haploid also Hom(A 1) is one-dimensionalThe counit ε is a nonzero element in this space and so gives a basis This implies firstlythat ε η = 0 and secondly that there is a constant β isin C such that

β middot ε = dA (idAor otimes m) (bA otimes idA) (216)

Composing with η from the right yields β εη = diml A The right-hand side is nonzeroand so β = 0 By [FRS Lem 311] A is special We have already proved (ii)hArr(iii)and so A is absolutely simple A special Frobenius algebra in a semi-simple categoryhas a semi-simple category of bimodules (apply [FS Prop 524] to the algebra tensoredwith its opposite algebra) For semi-simple C-linear categories simple and absolutelysimple are equivalent2 Thus A is simple Remark 212 For a Frobenius algebra A the morphisms (215) are invertible and henceA sim= Aor In this case one has diml A = dimr A [FS Rem 363] and so we could havestated the above lemma equivalently with the condition dimr A = 0

Let F C1 rarr C2 be a lax tensor functor between two tensor categories C1 C2 and

let (Am A ηA) be an algebra in C1 Define morphisms F(A)otimes F(A)m F(A)minusminusminusrarr F(A) and

12ηF(A)minusminusminusrarr F(A) as

m F(A) = F(m A) φF2 ηF(A) = F(ηA) φF

0 (217)

Then (F(A)m F(A) ηF(A)) is an algebra in C2 [JS Prop 55] If f A rarr B is analgebra homomorphism between two algebras A B isin C1 then F( f ) F(A) rarr F(B)is also an algebra homomorphism If (MmM ) is a left (or right) A-module in C1 then(F(M) F(mM )φF

2 ) is a left (or right) F(A)-module if M has a A-bimodule structurethen F(M) naturally has a F(A)-bimodule structure

Similarly if (AA εA) is a coalgebra in C1 and F C1 rarr C2 is a colax tensor func-

tor then F(A) with coproduct F(A)F(A)minusminusminusrarr F(A) otimes F(A) and counit F(A)

εF(A)minusminusminusrarr 12given by

F(A) = ψ F2 F(A) εF(A) = ψ F

0 F(εA) (218)

is a coalgebra in C2 If f A rarr B is a coalgebra homomorphism between two coalgebrasA B isin C1 then F( f ) F(A) rarr F(B) is also a coalgebra homomorphism

2 To see this note that if U is simple then the C-vector space Hom(UU ) is a division algebra and henceHom(UU ) = C idU Conversely if U is not simple then U = U1 oplus U2 and Hom(UU ) contains at leasttwo linearly independent elements namely idU1 and idU2

Cardy Algebras and Sewing Constraints I 881

Proposition 2133 If F C1 rarr C2 is a Frobenius functor and (Am A ηAA εA) aFrobenius algebra in C1 then (F(A)m F(A) ηF(A)F(A) εF(A)) is a Frobenius alge-bra in C2

Proof One Frobenius property (m F(A)otimes idF(A)) (idF(A)otimesF(A)) = F(A) m F(A)follows from the commutativity of the following diagram (we spell out the associativityisomorphisms)

F(A)otimes F(A)idF(A)otimesF(A)

φF2

F(A)otimes F(A otimes A)idF(A)otimesψF

2

φF2

F(A)otimes (F(A)otimes F(A))

αF(A)F(A)F(A)

F(A otimes A)

F(idAotimesA)

F(m A)

F(A otimes (A otimes A))

F(αAAA)

(F(A)otimes F(A))otimes F(A)

φF2 otimesidF(A)

F((A otimes A)otimes A)

F(m AotimesidA)

ψF2 F(A otimes A)otimes F(A)

F(m A)otimesidF(A)

F(A)

F(A) F(A otimes A)ψF

2 F(A)otimes F(A)

(219)

The commutativity of the upper-left subdiagram follows from the naturalness of φF2 that

of the upper-right subdiagram follows from (25) that of the lower-left subdiagram fol-lows from the Frobenius properties of A and that of the lower-right subdiagram followsfrom the naturalness of ψ F

2 The proof of the other Frobenius property is similar Proposition 214 If F C1 rarr C2 is a tensor functor and A a Frobenius algebra in C1then

(i) F(A) has a natural structure of Frobenius algebra as given in Proposition 213(ii) If A is (normalised-)special so is F(A)

Proof Part (i) follows from Propositions 25 and 213 Part (ii) is a straightforwardverification of the definition using ψ F

2 = (φF2 )

minus1 and ψ F0 = (φF

0 )minus1

Let C1 C2 be sovereign tensor categories and F C1 rarr C2 a Frobenius functor Wedefine two morphisms IF(Aor) I prime

F(Aor) F(Aor) rarr F(A)or for a Frobenius algebra A inC1 as follows

IF(Aor) = ((ψ F0 F(dA) φF

2 )otimes idF(A)or) (idF(Aor) otimes bF(A))

I primeF(Aor) = (idF(A)or otimes (ψ F

0 F(dA) φF2 )) (bF(A) otimes idF(Aor))

(220)

It is easy to see that these are isomorphisms

Lemma 215 If F C1 rarr C2 is a Frobenius functor and A a Frobenius algebra in C1then

F(A) = IF(Aor) F(A) primeF(A) = I prime

F(Aor) F(primeA) (221)

3 After the preprint of the present paper appeared we noticed that this proposition is also proved in[DP Cor 5]

882 L Kong I Runkel

Proof We only prove the first equality the second one can be seen in the same way Bydefinition we have

IF(Aor) F(A)

= ((ψ F0 F(dA) φF

2 )otimes idF(A)or) (idF(Aor) otimes bF(A)) F(A)

=[(ψ F

0 F(dA) φF2 ) (F(A)otimes idF(A))

]otimes idF(A)or

(idF(A) otimes bF(A))

For the term inside the square brackets we find

ψ F0 F(dA) φF

2 (F(A)otimes idF(A)) = ψ F0 F(dA) F(A otimes idA) φF

2

= ψ F0 F(dA (A otimes idA)) φF

2 = ψ F0 F(εA m A) φF

2 (222)

On the other hand by definitionF(A) = [((ψ F0 F(εA) (F(m A)φF

2 ))otimes idF(A)or] (idF(A) otimes bF(A)) This demonstrates the first equality in (221) Proposition 216 Let F C1 rarr C2 be a tensor functor G C2 rarr C1 a functor〈FG χ〉 an adjunction A a C1-algebra and B a C2-algebra Then f A rarr G(B)is an algebra homomorphism if and only if f = χminus1( f ) F(A) rarr B is an algebrahomomorphism

Proof We need to show that

mG(B) ( f otimes f ) = f m A and f ηA = ηG(B) (223)

is equivalent to

m B ( f otimes f ) = f m F(A) and f ηF(A) = ηBs (224)

We first prove that the first identity in (223) is equivalent to the first identity in (224)For the left-hand side of the first identity in (223) we have the following equalities

mG(B) ( f otimes f )(1)= G(m B) φG

2 ( f otimes f )(2)= G(m B) G(ρ otimes ρ) G(ψ F

2 ) δ ( f otimes f )(3)= G(m B) G(ρ otimes ρ) G(ψ F

2 ) G F( f otimes f ) δ(4)= G(m B) G(ρ otimes ρ) G(F( f )otimes F( f )) G(ψ F

2 ) δ(5)= G

(m B (ρ otimes ρ) (F( f )otimes F( f )) ψ F

2

) δ

(6)= χ(

m B (ρ otimes ρ) (F( f )otimes F( f )) ψ F2

) (225)

where (1) is the definition of mG(B) in (217) (2) is the second identity in (210) (3)and (4) are naturality of δ and ψ F

2 respectively step (5) is functoriality of G and finallystep (6) is (29) For the right hand side of the first identity in (223) we get

f m A(1)= Gρ δG ( f m A)

(2)= Gρ G F( f m A) δ(3)= G(ρ F( f m A)) δ(4)= χ(ρ F( f ) F(m A)) (226)

Cardy Algebras and Sewing Constraints I 883

where (1) is the adjunction property (28) (2) is naturality of δ (3) functoriality of Gand (4) amounts to (29) and functoriality of F

On the other hand we see that the first equality in (224) is equivalent to

m B (ρ otimes ρ) (F( f )otimes F( f )) = ρ F( f ) F(m A) φF2 (227)

Using that φF2 is invertible with inverse (φF

2 )minus1 = ψ F

2 and that χ is an isomorphismit follows that the statement that (225) is equal to (226) is equivalent to the identity(227)

Now we prove that the second identity in (223) is equivalent to the second identity in(224) Using (217) and (210) we can write ηG(B) = G(ηB)φF

0 = G(ηB)G(ψ F0 )δ1

Together with (29) this shows that the second identity in (223) is equivalent to

f ηA = χ(ηB ψ F0 ) (228)

On the other hand the second identity in (224) is equivalent to

ρ F( f ) F(ηA) φF0 = ηB (229)

which by φF0 = (ψ F

0 )minus1 and (29) is further equivalent to (228)

Definition 217 Let (Am A ηAA εA) and (Bm B ηBB εB) be two Frobeniusalgebras in a tensor category C For f A rarr B we define f lowast B rarr A by

f lowast = ((εB m B)otimes idA) (idB otimes f otimes idA) (idB otimes (A ηA)) (230)

The following lemma is immediate from the definition of (middot)lowast and the properties ofFrobenius algebras We omit the proof

Lemma 218 Let C be a tensor category let A BC be Frobenius algebras in C andlet f A rarr B and g B rarr C be morphisms

(i) (g f )lowast = f lowast glowast(ii) f is a monomorphism iff f lowast is an epimorphism

(iii) f is an algebra map iff f lowast is a coalgebra map(iv) If f is a homomorphism of Frobenius algebras then f lowast f = idA and f f lowast =

idB(v) If C is sovereign and if A and B are symmetric then f lowastlowast = f

Let C and D be tensor categories and let F C rarr D be a Frobenius functor GivenFrobenius algebras A B in C and a morphism f A rarr B the next lemma shows how(middot)lowast behaves under F

Lemma 219 F( f lowast) = F( f )lowast

Proof The definition of the structure morphisms of the Frobenius algebra F(A) is givenin (217) and (218) Substituting these definitions gives

F( f )lowast =[(ψ F

0 F(εB) F(m B) φF2

)otimes idF(A)

] [

idF(B) otimes F( f )otimes idF(A)]

[idF(B) otimes

(ψ F

2 F(A) F(ηA) φF0

)]

= (ψ F0 otimes idF(A)) [

F (εB m B (idB otimes f ))otimes idF(A)]

(φF2 otimes idF(A)) (idF(B) otimes ψ F

2 )

[idF(B) otimes F(A ηA)

] (idF(B) otimes φF0 ) (231)

884 L Kong I Runkel

In the middle line of the last expression we can use the defining property (25) of F namely we substitute (φF

2 otimes idF(A)) (idF(B) otimes ψ F2 ) = ψ F

2 φF2 Then ψ F

2 can bemoved to the left and φF

2 to the right until they can be omitted against ψ F0 and φF

0 respectively using (22) and (24) This results in

F( f )lowast = F ((εB m B (idB otimes f ))otimes idA) F (idB otimes (A ηA)) (232)

which is nothing but F( f lowast)

23 Modular tensor categories Let C be a modular tensor category [TBK] ie anabelian semi-simple finite C-linear ribbon category with simple tensor unit 1 and a non-degeneracy condition on the braiding (to be stated in a moment) We denote the set ofequivalence classes of simple objects in C by I elements in I by i j k isin I and theirrepresentatives by Ui U j Uk We also set U0 = 1 and for an index k isin I we define kby Uk

sim= Uork

Since the tensor unit is simple we shall for modular tensor categories identifyHom(1 1) sim= C (cf footnote 2) Define numbers si j isin C by4

si j = UiU j (233)

They obey si j = s ji and s0i = dim Ui see eg [BK Sect 31] (In a ribbon cate-gory the left and right dimension (214) of Ui coincide and are denoted by dim Ui )The non-degeneracy condition on the braiding of a modular tensor category is that the|I|times|I|-matrix s should be invertible In fact [BK Thm 317]

sumkisinI

sik sk j = Dim C δij (234)

where Dim C = sumiisinI(dim Ui )

2 One can show (even in the weaker context of fusioncategories over C) that Dim C ge 1 [ENO1 Thm 23] In particular Dim C = 0 We fixonce and for all a square root

radicDim C of Dim C

Let us fix a basis λα(i j)kN k

i jα=1 in HomC(Ui otimesU j Uk) and the dual basis ϒ(i j)k

α N ki j

α=1

in HomC(UkUi otimesU j ) The duality of the bases means that λα(i j)k ϒ(i j)kβ = δαβ idUk

We also fix λ(0i)i = λ(i0)i = idUi We denote the basis vectors graphically as follows

λα(i j)k = α

Uk

Ui U j

ϒ(i j)kα = α

Uk

Ui U j

(235)

4 In the graphical notation used below we have given an orientation to the ribbons indicated by the arrowsFor example it is understood that this orientation determines which of the duality morphisms in (213) to use

Cardy Algebras and Sewing Constraints I 885

For V isin C we also choose a basis b(iα)V of HomC(VUi ) and the dual basis bV(iβ)

of HomC(Ui V ) for i isin I such that b(iα)V bV(iβ) = δαβ idUi We use the graphical

notation

b(iα)V = α

Ui

V

bV(iα) = α

Ui

V

(236)

Given two modular tensor categories C and D by C D we mean the tensor productof additive categories over C [BK Def 1115] ie the category whose objects are directsums of pairs V times W of objects V isin C and W isin D and whose morphism spaces are

HomCD(V times W V prime times W prime) = HomC(V V prime)otimesC HomC(WW prime) (237)

for pairs and direct sums of these if the objects are direct sums of pairsIf we replace the braiding and the twist in C by the antibraiding cminus1 and the antitwist

θminus1 respectively we obtain another ribbon category structure on C In order to distin-guish these two distinct structures we denote (C c θ) and (C cminus1 θminus1) by C+ and Cminusrespectively As in the Introduction we will abbreviate

C2plusmn = C+ Cminus (238)

Note that a set of representatives of the simple objects in C2plusmn is given by Ui times U j fori j isin I

For the remainder of Sect 2 we fix a modular tensor category C

24 The functors T and R The tensor product bifunctor otimes can be naturally extendedto a functor T C2plusmn rarr C Namely T (oplusN

i=1Vi times Wi ) = oplusNi=1Vi otimes Wi for all Vi Wi isin C

and N isin N The functor T becomes a tensor functor as follows For φT0 1 rarr T (1 times 1)

take φT0 = id1 (or lminus1

1 in the non-strict case) Next notice that for U VW X isin C

T (U times V )otimes T (W times X)= (U otimes V )otimes (W otimes X)

T ((U times V )otimes (W times X))= (U otimes W )otimes (V otimes X)(239)

We define φT2 T (U times V )otimes T (W times X) rarr T ((U times V )otimes (W times X)) by

φT2 = idU otimes cminus1

W V otimes idX (240)

(In the non-strict case the appropriate associators have to be added) The above definitionof φT

2 can be naturally extended to a morphism φT2 T (M1 otimes M2) rarr T (M1)otimes T (M2)

for any pair of objects M1M2 in C2plusmn The following result can be checked by directcalculation [JS Prop 52]

Lemma 220 The triple (T φT2 φ

T0 ) gives a tensor functor

886 L Kong I Runkel

In particular (T φT2 φ

T0 ψ

T2 ψ

T0 ) where ψT

2 = (φT2 )

minus1 and ψT0 = (φT

0 )minus1 gives

a Frobenius functorDefine the functor R C rarr C2plusmn as follows for A isin C and f isin HomC(A B)

R(A) =oplusiisinI

(A otimes Uori )times Ui R( f ) =

oplusiisinI

( f otimes idUori)times idUi (241)

This functor was also considered in a slightly different context in [ENO2 Prop 23]The family of isomorphisms γ R

A = oplusiisinI DimCdim Ui

id(AotimesUori )timesUi

isin Aut(R(A)) defines a

natural isomorphism γ R R rarr ROur next aim is to show that R is left and right adjoint to T in other words R

and T form an ambidextrous adjunction (see eg [Ld] for a discussion of ambidextrousadjunctions) To this end we introduce two linear isomorphisms for A isin C and M isin C2plusmn

χ HomC(T (M) A) minusrarr HomC2plusmn(M R(A))

χ HomC(A T (M)) minusrarr HomC2plusmn(R(A)M)(242)

If we decompose M as M = oplusNn=1 Ml

n times Mrn then χ and χ are given by

χ Noplus

n=1

fn

Mln Mr

n

A

rarrNoplus

n=1

oplusiisinI

sumα

fn

Mln

Mrn

A

α

Uori

times α

Mrn

Ui

(243)

and

χ Noplus

n=1 gn

Mln Mr

n

A

rarrNoplus

n=1

oplusiisinI

sumα gn

Mln

Mrn

A

α

Uori

times α

Ui

Mrn

DimCdim Ui

(244)

Notice that χ and χ are independent of the choice of basis

Theorem 221 〈T R χ〉 and 〈R T χminus1〉 are adjunctions ie R is both left and rightadjoint of T

Proof Write M as M = oplusNn=1 Ml

n times Mrn The isomorphism χ amounts to the following

composition of natural isomorphisms

HomC(T (M) A) = oplusnHomC(Mln otimes Mr

n A)sim= oplusni HomC(Ml

n otimes Ui A)otimes HomC(Mrn Ui )

sim= oplusni HomC(Mln A otimes Uor

i )otimes HomC(Mrn Ui ) = HomC2plusmn(M R(A)) (245)

Cardy Algebras and Sewing Constraints I 887

Thus χ is natural Let (γ RA )

lowast HomC2plusmn(R(A)M) rarr HomC2plusmn(R(A)M) denote the

pull-back of γ RA The isomorphism χ is equal to the composition of (γ R

A )lowast and the

following sequence of natural isomorphisms

HomC(A T (M)) = oplusn HomC(AMln otimes Mr

n)

sim= oplusni HomC(AMln otimes Ui )otimes HomC(Ui Mr

n)

sim= oplusni HomC(A otimes Uori Ml

n)otimes HomC(Ui Mrn) = HomC2plusmn(R(A)M) (246)

We have proved that both χ and χ are natural isomorphisms There are four natural transformations associated to χ and χ namely

idC2plusmnδminusrarr RT

ρminusrarr idC2plusmn and idCδminusrarr T R

ρminusrarr idC (247)

defined by for A isin C M isin C2plusmn

δM = χ(idT (M)) ρA = χminus1(idR(A))

ρM = χ (idT (M)) δA = χminus1(idR(A))(248)

They can be expressed graphically as follows with M = oplusNn=1 Ml

n times Mrn

δM =oplusni

sumα

Mln

Mln Mr

n Uori

αtimes

Ui

Mrn

α

ρA =oplusiisinI

A

A

Uori Ui

(249)

ρM =oplusni

sumα

Mln

Mln

Mrn Uor

i

α times

Ui

Mrn

α DimCdim Ui

δA =oplusiisinI

A

A Uori Ui

dim UiDimC

Note that

ρM δM = Dim C middot idM and ρA δA = idA (250)

Lemma 222 The functors T and R as maps on the sets of morphisms have left inversesand thus are injective

Proof Let f A rarr B be a morphism in C We define a map Q R HomC2plusmn(R(A)

R(B)) rarr HomC(A B) by f prime rarr ρB T ( f prime) δA Then we have

Q R R( f ) = ρB T R( f ) δA = ρB δB f = f (251)

888 L Kong I Runkel

where we used naturality of δ and (250) in the second and third equalities respectivelySo Q R is a left inverse of R on morphisms Thus R is injective on morphisms Similarlylet g M rarr N be a morphism in C2plusmn We define a map QT HomC(T (M) T (N )) rarrHomC2plusmn(M N ) by gprime rarr (Dim C)minus1 middot ρN R(gprime) δM Then we have

QT T (g) = (Dim C)minus1 middot ρN RT (g) δM = (Dim C)minus1 middot ρN δN g = g (252)

So QT is a left inverse of T on morphisms Thus T is injective on morphisms Using (29) and (250) one can express the two inverse maps χminus1 χminus1 as follows

for f isin HomC2plusmn(M R(A)) and g isin HomC2plusmn(R(A)M)

χminus1( f ) = ρ T ( f ) χminus1(g) = T (g) δ (253)

By Proposition 25 and Lemma 27 R is both a lax and colax tensor functor In particularφR

0 1 times 1 rarr R(1) is given by

φR0 = χ (ψT

0 ) = R(ψT0 ) δ1 times 1 = id1times1 (254)

and φR2 R(A)otimes R(B) rarr R(A otimes B) by φR

2 = R(ρA otimes ρB) R(ψT2 ) δ which can

be expressed graphically as

φR2 =

oplusi jkisinI

sumα

A

A

Uori B

B

Uorj

Uork

α

times

Ui U j

α

Uk

(255)

Similarly ψ R0 R(1) rarr 1 times 1 is given by

ψ R0 = ρ1 R(φT

0 ) = Dim C id1times1 (256)

and ψ R2 R(A otimes B) rarr R(A) otimes R(B) by ψ R

2 = ρ R(φT2 ) R(δA otimes δB) which in

graphical notation reads

ψ R2 =

oplusi jkisinI

sumα

A

A Uori

B

B Uorj

Uork

α

times

Ui U j

α

Uk

dim Ui dim U j

dim Uk Dim C

(257)

If C has more than one simple object then R does not take the tensor unit of C to thetensor unit of C2plusmn and so is clearly not a tensor functor However we will show that R isstill a Frobenius functor This will imply that if A is a Frobenius algebra in C then

R(A) = (R(A)m R(A) ηR(A)R(A) εR(A)) (258)

Cardy Algebras and Sewing Constraints I 889

is a Frobenius algebra in C2plusmn where the structure morphisms were given in (217) and(218) In the case A = 1 it was proved in [Muuml1 Prop 41] (see also [Fr Lem 619]and [K1 Thm 52]) that (258) is a commutative simple symmetric normalised-specialFrobenius algebra in C2plusmn In fact given a Frobenius algebra A in C it is straightforwardto verify that the structure morphisms in (258) are precisely those of (A times 1)otimes R(1)cf Sect 22

Proposition 223 (R φR2 φ

R0 ψ

R2 ψ

R0 ) is a Frobenius functor

Proof Using the explicit graphical expression of φR2 φ

R0 ψ

R2 ψ

R0 it is easy to see that

the commutativity of the diagrams (25) and (26) are equivalent to the statement thatR(1) with structure morphisms as in (258) is a Frobenius algebra in C2plusmn The latterstatement is true by [Muuml1 Prop 41]

From Lemma 220 and Proposition 223 we see that T and R take Frobenius alge-bras to Frobenius algebras The following two propositions show how the properties ofFrobenius algebras are transported

Proposition 224 Let A be a Frobenius algebra in C2plusmn Then T (A) is a Frobenius algebrain C and

(i) A is symmetric iff T (A) is symmetric(ii) A is (normalised-)special iff T (A) is (normalised-)special

Proof For part (i) write A as a direct sum oplusNn=1 Al

n times Arn Then the maps IT (A) I prime

T (A) T (Aor) rarr T (A)or defined in (220) are given by

IT (A) = I primeT (A) = oplusN

n=1c(Aln)

or(Arn)

or (259)

Therefore by (221) T (A) = primeT (A) is equivalent to T (A) = T (prime

A) Since byLemma 222 T is injective on morphisms this proves part (i) Part (ii) can be checkedin the same way for example the condition mT (A) T (A) = ζ idT (A) is easily checkedto be equivalent to T (m A A) = ζ T (idA) Proposition 225 Let A be a Frobenius algebra A in C Then R(A) is a Frobeniusalgebra in C2plusmn and

(i) A is symmetric iff R(A) is symmetric(ii) A is (normalised-)special iff R(A) is (normalised-)special

Proof Recall that the structure morphisms of the Frobenius algebra R(A) are equal tothose of (A times 1) otimes R(1) Using this equality part (i) and (ii) follow because R(1) issymmetric and normalised-special For example

m R(A) R(A) = [(m A A)times id1] otimes idR(1) = R(m A A) (260)

so that m R(A) R(A) = ζ idR(A) is equivalent to R(m A A) = ζ R(idA) which byLemma 222 is equivalent to m A A = ζ idA

The functor R has one additional property not shared by T namely R takes absolutelysimple algebras to absolutely simple algebras We will see explicitly in Sect 33 thatthis is not true for T

890 L Kong I Runkel

Lemma 226 For a C-algebra A the map

R HomA|A(A A) rarr HomR(A)|R(A)(R(A) R(A)) (261)

given by f rarr R( f ) is well-defined and an isomorphism

Proof Since R is a lax tensor functor R(A) is naturally a R(A)-bimodule It is easy tosee that R in (261) is a well-defined map R(A) is also naturally a R(1)-bimodule whichcan be identified with the induced R(1)-bimodule structure on (A times 1)otimes R(1) wherethe left R(1) action on (Atimes1)otimes R(1) is given by (idAtimes1 otimesm R(1))(cminus1

Atimes1R(1)otimes idR(1))We have the following natural isomorphisms

HomR(1)|R(1)(R(A) R(A))sim=minusrarr HomC2plusmn(A times 1 R(A))

χminus1

minusminusrarr HomC(A A) (262)

which by (253) are given by for f isin HomR(1)|R(1)(R(A) R(A))

f rarr f prime = f (idAtimes1 otimes ηR(1)) rarr f primeprime = ρ T ( f prime) (263)

and its inverse is given by for g isin HomC(A A)

g rarr gprime = R(g) δ rarr gprimeprime = (idAtimes1 otimes m R(1)) (gprime otimes idR(1)) (264)

where gprimeprime is indeed a R(1)-bimodule map due to the commutativity of R(1) It is easy tocheck that gprimeprime = R(g) in (264) Therefore R gives an isomorphism from HomC(A A)to HomR(1)|R(1)(R(A) R(A)) Moreover one verifies that R(g) is an R(A)-bimodulemap iff g is an A-bimodule map In other words R g rarr R(g) gives an isomorphism

HomA|A(A A)sim=minusrarr HomR(A)|R(A)(R(A) R(A))

Corollary 227 Let A be a C-algebra

(i) A is absolutely simple iff R(A) is absolutely simple(ii) Let A be in addition Frobenius Then A is simple and special iff R(A) is simple and

special

Proof Part (i) immediately follows from Lemma 226 The statement of part (ii) withoutthe qualifier lsquosimplersquo is proved in Proposition 225 But as in the proof of (iii)rArr(i) inLemma 211 a special Frobenius algebra in a semi-simple category has a semi-simplecategory of bimodules and for a semi-simple C-linear category simple and absolutelysimple are equivalent Part (ii) then follows from part (i)

The following lemma will be needed in Sect 32 below to discuss the properties ofCardy algebras

Lemma 228 Let A be a Frobenius algebra in C Then (δA)lowast = ρA

Proof Recall from (250) that δA is a morphism A rarr T R(A) Since T and R are bothFrobenius functors T R(A) is a Frobenius algebra in C Substituting the definitions aftera short calculation one finds

Cardy Algebras and Sewing Constraints I 891

εT R(A) mT R(A) = Dim(C)oplusiisinI

A Uori Ui A U or

ı Uı

(265)

Substituting this in the definition of (δA)lowast gives again after a short calculation the mor-

phism ρA At an intermediate step one uses that the part of the morphism (δA)lowast which

is made up of Ui and Uı ribbons their duals and the basis morphisms λ(iı)0 andϒ(iı)0can be replaced by 1

dim Uimiddot dUi

3 Cardy Algebras

In this section we start by investigating the properties of Frobenius algebras which sat-isfy the so-called modular invariance condition We then give two definitions of a Cardyalgebra and prove their equivalence Finally in Sect 33 we study the properties of thesealgebras and state our main results

We fix a modular tensor category C Recall that C2plusmn is an abbreviation for C+ Cminus

31 Modular invariance In C2plusmn we define the object K and the morphism ω K rarr Kas

K =oplus

i jisinIUi times U j ω =

sumi jisinI

dim Ui dim U j

Dim C idUi timesU j (31)

They have the property (see eg [BK Cor 3111])

ω

K

(Uk times Ul )or

(Uk times Ul )or

Ui times U j

Ui times U j

= δik δ jlDim C

dim Ui dim U jbUi timesU j dUi timesU j (32)

892 L Kong I Runkel

Definition 31

(i) Let A B be objects of C2plusmn A morphism f A otimes B rarr B is called S-invariant iff

A

B

W

W

f =

A

B

W

W

f

K

ω(33)

holds for all for W isin C2plusmn(ii) A C2plusmn-algebra (Aclmcl ηcl) is called modular invariant iff θAcl = idAcl and mcl

is S-invariant

Lemma 32 The morphism f A otimes B rarr B is S-invariant if and only if

A

B

Ui times U j

Ui times U j

f = Dim Cdim Ui dim U j

sumα

A

B

B

f

Ui times U j

Ui times U j

α

α

(34)

holds for all i j isin I

Proof Condition (33) holds for all W iff it holds for all W = Ui times U j i j isin I so itis enough to show that the right-hand side of (33) with W = Ui times U j is equal to theright-hand side of (34) Recall the notation for basis morphisms in (236) Starting from(33) write

idBor otimes idUi timesU j =⎛⎝sum

klα

(b(ktimeslα)

B

)or (

bB(ktimeslα)

)or⎞⎠ otimes idUi timesU j (35)

and then apply (32) The graphical representation of the resulting morphism can bedeformed to give (34) Remark 33 As shown in [K3 Sect 61] the modular invariance condition of a C2plusmn-alge-bra exactly coincides with the modular invariance condition for torus 1-point correlationfunctions of a genus-01 closed CFT In particular the condition θAcl = idAcl is equiv-alent to invariance under the modular transformation T τ rarr τ + 1 and the condition(34) with f = mcl is equivalent to invariance under S τ rarr minus 1

τ Combining the

modular invariance condition with the genus zero properties of a genus-01 closed CFTresults in a modular invariant commutative symmetric Frobenius algebra in C2plusmn

Cardy Algebras and Sewing Constraints I 893

Let Acl be a modular invariant C2plusmn-algebra Evaluating (34) for f = mcl composingit with ηcl otimes idUi timesU j and taking the trace implies the following identity

Zi j = 1

DimCAclUi timesU j where

(36)

Zi j = dimC HomC2plusmn(UitimesU j Acl)

Decomposing Acl into simple objects this gives

Zi j =sum

klisinISik Zkl Sminus1

l j where Si j = si jradic

DimC (37)

which in CFT terms is of course nothing but the invariance of the torus partition functionunder the modular S-transformation

The following theorem gives a simple criterion for modular invariance

Theorem 34 Let Acl be a haploid commutative symmetric Frobenius algebra in C2plusmn

(i) If Acl is modular invariant then dim Acl = Dim C(ii) If dim Acl = Dim C then Acl is special and modular invariant

Proof Part (i) Since Acl is haploid for i = j = 0 Equation (36) reduces to 1 =dim AclDimCPart (ii) By the same reasoning as in the proof of (iii)rArr(i) in Lemma 211 one showsthat Acl is special Thus mcl cl = ζclidAcl for some ζcl = 0

By [KO Thm 45] the category (C2plusmn)locAcl

of local Acl-modules is again a mod-

ular tensor category and Dim (C2plusmn)locAcl

= (Dim C dim Acl)2 (see [Fr Prop 321 amp

Rem 323] for the same statement in the notation used here) Thus by assumption wehave Dim(C2plusmn)loc

Acl= 1 It then follows from [ENO1 Thm 23] that up to isomorphism

(C2plusmn)locAcl

has a unique simple object (namely the tensor unit) In other words every simplelocal Acl-module is isomorphic to Acl (seen as a left-module over itself)

We have the following isomorphisms between morphism spaces [FS Prop 47 amp 411]

HomAcl(Acl otimes (Ui times U j ) Acl) sim= HomC2plusmn(Ui times U j Acl)

HomAcl(Acl Acl otimes (Ui times U j )) sim= HomC2plusmn(AclUi times U j )

Using these to transport the bases (236) from the right to the left we obtain bases bα(i j)αof HomAcl(Acl otimes (Ui times U j ) Acl) and b(i j)

β β of HomAcl(Acl Acl otimes (Ui times U j )) Thesecan be expressed graphically as

894 L Kong I Runkel

bα(i j) =

Acl

Acl

Acl

Ui times U j

α

b(i j)β = DimC

dim Ui dim U j

1

ζcl

Acl

Ui times U jAcl

Acl

β

(38)

where the nonzero factor in b(i j)β is included for convenience Notice that bα(i j) b(i j)

β isa left Acl-module map Since Acl is simple as a left module over itself we have

bα(i j) b(i j)β = λαβ idAcl (39)

for some λαβ isin C By computing tr(bα(i j) b(i j)β ) it is easy to verify that λαβ = δαβ

We will now prove the following identity

1

ζcl

Acl

Acl

Ui times U j

Ui times U j

Acl

=sumα

DimCdim Ui dim U j

1

ζcl

Acl

Ui times U j

Ui times U j

Acl

Acl

Acl

α

α

(310)

One checks that the left-hand side of this equation is an idempotent which we denote byζminus1

cl PlAcl(Ui timesU j ) cf [Fr Sect 31] By [Fr Prop 41] the image Im(ζminus1

cl PlAcl(Ui timesU j ))

is a local Acl-module and hence isomorphic to AoplusNcl for some N isin Zge0

All left-module morphisms from Acl to Aclotimes(Ui timesU j ) are linear combinations of the

b(i j)β Furthermore one verifies that ζminus1

cl PlAcl(Ui timesU j )b(i j)

β = b(i j)β Therefore the b(i j)

β

describe precisely the image of the idempotent ie ζminus1cl Pl

Acl(Ui timesU j ) = sum

α b(i j)α bα(i j)

which is nothing but (310) Composing (310) with ζcl middot εcl otimes idUi timesU j from the left (iefrom the top) produces (34)

In addition since Acl is commutative symmetric Frobenius it satisfies θAcl = idAcl

[Fr Prop 225] Altogether this shows that Acl is modular invariant Remark 35 As we were writing this paper we heard that the results in Theorem 34were obtained independently by Kitaev and Muumlger [Ki]

Remark 36 Setting i = j = 0 in (36) gives the identity dim Acl = Z00 Dim C [KRProp 23] Combining this with Theorem 34 (ii) one may wonder if a general modularinvariant commutative symmetric Frobenius algebra Acl in C2plusmn is isomorphic to a directsum of simple such algebras However this is not so For example one can take the com-mutative symmetric Frobenius algebra Acl = C[x]x2 in the category of vector spacesequipped with the non-degenerate trace ε(ax + b) = a In this case the modular invari-ance condition holds automatically but Acl is clearly not a direct sum of two algebras

Cardy Algebras and Sewing Constraints I 895

For a general modular tensor category C the algebra C[x]x2 R(1) understood as analgebra in C2plusmn via the braided monoidal isomorphism Vect f (C) C2plusmn rarr C2plusmn providesanother counter-example

32 Two definitions Define a morphism PlA A rarr A for a Frobenius algebra A in C

or C2plusmn as follows [Fr Sect 24]

PlA =

A

A

A

m

(311)

If A is also commutative and obeys m A A = ζA idA we have PlA = ζA idA In partic-

ular this holds if A is commutative and special Using the fact that the Frobenius algebraR(1) is commutative and normalised-special one can check that Pl

R(A) R(A) rarr R(A)takes the following form

PlR(A) =

oplusiisinI

A

A

Uori

Uori

A

m

times

Ui

Ui

(312)

With these ingredients we can now give the first definition of a Cardy C|C2plusmn-algebrawhich was introduced in [K3 Def 514] cf Remark 315 below

Definition 37 (CardyC|C2plusmn-algebra I) A CardyC|C2plusmn-algebra is a triple (Aop|Acl ιcl-op)where (Aclmcl ηclcl εcl) is a modular invariant commutative symmetric Frobe-nius C2plusmn-algebra (Aopmop ηopop εop) is a symmetric Frobenius C-algebra andιcl-op Acl rarr R(Aop) an algebra homomorphism such that the following conditionsare satisfied

(i) Centre condition

Acl R(Aop)

R(Aop)

R(Aop)

ιcl-op

m R(Aop)

=

Acl

ιcl-op

R(Aop)

R(Aop)

R(Aop)

m R(Aop)

(313)

896 L Kong I Runkel

(ii) Cardy condition

ιcl-op ιlowastcl-op =

R(Aop)

R(Aop)

R(Aop)

(314)

Remark 38

(i) The name ldquoCardy C|C2plusmn-algebrardquo in Definition 37 was chosen because many ofthe important ingredients were first studied by Cardy the modular invariance ofthe closed theory [C1] the consistency of the annulus amplitude [C2] and thebulk-boundary OPE [CL] On the other hand the boundary-boundary OPE andthe OPE analogue of the centre condition were first considered in [Lw]

(ii) One can easily see that in the special case that C is the category Vect f (C) of finite-dimensional C-vector spaces a Cardy C|C2plusmn-algebra gives exactly the algebraicformulation of two-dimensional open-closed topological field theory over C (cfRemark 614 in [K3]) see [Lz Sect 48] [Mo Thm 11] [AN Thm 45] [LPCor 43] [MS Sect 22] When passing to a general modular tensor category Cthere are two important differences to the two-dimensional topological field the-ory Firstly the algebras Acl and Aop now live in different categories which inparticular affects the formulation of the centre condition and the Cardy conditionSecondly the modular invariance condition has to be imposed on Acl In the caseC = Vect f (C) modular invariance holds automatically

Definition 39 A homomorphism of Cardy C|C2plusmn-algebras (A(1)op |A(1)cl ι(1)cl-op) rarr (A(2)op

|A(2)cl ι(2)cl-op) is a pair ( fop fcl) of Frobenius algebra homomorphisms fop A(1)op rarr A(2)op

and fcl A(1)cl rarr A(2)cl such that the diagram

A(1)cl

fcl

ι(1)cl-op

A(2)cl

ι(2)cl-op

R(A(1)op )

R( fop) R(A(2)op )

(315)

commutes

Remark 310 Since a homomorphism of Frobenius algebras is invertible (cf Lemma218 (iv)) a homomorphism of Cardy algebras is always an isomorphism

For a homomorphism ( fop fcl) of Cardy C|C2plusmn-algebras using the commutativity of(315) and the fact that fcl and fop are both algebra and coalgebra homomorphisms itis easy to show that (315) commutes iff

R(A(1)op )R( fop)

(ι(1)cl-op)

lowast

R(A(2)op )

(ι(2)cl-op)

lowast

A(1)cl

fcl A(2)cl

(316)

Cardy Algebras and Sewing Constraints I 897

commutesLet (Aop|Acl ιcl-op) be a Cardy C|C2plusmn-algebra Define the morphism

ιcl-op = χminus1(ιcl-op) T (Acl) minusrarr Aop (317)

Decompose Acl as Acl = oplusNn=1Cl

n times Crn such that Cl

1 times Cr1 = ηcl(1 times 1) We use ι(n)cl-op

to denote the restriction of ιcl-op to Cln times Cr

n and ι(n)cl-op to denote the restriction of ιcl-op

to Cln otimes Cr

n We introduce the following graphical notation

ι(n)cl-op =

Cln Cr

n

Aop

(318)

By (243) ιcl-op can be expressed in terms of ιcl-op as follows

ιcl-op =Noplus

n=1

oplusiisinI

sumα

Cln

Crn

Aop

α

Uori

times α

Crn

Ui

(319)

Lemma 311 The centre condition (313) is equivalent to the following condition in C

Cln Cr

n

Aop

Aop

Aop

=

Cln Cr

n

Aop

Aop

Aop

for n = 1 N (320)

Proof First insert (319) and the definition (217) of m R(Aop) into (313) Then apply thecommutativity of R(1) to the left hand side of (313) The equivalence between (313)and (320) follows immediately Remark 312 The centre condition (320) is very natural from the open-closed confor-mal field theory point of view Correlators on the upper half plane are expressed in termsof conformal blocks on the full complex plane The objects Cl

n and Crn are associated to

the field insertion at a point z in the upper half plane and at the complex conjugate pointz in the lower half plane respectively The object Aop corresponds to a field inserted ata point r on the real axis The centre condition (320) simply says that the correlationfunctions in the disjoint domains |z| gt r gt 0 and r gt |z| gt 0 are analytic continuationsof each other see [K2 Prop 118]

898 L Kong I Runkel

Recall that we define ιlowastcl-op Aop rarr T (Acl) as in (230) We introduce the graphicalnotation

ιlowastcl-op =Noplus

n=1

Cln Cr

n

Aop

=Noplus

n=1

Cln Cr

n

Aop

Aop T (minus1Acl) (321)

where the second equality follows from (215) (221) and (259)

Lemma 313 The Cardy condition (314) is equivalent to the following identity in C

Noplusn=1

DimCdim Ui

sumα

Cln

Crn

Crn

α

α

Aop

Aop Uori

Uori

=Aop

Aop

Aop

Uori

Uori

op

mop for all i isin I (322)

Proof By (244) (312) and (319) it is easy to see that (322) is equivalent to thefollowing identity

ιcl-op χ(ιlowastcl-op) = PlR(Aop)

(323)

Therefore it is enough to show that

χ(ιlowastcl-op) = ιlowastcl-op (324)

We have

χminus1(ιlowastcl-op)(1)= T (ιlowastcl-op) δ (2)= T (ιcl-op)

lowast δlowastlowast (3)= (ρ T (ιcl-op)

)lowast (4)= (ιcl-op

)lowast (325)

In step (1) we use the expression (253) for χminus1 step (2) follows from Lemma 218 (v)and Lemma 219 Step (3) is Lemma 218 (i) and Lemma 228 and finally step (4)amounts to substituting (253) and (317) Acting with χ on both sides of the aboveequality produces (324)

Combining Lemmas 311 and 313 and Proposition 216 we obtain the followingequivalent definition of Cardy C|C2plusmn-algebra (recall the graphical notation (318) forιcl-op and (321) for ιlowastcl-op)

Cardy Algebras and Sewing Constraints I 899

Definition 314 (CardyC|C2plusmn-algebra II) A CardyC-algebra is a triple (Aop|Acl ιcl-op)where Acl is a commutative symmetric Frobenius C2plusmn-algebra satisfying property (34)with f = mcl Aop is a symmetric Frobenius C-algebra and ιcl-op T (Acl) rarr Aop isan algebra homomorphism satisfying the conditions (320) and (322)

Remark 315 Up to a choice of normalisation Definition 314 is the same as the orig-inal one in [K3 Def 613] The difference between the two definitions is the factorDim C dim Ui on the left hand side of (322) which in [K3 Def 613] is given byradic

Dim C dim Ui The two definitions are related by rescaling the coproduct cl andcounit εcl of Acl by 1

radicDim C and

radicDim C respectively We chose the convention in

(322) to remove all dimension factors from the expression (314) for the Cardy condition

33 Uniqueness and existence theorems In this subsection we investigate the structureof Cardy algebras We start with the following proposition which when combined withthe results of part II provides an alternative proof of [Fj Prop 422]

Proposition 316 Let (Aop|Acl ιcl-op) be a Cardy C|C2plusmn-algebra If Acl is simple anddim Aop = 0 then Aop is simple and special

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis therefore haploid Restricting the Cardy condition (322) to the case Ui = 1 andcomposing both sides with εop from the left we see that εop kills all terms associated toU j times 1 isin Acl in the sum except for a single 1 times 1 term Thus we obtain the followingidentity

β εop = dAop (mop otimes idAorop) (idAop otimes bAop) (326)

where β isin C Composing with ηop from the right in turn implies that βεop ηop =dim Aop which is nonzero by assumption Thus also β = 0 and εop is a nonzero multipleof the morphism on the right hand side of (326) By [FRS Lem 311] Aop is special

Since Aop is a special Frobenius algebra Aop is semi-simple as an Aop-bimodule(apply [FS Prop 524] to Aop tensored with its opposite algebra) Suppose Aop is not

simple so that we can write Aop = A(1)op oplus A(2)op for nonzero Aop-bimodules A(1)op and A(2)op We denote the canonical embeddings and projections associated to this decompositionas ι12 and π12 We have the identities

mop (ι1 otimes ι2) = 0 εop ηop =2sum

i=1

εop ιi πi ηop (327)

The first identity follows since π1 mop (ι1 otimes ι2) = 0 (as mop gives the left action of

Aop on Aop and hence it preserves A(2)op ) and similarly π2 mop (ι1 otimes ι2) = 0 Thesecond identity is just the completeness of ι12 π12

Since εop ηop = 0 without losing generality we can assume εop ι1 π1 ηop = 0Using that π2 is a bimodule map we compute

π2 [LHS of (322)]Ui =1 ι1 π1 ηop = π2 [RHS of (322)]Ui =1 ι1 π1 ηop

= PlA(2)op

π2 ι1 π1 ηop = 0 (328)

900 L Kong I Runkel

On the other hand using that Acl is haploid that ιcl-op is an algebra map and thatι lowastcl-op is a coalgebra map one can check that the left-hand side of (328) is equal toλ(εop ι1 π1 ηop) π2 ηop for some λ = 0 This implies that π2 ηop = 0 Thusηop = sum

i ιi πi ηop = ι1 π1 ηop Hence we have

0 = π2 ι2 = π2 mop (ηop otimes ι2) = π2 mop ((ι1 π1 ηop)otimes ι2) (329)

However the right-hand side is zero by (327) This is a contradiction and hence Aopmust be simple

To formulate the next theorem we need the notion of the full centre of an algebra [FjDef 49] Recall that an algebra A in a braided tensor category has a left centre and aright centre [VZO] both of which are sub-algebras of A Of these two we will onlyneed the left centre The following definition is [Fr Def 231] which in our setting isequivalent to that of [VZO]

Definition 317 Let A be a symmetric special Frobenius algebra such that m A A =ζA idA

(i) The left centre Cl(A) of A is the image of the idempotent ζminus1A Pl

A(ii) The full centre Z(A) is Cl(R(A))

That ζminus1A Pl

A is an idempotent follows from [FRS Lem 52] when keeping track ofthe factors ζA ([FRS] assumes normalised-special ie ζA = 1) Note that Cl(A) is againan object of C while Z(A) is an object of C2plusmn Let el

A Cl(A) rarr A be the embeddingof Cl(A) into A The left centre is in fact the maximal subobject of A such that

m A cAA (elA otimes idA) = m A (330)

see [Fr Lem 232] This observation explains the name left centre and also makes theconnection to [O Def 15]

The full centre is by definition the image of the idempotent ζminus1A Pl

R(A) R(A) rarrR(A) Since C2plusmn is abelian the idempotent splits and we obtain the embedding andrestriction morphisms

e Z(A) rarr R(A) and r R(A) Z(A) (331)

which obey r e = idZ(A) and e r = ζminus1A Pl

R(A) It follows from Proposition 225 and

[Fr Prop 237] that Z(A) is a commutative symmetric Frobenius algebra in C2plusmn withstructure morphisms5

m Z(A) = r m R(A) (e otimes e) ηZ(A) = r ηR(A)

Z(A) = ζA middot (r otimes r) R(A) e εZ(A) = ζminus1A middot εR(A) e

(332)

Moreover if A is simple then Z(A) is simple and if A is simple and dim A = 0 thenZ(A) is simple and special The normalisation of the counit is such that

εZ(A) ηZ(A) = ζminus2A dim A DimC (333)

5 The normalisation of product and unit is the standard one The factors in the coproduct and counit haveto be included in order for (A|Z(A) e) to be a Cardy algebra see Theorem 318 below The normalisation ofthe counit enters the Cardy condition (314) through the definition of ( middot )lowast

Cardy Algebras and Sewing Constraints I 901

Theorem 318 Let A be a special symmetric Frobenius C-algebra Then (A|Z(A) e)is a Cardy C|C2plusmn-algebra

The proof of this theorem makes use the following two lemmas

Lemma 319 e Z(A) rarr R(A) is an algebra map and elowast = ζA middot r

Proof It follows from [Fr Lem 229] (or by direct calculation using in particularm R(A) R(A) = ζAidR(A)) that

m R(A) (e otimes e) = ζminus1A middot Pl

R(A) m R(A) (e otimes e) (334)

Substituting e r = ζminus1A Pl

R(A) shows that e is compatible with multiplication For theunit one finds

e ηZ(A) = e r ηR(A) = ζminus1A Pl

R(A) ηR(A) = ηR(A) (335)

Thus e is an algebra map For the second statement one computes

elowast (1)= ζA

R(A)

r

R(A)

R(A)

Z(A)

R(A)

Z(A)

r

e

e

(2)=

R(A)

r

R(A)

R(A)

R(A)

Z(A)

(3)=

R(A)

r

R(A)

Z(A)

(4)= ζA middot r (336)

where in (1) the definitions (230) and (332) have been substituted step (2) is e r = ζminus1

A PlR(A) step (3) uses that R(A) is symmetric Frobenius and step (4) is again

e r = ζminus1A Pl

R(A) Lemma 320 Let A be a symmetric Frobenius algebra in C The morphism

PlR(A) m R(A)

(Pl

R(A) otimes PlR(A)

) R(A)otimes R(A) minusrarr R(A) (337)

is S-invariant

The proof of this lemma is a slightly lengthy explicit calculation and has been deferredto Appendix A2

902 L Kong I Runkel

Proof of Theorem 318 That e is an algebra map was proved in Lemma 319 The centrecondition (313) holds by property (330) of the left centre The Cardy condition (314)also is an immediate consequence of Lemma 319

ιcl-op ιlowastcl-op = e (ζA r) = PlR(A) (338)

The full centre Z(A) is a commutative symmetric Frobenius algebra It remains to provemodular invariance That θZ(A) = idZ(A) is implied by commutativity and symmetryof Z(A) [Fr Prop 225] The S-invariance condition (33) follows from Lemma 320In (337) substitute Pl

R(A) = ζA e r and then put the resulting morphism into (33)Compose the resulting equation with e otimes idW from the right (ie from the bottom) andsubstitute the definition (332) of m Z(A) This results in the statement that m Z(A) isS-invariant

The following theorem is analogous to [LR Prop 29] and [Fj Thm 426] whichroughly speaking answer the question under which circumstances the restriction of atwo-dimensional conformal field theory to the boundary already determines the entireconformal field theory The first work is set in Minkowski space and uses operator alge-bras and subfactors while the second work is set in Euclidean space and uses modulartensor categories

Theorem 321 Let (A|Acl ιcl-op) be a Cardy C|C2plusmn-algebra such that dim A = 0 andAcl is simple Then A is special and (A|Acl ιcl-op) sim= (A|Z(A) e) as Cardy algebras

Proof By Proposition 316 A is simple and special Since Acl is simple the algebramap ιcl-op Acl rarr R(A) is either zero or a monomorphism But ιcl-op ηcl = ηR(A)and so ιcl-op = 0 Thus ιcl-op is monic By Lemma 218 (ii) ζminus1

A ιlowastcl-op is epi The Cardycondition (314) implies

ιcl-op ζminus1A ιlowastcl-op = ζminus1

A PlR(A) = e r (339)

Composing this with e r from the left yields e r ιcl-op ζminus1A ιlowastcl-op = e r =

ιcl-op ζminus1A ιlowastcl-op Since ζminus1

A ιlowastcl-op is epi we have

e r ιcl-op = ιcl-op (340)

Actually (340) also follows from (313) and specialness of R(A) We will prove that

( fop fcl) (A|Acl ιcl-op) minusrarr (A|Z(A) e) where fop = idA fcl = r ιcl-op

(341)

is an isomorphism of Cardy algebrasfcl is an algebra map Compatibility with the units follows since ιcl-op is an algebramap

fcl ηcl = r ιcl-op ηcl = r ηR(A) = ηZ(A) (342)

Compatibility with the multiplication also follows since ιcl-op is an algebra map

m Z(A) ( fcl otimes fcl) = r m R(A) (e otimes e) (r otimes r) (ιcl-op otimes ιcl-op)

= r m R(A) (ιcl-op otimes ιcl-op) = r ιcl-op mcl = fcl mcl (343)

where in the second step we used (340)

Cardy Algebras and Sewing Constraints I 903

fcl is an isomorphism As above since fcl is an algebra map and since Acl is simple fcl

has to be monic By Lemma 319 rlowast = ζminus1A e Thus f lowast

cl = ιlowastcl-op rlowast = ζminus1A ιlowastcl-op e and

fcl f lowastcl = r ιcl-op ζminus1

A ιlowastcl-op e = r e r e = idZ(A) (344)

and so fcl is also epi and hence isofcl is a coalgebra map Since fcl is an algebra map so is f minus1

cl By (344) f minus1cl = f lowast

cland by Lemma 218 (iii) this implies that fcl is a also coalgebra mapThe diagram (315) commutes Commutativity of (315) is equivalent to e fcl = ιcl-opwhich holds by (340)

Let A be a special symmetric Frobenius algebra So far we have seen that (A|Z(A) e)is a Cardy algebra and that all Cardy algebras with Aop = A and simple Acl are of thisform It is now natural to ask if every simple Acl does occur as part of a Cardy algebraThe following theorem provides an affirmative answer Recall that for an A-left moduleM the object Mor otimesA M is an algebra (see eg [KR Lem 42])

Theorem 322 If Acl is a simple modular invariant commutative symmetric FrobeniusC2plusmn-algebra then there exist a simple special symmetric Frobenius C-algebra A and amorphism ιcl-op Acl rarr R(A) such that

(i) Acl sim= Z(A) as Frobenius algebras(ii) (A|Acl ιcl-op) is a Cardy C|C2plusmn-algebra

(iii) T (Acl) sim= oplusκisinJ Morκ otimesA Mκ as algebras where MκκisinJ is a set of representa-

tives of the isomorphism classes of simple A-left modules

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis haploid It then follows from Theorem 34 that Acl is special By Proposition 224T (Acl) is a special symmetric Frobenius algebra in C Thus T (Acl) = oplusi Ai where theAi are simple symmetric Frobenius algebras We will show that at least one of the Aiis special Since T (Acl) is special we have mT (Acl) T (Acl) = ζ idT (Acl) for someζ isin C

times Restricting this to the summand Ai shows mi i = ζ idAi FurthermoreεT (Acl) ηT (Acl) = ξ id1 for some ξ isin C

times But εT (Acl) ηT (Acl) = sumi εi ηi and so

at least one of the εi ηi has to be nonzero Therefore at least one of the Ai is speciallet A equiv Ai be this summand We denote the embedding A rarr T (Acl) by e0 and therestriction T (Acl) A by r0 Notice that r0 is an algebra homomorphism Define

ιcl-op = χ (r0) Acl minusrarr R(A) (345)

By Proposition 216 ιcl-op is an algebra homomorphism Next we verify the centre con-dition (313) or rather its equivalent form (320) By substituting the definitions onecan convince oneself that the commutativity mcl cAclAcl = mcl of Acl in C2plusmn implies thecondition mT (Acl) = mT (Acl) in C see [K2 Prop 36] Here T (Acl)otimes T (Acl) rarrT (Acl)otimes T (Acl) is given by

=oplusmn

(idCl

notimes cCl

m Crnotimes idCr

m

)

(cCl

m Clnotimes cminus1

Crn C

rm

)

(idCl

motimes cminus1

Cln C

rm

otimes idCrn

)

(346)

and we decomposed Acl as Acl = oplusnCln times Cr

n As a consequence we obtain the identity

r0 mT (Acl) (idAcl otimes e0) = r0 mT (Acl) (idAcl otimes e0) (347)

904 L Kong I Runkel

Using that r0 is an algebra map and that by definition ιcl-op = r0 we obtain (320)In order to show that (A|Acl ιcl-op) is a Cardy algebra it remains to show that the

Cardy condition (314) is satisfied We will demonstrate this via a detour by first provingthat Acl sim= Z(A) as Frobenius algebras

Recall the notations e and r given in (331) Using the centre condition (320) onecan check that Pl

R(A) ιcl-op = m R(A) R(A) ιcl-op By specialness of A we have

m A A = ζA idA and so together with e r = ζminus1A Pl

R(A) we get

e r ιcl-op = ιcl-op (348)

Next consider the morphism

fcl = r ιcl-op Acl minusrarr Z(A) (349)

By the same derivation as in (342) and (343) one sees that fcl is an algebra map Inparticular fcl ηcl = ηZ(A) = 0 and so fcl = 0 Since Acl is simple fcl has to be amonomorphism

By the same argument as used in the proof of Theorem 34 (ii) up to isomorphismAcl is the unique simple local Acl-(left-)module The algebra monomorphism fcl turnsZ(A) into an Acl-module Since Z(A) is commutative it is local as an Acl-module andso Z(A) sim= AoplusN

cl for some N ge 1 By construction A is a simple special symmetricFrobenius algebra Proposition 225 and Corollary 227 show that R(A) inherits all theseproperties and thus Z(A) is simple (see the comment below Eq (332)) By Theorem318 Z(A) is modular invariant and then by Theorem 34 (i) dim Z(A) = Dim C Thisimplies that N = 1 in Z(A) sim= AoplusN

cl and so fcl is in fact an isomorphismSince Acl and Z(A) are both haploid we have εZ(A) fcl = ξ εcl for some ξ isin C

timesThe counit uniquely determines the Frobenius structure on Acl and Z(A) (see eg [FRSLemma 37]) so that fcl is a coalgebra isomorphism iff ξ = 1 To compute ξ we composethe above identity with ηcl from the right Defining ζcl via εcl ηcl = ζminus1

cl Dim C middot id1 andusing (333) gives ξ = dim A ζclζ

2A By rescaling the comultiplication and the counit

of A and consequently changing ζA we can always achieve ξ = 1 This proves part (i)of the theorem

Equation (348) implies that ιcl-op = e fcl Since fcl is an isomorphism of Frobeniusalgebras by Lemmas 218 and 319 we have

ιcl-op ιlowastcl-op = e fcl f lowastcl elowast = ζA e r = Pl

R(A) (350)

Thus (A|Acl ιcl-op) is a Cardy algebra This proves part (ii) of the theoremPart (iii) can be seen as follows By [KR Prop 43] T Z(A) sim= oplusκisinJ Mor

κ otimesA Mκ asalgebras Together with the observation that T ( fcl) T (Acl) rarr T Z(A) is an isomor-phism of algebras this proves part (iii) Remark 323 Part (i) of Theorem 322 was announced by Muumlger [Muuml2] We provide anindependent proof in the setting of Cardy algebras

The above theorem together with Lemma 211 and Theorem 34 shows that a sim-ple commutative symmetric Frobenius C2plusmn-algebra Acl with dim Acl = Dim C is alwayspart of a Cardy algebra (Aop|Acl ιcl-op) for some simple special symmetric Frobeniusalgebra Aop in C However the above proof also illustrates that Aop is not unique Thisraises the question how two Cardy algebras with a given Acl can differ This question isanswered by [KR Thm 11] which in the present framework can be restated as follows

Cardy Algebras and Sewing Constraints I 905

Theorem 324 If (A(i)op |A(i)cl ι(i)cl-op) i = 1 2 are two Cardy C|C2plusmn-algebras such that

A(i)cl is simple and dim A(i)op = 0 for i = 1 2 then A(1)clsim= A(2)cl as algebras if and only

if A(1)op and A(2)op are Morita equivalent

Proof Theorem 11 in [KR] is stated for A(i)op being non-degenerate algebras and A(i)cl =Z(A(i)op ) for i = 1 2 By Proposition 316 A(i)op are simple and special for i = 1 2

Then by [KR Lem 21] A(i)op are non-degenerate algebras By Theorem 321 we have

A(i)clsim= Z(A(i)op ) as Frobenius algebras Finally by [KR Thm 11] Z(A(1)op ) sim= Z(A(2)op )

as algebras iff A(1)op and A(2)op are Morita equivalent Let Cmax(C2plusmn) be the set of equivalence classes [B] of simple modular invariant

commutative symmetric Frobenius algebras B in C2plusmn Two such algebras B and B prime areequivalent if B and B prime are isomorphic as algebras (but not necessarily as Frobenius alge-bras) Let Msimp(C) be the set of Morita classes of simple special symmetric Frobeniusalgebras in C Define the map z Msimp(C) rarr Cmax(C2plusmn) by z A rarr [Z(A)] whereA denotes the Morita class of A From Theorem 322 (i) and [KR Thm 11] we learn

Corollary 325 The map z Msimp(C) rarr Cmax(C2plusmn) is a bijection

A Appendix

A1 Proof of Lemma 27 We will show that if (F ψ F2 ψ

F0 ) is a colax tensor functor

from C1 to C2 then (G φG2 φ

G0 ) is a lax tensor functor from C2 to C1 Applying this

result to the opposed categories then gives the converse statementWe need to show that φG

0 and φG2 make the diagrams (21) and (22) commute We

first prove the commutativity of (21) Consider the following diagram

G(A)otimes (G(B)otimes G(C))

GψF2 δ

idG(A)otimesφG2 G(A)otimes G(B otimes C)

GψF2 δ

G(FG(A)otimes F(G(B)otimes G(C)))

G(F(idG(A))otimesF(φG2 ))

G(idFG(A)otimesψ F2 )

G(FG(A)otimes FG(B otimes C))

G(ρAotimesρBotimesC )

G(FG(A)otimes (FG(B)otimes FG(C)))

G(ρAotimes(ρBotimesρC )) G(A otimes (B otimes C))

(A1)

The top subdiagram is commutative because of the naturality of Gψ F2 δ The commu-

tativity of the bottom subdiagram follows from the following identities

(ρB otimes ρC ) ψ F2 = (ρB otimes ρC ) ψ F

2 ρF Fδ

= ρBotimesC FG(ρB otimes ρC ) FG(ψ F2 ) Fδ

= ρBotimesC F(φG2 ) (A2)

as a map F(G(B) otimes G(C)) rarr B otimes C The commutativity of (A1) implies that thecomposition of maps in the left column in (21) can be replaced by

G(ρA otimes (ρB otimes ρC )) G((idFG(A) otimes ψ F

2 ) ψ F2

) δ (A3)

906 L Kong I Runkel

Similarly we can show that the composition of maps in the right column in (21) can bereplaced by

G((ρA otimes ρB)otimes ρC ) G((ψ F

2 otimes idFG(C)) ψ F2

) δ (A4)

Using the commutativity of (23) it is easy to see that (21) with the left and right col-umns of (21) replaced by (A3) and (A4) respectively is commutative Hence (21) iscommutative

Now we prove the commutativity of the first diagram in (22)

φG2 (φG

0 otimes idG(A))

(1)= G(ρ12 otimes ρA) Gψ F2 δ

[(Gψ F

0 δ11)otimes idG(A)

]

(2)= G(ρ12 otimes ρA) Gψ F2 G F(Gψ0 otimes idG(A)) G F(δ11 otimes idG(A)) δ

(3)= G(ρ12 otimes ρA) G(FG(ψ F0 )otimes idFG(A)) Gψ F

2 G F(δ11 otimes idG(A)) δ(4)= G(id12 otimes ρA) G

([ψ F

0 ρF(11) (Fδ)11] otimes idFG(A)

) Gψ F

2 δ(5)= G(id12 otimes ρA) G(ψ F

0 otimes idFG(A)) Gψ F2 δ

(6)= G(id12 otimes ρA) G(lminus1FG(A)) G F(lG(A)) δ

(7)= G(lminus1A ) GρA δG lG(A)

(8)= G(lminus1A ) lG(A) (A5)

where in step (1) we substituted the definition of φG0 φ

G2 given in (210) in step (2)

we used the naturality of δ in step (3) we used the naturality of Gψ F2 in step (4) we

switched the position between Gρ12 and G FG(ψ F0 ) and the position between Gψ2 and

G F(δ12 otimes idG(A)) using the naturality of ρ and FψG2 respectively in step (5) we applied

the second identity in (28) in step (6) we used (24) in step (7) we used the naturalityof lminus1 and δ in step (8) we used the first identity in (28)

The proof of the commutativity of the second diagram in (22) is similar Thus wehave shown that G is a lax tensor functor

A2 Proof of Lemma 320 To prepare the proof recall that for a given object B isin Cthe modular group P SL(2Z) acts on the space oplusi HomC(B otimes Ui Ui ) see eg [BKSect 31] and [K3 Eq (455)] We will only need the action of S and Sminus1 Let f isinoplusi HomC(B otimes Ui Ui ) Then

S oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A6)

Cardy Algebras and Sewing Constraints I 907

Sminus1 oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A7)

By Lemma 32 to establish that (337) is S-invariant it is enough to prove the identity(34) when f is given by (337) Using (A6) and (A7) we can see that Eq (34) simplysays that oplusi j [RHS of (34)] is invariant under the action of S times S Consider the elementg of oplus jkisinIHomC2plusmn(R(A)otimes (Uor

j times Uk)Uorj times Uk) given by

g =oplusjkisinI

sumα

R(A)

R(A)

R(A)

R(A)

Uorj times Uk

Uorj times Uk

α

α

(A8)

By the above arguments proving S-invariance of (337) is equivalent to proving invari-ance of g under the action of S times S

For i isin I we denote by gi the component of g in(oplus jisinIHomC(A otimes Uor

i otimes Uorj U

orj )

)otimes (opluskisinIHomC(Ui otimes UkUk))

We view the second Hom-space in the above tensor product as a Hom-space in C+ insteadof Cminus It is enough to show that gi is invariant under the action of S times Sminus1 Note thatthe action of Sminus1 in C+ is equivalent to that of S in Cminus

The morphism gi can be canonically identified with a bilinear pairing

( middot middot )i ⎛⎝oplus

jisinIHomC(Uor

j A otimes Uori otimes Uor

j )

⎞⎠

times(oplus

kisinIHomC(UkUi otimes Uk)

)minusrarr C (A9)

as follows For h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) we set

(h1 h2)i = (dim U j dim Uk)minus1 trUor

j timesUk[gi (h1 times h2)] (A10)

908 L Kong I Runkel

When substituting the explicit form of the product m R(A) of R(A) = (A times 1) otimes R(1)after a short calculation one finds

(h1 h2)i =sumα

1

dim U j

A

A

A

A

A A

A α

α

h1

h2

U jUk

Uorj

Ui

Uk (A11)

Here the top morphism PlR(A) has been simplified with the help of the identity

PR(A) m R(A) (PR(A) otimes PR(A))

=(oplus

iisinI((m A A)otimes idUor

i)times idUi

) m R(A) (PR(A) otimes PR(A)) (A12)

which can be checked by direct calculation along the same lines as in the proof of [FrLem 310]

The action of the modular transformation S on oplusiisinI(B otimes Ui Ui ) for B isin C nat-urally induces an action on oplusiisinI(Ui B otimes Ui ) [K3 Prop 514] which we denote bySlowast In the present case we get an action of Slowast on oplus jisinIHomC(Uor

j A otimes Uori otimes Uor

j ) and

opluskisinIHomC(UkUi otimes Uk) Then to show gi is invariant under the action of S times Sminus1

amounts to showing that

(h1 h2)i = ((Sminus1)lowast h1 Slowast h2)i (A13)

Cardy Algebras and Sewing Constraints I 909

for all h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) We have

((Sminus1)lowast h1 Slowast h2)i =sum

mnα

dim Un

DimC

A

A

A

A

A A

A

Uk

α

α

h1

h2

Ui

U j

UmUn

(A14)

Now drag the upper vertex indexed by α in the above graph along its Uorm -leg until it

meets the lower vertex also indexed by α then sum over α and m This gives

((Sminus1)lowast h1 Slowast h2)i =sum

n

dim Un

DimC

A

A

A

A

A A

A

Uk

Un

h1

h2

Ui

U j

(A15)

910 L Kong I Runkel

If we just look at the neighbourhood of the Un-loop in the above graph we see thefollowing subgraphs

sumn

dim Un

DimC

A Aor

Uorj U j

Uk Uork

Un=

sumα

1

dim U j

A Aor Uk Uork

Uorj U j

αα

(A16)

where we have applied [BK Cor 3111] Substituting this subgraph back to the originalgraph in (A15) we obtain

((Sminus1)lowast h1 Slowast h2)i =sumα

1

dim U j

A

A

A

A

A A

A

A

A

Uorj

α

α

h1

h2m A

Ui

U jUk

Uk

(A17)

The graph in (A17) is equal to that in (A11) In order to see this we first drag theldquobubblerdquo (m A A) along A lines and through the m A vertex (because m A A is abimodule map) until it reaches the lower-left leg of the upper vertex indexed by α Thendrag the m A vertex along the (red) dotted line in the above graph Finally we applythe associativity of A (A12) and [Fr Lem 311] Then we see that the graph in (A17)exactly matches with the one in (A11)

Acknowledgements We would like to thank the organisers of the Oberwolfach Arbeitsgemeinschaft ldquoAlge-braic structures in conformal field theoriesrdquo (April 2007) where this work was started for an inspiring meetingWe would further like to thank the Hausdorff Institute for Mathematics in Bonn and the organisers of the stim-ulating meeting ldquoGeometry and Physicsrdquo (May 2008) We are indebted to Alexei Davydov Jens FjelstadJuumlrgen Fuchs Yi-Zhi Huang Alexei Kitaev Urs Schreiber Christoph Schweigert Stephan Stolz and PeterTeichner for helpful discussions andor comments on a draft of this paper The research of IR was partiallysupported by the EPSRC First Grant EPE0050471 the PPARC rolling grant PPC5071451 and the MarieCurie network lsquoSuperstring Theoryrsquo (MRTN-CT-2004-512194)

Cardy Algebras and Sewing Constraints I 911

References

[AN] Alexeevski A Natanzon SM Noncommutative two-dimensional topological field theories andhurwitz numbers for real algebraic curves Sel Math New Ser 12 307ndash377 (2006)

[Bi] Bichon J Cosovereign hopf algebras J Pure Appl Alg 157 121ndash133 (2001)[BK] Bakalov B Kirillov AA Lectures on Tensor Categories and Modular Functors Providence

RI Amer Math Soc 2001[C1] Cardy JL Operator content of two-dimensional conformal invariant theories Nucl Phys

B 270 186ndash204 (1986)[C2] Cardy JL Boundary conditions fusion rules and the verlinde formula Nucl Phys B 324

581ndash596 (1989)[CL] Cardy JL Lewellen DC Bulk and boundary operators in conformal field theory Phys Lett

B 259 274ndash278 (1991)[DMZ] Dong C-Y Mason G Zhu Y-C Discrete series of the Virasoro algebra and the moonshine

module In Algebraic Groups and Their Generalizations Quantum and infinite-dimensional Meth-ods Proc Symp Pure Math 56 Part 2 Providence RI Amer MathSoc 1994 pp 295ndash316

[DP] Day B Pastro C Note on frobenius monoidal functors New York J Math 14 733ndash742 (2008)[ENO1] Etingof PI Nikshych D Ostrik V On fusion categories Ann Math 162 581ndash642 (2005)[ENO2] Etingof PI Nikshych D Ostrik V An analogue of radfordrsquos s4 formula for finite tensor

categories Int Math Research Notices 54 2915ndash2933 (2004)[Fe] Felder G Froumlhlich J Fuchs J Schweigert C Correlation functions and boundary conditions in

rational conformal field theory and three-dimensional topology Comp Math 131 189ndash237 (2002)[FHL] Frenkel IB Huang Y-Z Lepowsky J On axiomatic approaches to vertex operator algebras

and modules Mem Amer Math Soc 104 (1993)[Fj] Fjelstad J Fuchs J Runkel I Schweigert C Uniqueness of openclosed rational cft with

given algebra of open states Adv Theor Math Phys 12 1283ndash1375 (2008)[FK] Froumlhlich J King C The chern-simons theory and knot polynomials Commun Math

Phys 126 167ndash199 (1989)[Fr] Froumlhlich J Fuchs J Runkel I Schweigert C Correspondences of ribbon categories Adv

Math 199 192ndash329 (2006)[FRS] Fuchs J Runkel I Schweigert C Tft construction of rcft correlators I partition functions Nucl

Phys B 646 353ndash497 (2002)[FS] Fuchs J Schweigert C Category theory for conformal boundary conditions Fields Inst Com-

mun 39 25ndash71 (2003)[H1] Huang Y-Z Two-dimensional conformal geometry and vertex operator algebras Progress in

Mathematics Vol 148 Boston Birkhaumluser 1997[H2] Huang Y-Z Rigidity and modularity of vertex tensor categories Comm Contemp Math 10

871ndash911 (2008)[HK1] Huang Y-Z Kong L Open-string vertex algebra category and operads Commun Math

Phys 250 433ndash471 (2004)[HK2] Huang Y-Z Kong L Full field algebras Commun Math Phys 272 345ndash396 (2007)[HK3] Huang Y-Z Kong L Modular invariance for conformal full field algebras httparxivorgabs

math0609570v2[mathQA] 2006[HL] Huang Y-Z Lepowsky J Tensor products of modules for a vertex operator algebra and vertex

tensor categories In Lie Theory and Geometry in honor of Bertram Kostant ed R BrylinskiJ-L Brylinski V Guillemin V Kac Boston Birkhaumluser 1994 pp 349ndash383

[JS] Joyal A Street R Braided tensor categories Adv Math 102 20ndash78 (1993)[K1] Kong L Full field algebras operads and tensor categories Adv Math 213 271ndash340 (2007)[K2] Kong L Open-closed field algebras Commun Math Phys 280 207ndash261 (2008)[K3] Kong L Cardy condition for open-closed field algebras Commun Math Phys 283 25ndash92 (2008)[Ki] Kitaev A Private communication[KO] Kirillov AA Ostrik V On q-analog of mckay correspondence and ade classification of sl(2)

conformal field theories Adv Math 171 183ndash227 (2002)[KR] Kong L Runkel I Morita classes of algebras in modular tensor categories Adv Math 219

1548ndash1576 (2008)[Ld] Lauda AD Frobenius algebras and ambidextrous adjunctions Theo Appl Cat 16 84ndash

122 (2006)[Li1] Li H-S Regular representations of vertex operator algebras Commun Contemp Math 4

639ndash683 (2002)[Li2] Li H-S Regular representations and huang-lepowsky tensor functors for vertex operator

algebras J Alge 255 423ndash462 (2002)

878 L Kong I Runkel

φG0 11 rarr G(12) and a natural transformation φG

2 otimes (G times G) rarr G otimes by forA B isin C2

φG0 = χ(ψ F

0 ) = 11δ11minusrarr G F(11)

G(ψ F0 )minusminusminusminusrarr G(12)

φG2 = χ((ρA otimes ρB) ψ F

2 ) = G(A)otimes G(B)δminusrarr G F (G(A)otimes G(B))

G(ψ F2 )minusminusminusminusrarr G (FG(A)otimes FG(B))

G(ρAotimesρB )minusminusminusminusminusminusrarr G(A otimes B) (210)

where we have used the first identity in (29) Notice that φG2 is natural because it is a

composition of natural transformations One can easily show that ψ F0 and ψ F

2 can bere-obtained from φG

0 and φG2 as follows

ψ F0 = χminus1(φG

0 ) = F(11)FφG

0minusminusrarr FG(12)ρminusrarr 12

ψ F2 = χminus1(φG

2 (δ otimes δ)) = F(U otimes V )F(δotimesδ)minusminusminusminusrarr F (G F(U )otimes G F(V ))

FφG2minusminusrarr FG (F(U )otimes F(V ))

ρminusrarr F(U )otimes F(V ) (211)

for U V isin C1 The following result is standard for the sake of completeness we givea proof in Appendix A1

Lemma 27 (F ψ F2 ψ

F0 ) is a colax tensor functor iff (G φG

2 φG0 ) is a lax tensor

functor

22 Algebras in tensor categories An algebra in a tensor category C or a C-algebra isa triple A = (Am η) where A is an object of C m (the multiplication) is a morphismA otimes A rarr A such that m (m otimes idA) αAAA = m (idA otimes m) and η (the unit) is amorphism 1 rarr A such that m (idA otimes η) = idA rA and m (ηotimes idA) = idA lAIf C is braided and m cAA = m then A is called commutative

A left A-module is a pair (MmM ) where M isin C and m M is a morphism AotimesM rarr Msuch that mM(idAotimesm M ) = m M(m AotimesidM )αAAM and m M(ηAotimesidM ) = idMlM Right A-modules and A-bimodules are defined similarly

Definition 28 Let C be a tensor category and let A be an algebra in C

(i) A is called simple iff it is simple as a bimodule over itselfLet C be in addition k-linear for k a field

(ii) A is called absolutely simple iff the space of A-bimodule maps from A to itself isone-dimensional dimk HomA|A(A A) = 1

(iii) A is called haploid iff dimk Hom(1 A) = 1 [FS Def 43]

In the following we will assume that all tensor categories are strict to avoid spelling outassociators and unit constraints

A C-coalgebra A = (A ε) is defined analogously to a C-algebra ie A rarrA otimes A and ε A rarr 1 obey coassociativity and counit conditions

If C is braided and if A and B are C-algebras there are two in general non-isomorphicalgebra structures on A otimes B We choose A otimes B to be the C-algebra with multiplicationm AotimesB = (m Aotimesm B)(idAotimescminus1

ABotimesidB) and unitηAotimesB = ηAotimesηB Similarly if A and Bare C-coalgebras then A otimes B becomes a C-coalgebra if we choose the comultiplicationAotimesB = (idA otimes cAB otimes idB) (A otimesB) and the counit εAotimesB = εA otimes εB

Cardy Algebras and Sewing Constraints I 879

Definition 29 A Frobenius algebra A = (Am η ε) is an algebra and a coalge-bra such that the coproduct is an intertwiner of A-bimodules

(idA otimes m) (otimes idA) = otimes m = (m otimes idA) (idA otimes)

We will use the following graphical representation for the morphisms of a Frobeniusalgebra

m =A A

A

η =A

=A A

A

ε =A

(212)

A Frobenius algebra A in a k-linear tensor category for k a field is called special iffm = ζ idA and ε η = ξ id1 for nonzero constants ζ ξ isin k If ζ = 1 we call A nor-malised-special A Frobenius algebra homomorphism between two Frobenius algebrasis both an algebra homomorphism and a coalgebra homomorphism

A (strictly) sovereign tensor category is a tensor category equipped with a left and aright duality which agree on objects and morphisms (see eg [BiFS] for more details)We will write the dualities as

Uor U

= dU Uor otimes U rarr 1U Uor

= dU U otimes Uor rarr 1

U Uor

= bU 1 rarr U otimes UorUor U

= bU 1 rarr Uor otimes U

(213)

In terms of these we define the left and right dimension of an object U as

diml U = dU bU dimr U = dU bU (214)

both of which are elements of Hom(1 1)Let now C be a sovereign tensor category For a Frobenius algebra A in C we define

two morphisms

A =

A

Aor

primeA =

A

Aor

(215)

Definition 210 A Frobenius algebra A is symmetric iff A = primeA

The following lemma shows that under certain conditions we do not need to distin-guish the various notions of simplicity in Definition 28

Lemma 211 Let A be a commutative symmetric Frobenius algebra in a C-linear semi-simple sovereign braided tensor category C and suppose that diml A = 0 Then thefollowing are equivalent

880 L Kong I Runkel

(i) A is simple(ii) A is absolutely simple

(iii) A is haploid

Proof (ii)hArr(iii) A is haploid iff it is absolutely simple as a left module over itself[FS Eq (417)] Furthermore for a commutative algebra we have HomA(A A) =HomA|A(A A) and so A is haploid iff it is absolutely simple(i)rArr(ii) If A is simple then every nonzero element of HomA|A(A A) is invertibleHence this space forms a division algebra over C and is therefore isomorphic to C(iii)rArr(i) Since C is semi-simple and A is haploid also Hom(A 1) is one-dimensionalThe counit ε is a nonzero element in this space and so gives a basis This implies firstlythat ε η = 0 and secondly that there is a constant β isin C such that

β middot ε = dA (idAor otimes m) (bA otimes idA) (216)

Composing with η from the right yields β εη = diml A The right-hand side is nonzeroand so β = 0 By [FRS Lem 311] A is special We have already proved (ii)hArr(iii)and so A is absolutely simple A special Frobenius algebra in a semi-simple categoryhas a semi-simple category of bimodules (apply [FS Prop 524] to the algebra tensoredwith its opposite algebra) For semi-simple C-linear categories simple and absolutelysimple are equivalent2 Thus A is simple Remark 212 For a Frobenius algebra A the morphisms (215) are invertible and henceA sim= Aor In this case one has diml A = dimr A [FS Rem 363] and so we could havestated the above lemma equivalently with the condition dimr A = 0

Let F C1 rarr C2 be a lax tensor functor between two tensor categories C1 C2 and

let (Am A ηA) be an algebra in C1 Define morphisms F(A)otimes F(A)m F(A)minusminusminusrarr F(A) and

12ηF(A)minusminusminusrarr F(A) as

m F(A) = F(m A) φF2 ηF(A) = F(ηA) φF

0 (217)

Then (F(A)m F(A) ηF(A)) is an algebra in C2 [JS Prop 55] If f A rarr B is analgebra homomorphism between two algebras A B isin C1 then F( f ) F(A) rarr F(B)is also an algebra homomorphism If (MmM ) is a left (or right) A-module in C1 then(F(M) F(mM )φF

2 ) is a left (or right) F(A)-module if M has a A-bimodule structurethen F(M) naturally has a F(A)-bimodule structure

Similarly if (AA εA) is a coalgebra in C1 and F C1 rarr C2 is a colax tensor func-

tor then F(A) with coproduct F(A)F(A)minusminusminusrarr F(A) otimes F(A) and counit F(A)

εF(A)minusminusminusrarr 12given by

F(A) = ψ F2 F(A) εF(A) = ψ F

0 F(εA) (218)

is a coalgebra in C2 If f A rarr B is a coalgebra homomorphism between two coalgebrasA B isin C1 then F( f ) F(A) rarr F(B) is also a coalgebra homomorphism

2 To see this note that if U is simple then the C-vector space Hom(UU ) is a division algebra and henceHom(UU ) = C idU Conversely if U is not simple then U = U1 oplus U2 and Hom(UU ) contains at leasttwo linearly independent elements namely idU1 and idU2

Cardy Algebras and Sewing Constraints I 881

Proposition 2133 If F C1 rarr C2 is a Frobenius functor and (Am A ηAA εA) aFrobenius algebra in C1 then (F(A)m F(A) ηF(A)F(A) εF(A)) is a Frobenius alge-bra in C2

Proof One Frobenius property (m F(A)otimes idF(A)) (idF(A)otimesF(A)) = F(A) m F(A)follows from the commutativity of the following diagram (we spell out the associativityisomorphisms)

F(A)otimes F(A)idF(A)otimesF(A)

φF2

F(A)otimes F(A otimes A)idF(A)otimesψF

2

φF2

F(A)otimes (F(A)otimes F(A))

αF(A)F(A)F(A)

F(A otimes A)

F(idAotimesA)

F(m A)

F(A otimes (A otimes A))

F(αAAA)

(F(A)otimes F(A))otimes F(A)

φF2 otimesidF(A)

F((A otimes A)otimes A)

F(m AotimesidA)

ψF2 F(A otimes A)otimes F(A)

F(m A)otimesidF(A)

F(A)

F(A) F(A otimes A)ψF

2 F(A)otimes F(A)

(219)

The commutativity of the upper-left subdiagram follows from the naturalness of φF2 that

of the upper-right subdiagram follows from (25) that of the lower-left subdiagram fol-lows from the Frobenius properties of A and that of the lower-right subdiagram followsfrom the naturalness of ψ F

2 The proof of the other Frobenius property is similar Proposition 214 If F C1 rarr C2 is a tensor functor and A a Frobenius algebra in C1then

(i) F(A) has a natural structure of Frobenius algebra as given in Proposition 213(ii) If A is (normalised-)special so is F(A)

Proof Part (i) follows from Propositions 25 and 213 Part (ii) is a straightforwardverification of the definition using ψ F

2 = (φF2 )

minus1 and ψ F0 = (φF

0 )minus1

Let C1 C2 be sovereign tensor categories and F C1 rarr C2 a Frobenius functor Wedefine two morphisms IF(Aor) I prime

F(Aor) F(Aor) rarr F(A)or for a Frobenius algebra A inC1 as follows

IF(Aor) = ((ψ F0 F(dA) φF

2 )otimes idF(A)or) (idF(Aor) otimes bF(A))

I primeF(Aor) = (idF(A)or otimes (ψ F

0 F(dA) φF2 )) (bF(A) otimes idF(Aor))

(220)

It is easy to see that these are isomorphisms

Lemma 215 If F C1 rarr C2 is a Frobenius functor and A a Frobenius algebra in C1then

F(A) = IF(Aor) F(A) primeF(A) = I prime

F(Aor) F(primeA) (221)

3 After the preprint of the present paper appeared we noticed that this proposition is also proved in[DP Cor 5]

882 L Kong I Runkel

Proof We only prove the first equality the second one can be seen in the same way Bydefinition we have

IF(Aor) F(A)

= ((ψ F0 F(dA) φF

2 )otimes idF(A)or) (idF(Aor) otimes bF(A)) F(A)

=[(ψ F

0 F(dA) φF2 ) (F(A)otimes idF(A))

]otimes idF(A)or

(idF(A) otimes bF(A))

For the term inside the square brackets we find

ψ F0 F(dA) φF

2 (F(A)otimes idF(A)) = ψ F0 F(dA) F(A otimes idA) φF

2

= ψ F0 F(dA (A otimes idA)) φF

2 = ψ F0 F(εA m A) φF

2 (222)

On the other hand by definitionF(A) = [((ψ F0 F(εA) (F(m A)φF

2 ))otimes idF(A)or] (idF(A) otimes bF(A)) This demonstrates the first equality in (221) Proposition 216 Let F C1 rarr C2 be a tensor functor G C2 rarr C1 a functor〈FG χ〉 an adjunction A a C1-algebra and B a C2-algebra Then f A rarr G(B)is an algebra homomorphism if and only if f = χminus1( f ) F(A) rarr B is an algebrahomomorphism

Proof We need to show that

mG(B) ( f otimes f ) = f m A and f ηA = ηG(B) (223)

is equivalent to

m B ( f otimes f ) = f m F(A) and f ηF(A) = ηBs (224)

We first prove that the first identity in (223) is equivalent to the first identity in (224)For the left-hand side of the first identity in (223) we have the following equalities

mG(B) ( f otimes f )(1)= G(m B) φG

2 ( f otimes f )(2)= G(m B) G(ρ otimes ρ) G(ψ F

2 ) δ ( f otimes f )(3)= G(m B) G(ρ otimes ρ) G(ψ F

2 ) G F( f otimes f ) δ(4)= G(m B) G(ρ otimes ρ) G(F( f )otimes F( f )) G(ψ F

2 ) δ(5)= G

(m B (ρ otimes ρ) (F( f )otimes F( f )) ψ F

2

) δ

(6)= χ(

m B (ρ otimes ρ) (F( f )otimes F( f )) ψ F2

) (225)

where (1) is the definition of mG(B) in (217) (2) is the second identity in (210) (3)and (4) are naturality of δ and ψ F

2 respectively step (5) is functoriality of G and finallystep (6) is (29) For the right hand side of the first identity in (223) we get

f m A(1)= Gρ δG ( f m A)

(2)= Gρ G F( f m A) δ(3)= G(ρ F( f m A)) δ(4)= χ(ρ F( f ) F(m A)) (226)

Cardy Algebras and Sewing Constraints I 883

where (1) is the adjunction property (28) (2) is naturality of δ (3) functoriality of Gand (4) amounts to (29) and functoriality of F

On the other hand we see that the first equality in (224) is equivalent to

m B (ρ otimes ρ) (F( f )otimes F( f )) = ρ F( f ) F(m A) φF2 (227)

Using that φF2 is invertible with inverse (φF

2 )minus1 = ψ F

2 and that χ is an isomorphismit follows that the statement that (225) is equal to (226) is equivalent to the identity(227)

Now we prove that the second identity in (223) is equivalent to the second identity in(224) Using (217) and (210) we can write ηG(B) = G(ηB)φF

0 = G(ηB)G(ψ F0 )δ1

Together with (29) this shows that the second identity in (223) is equivalent to

f ηA = χ(ηB ψ F0 ) (228)

On the other hand the second identity in (224) is equivalent to

ρ F( f ) F(ηA) φF0 = ηB (229)

which by φF0 = (ψ F

0 )minus1 and (29) is further equivalent to (228)

Definition 217 Let (Am A ηAA εA) and (Bm B ηBB εB) be two Frobeniusalgebras in a tensor category C For f A rarr B we define f lowast B rarr A by

f lowast = ((εB m B)otimes idA) (idB otimes f otimes idA) (idB otimes (A ηA)) (230)

The following lemma is immediate from the definition of (middot)lowast and the properties ofFrobenius algebras We omit the proof

Lemma 218 Let C be a tensor category let A BC be Frobenius algebras in C andlet f A rarr B and g B rarr C be morphisms

(i) (g f )lowast = f lowast glowast(ii) f is a monomorphism iff f lowast is an epimorphism

(iii) f is an algebra map iff f lowast is a coalgebra map(iv) If f is a homomorphism of Frobenius algebras then f lowast f = idA and f f lowast =

idB(v) If C is sovereign and if A and B are symmetric then f lowastlowast = f

Let C and D be tensor categories and let F C rarr D be a Frobenius functor GivenFrobenius algebras A B in C and a morphism f A rarr B the next lemma shows how(middot)lowast behaves under F

Lemma 219 F( f lowast) = F( f )lowast

Proof The definition of the structure morphisms of the Frobenius algebra F(A) is givenin (217) and (218) Substituting these definitions gives

F( f )lowast =[(ψ F

0 F(εB) F(m B) φF2

)otimes idF(A)

] [

idF(B) otimes F( f )otimes idF(A)]

[idF(B) otimes

(ψ F

2 F(A) F(ηA) φF0

)]

= (ψ F0 otimes idF(A)) [

F (εB m B (idB otimes f ))otimes idF(A)]

(φF2 otimes idF(A)) (idF(B) otimes ψ F

2 )

[idF(B) otimes F(A ηA)

] (idF(B) otimes φF0 ) (231)

884 L Kong I Runkel

In the middle line of the last expression we can use the defining property (25) of F namely we substitute (φF

2 otimes idF(A)) (idF(B) otimes ψ F2 ) = ψ F

2 φF2 Then ψ F

2 can bemoved to the left and φF

2 to the right until they can be omitted against ψ F0 and φF

0 respectively using (22) and (24) This results in

F( f )lowast = F ((εB m B (idB otimes f ))otimes idA) F (idB otimes (A ηA)) (232)

which is nothing but F( f lowast)

23 Modular tensor categories Let C be a modular tensor category [TBK] ie anabelian semi-simple finite C-linear ribbon category with simple tensor unit 1 and a non-degeneracy condition on the braiding (to be stated in a moment) We denote the set ofequivalence classes of simple objects in C by I elements in I by i j k isin I and theirrepresentatives by Ui U j Uk We also set U0 = 1 and for an index k isin I we define kby Uk

sim= Uork

Since the tensor unit is simple we shall for modular tensor categories identifyHom(1 1) sim= C (cf footnote 2) Define numbers si j isin C by4

si j = UiU j (233)

They obey si j = s ji and s0i = dim Ui see eg [BK Sect 31] (In a ribbon cate-gory the left and right dimension (214) of Ui coincide and are denoted by dim Ui )The non-degeneracy condition on the braiding of a modular tensor category is that the|I|times|I|-matrix s should be invertible In fact [BK Thm 317]

sumkisinI

sik sk j = Dim C δij (234)

where Dim C = sumiisinI(dim Ui )

2 One can show (even in the weaker context of fusioncategories over C) that Dim C ge 1 [ENO1 Thm 23] In particular Dim C = 0 We fixonce and for all a square root

radicDim C of Dim C

Let us fix a basis λα(i j)kN k

i jα=1 in HomC(Ui otimesU j Uk) and the dual basis ϒ(i j)k

α N ki j

α=1

in HomC(UkUi otimesU j ) The duality of the bases means that λα(i j)k ϒ(i j)kβ = δαβ idUk

We also fix λ(0i)i = λ(i0)i = idUi We denote the basis vectors graphically as follows

λα(i j)k = α

Uk

Ui U j

ϒ(i j)kα = α

Uk

Ui U j

(235)

4 In the graphical notation used below we have given an orientation to the ribbons indicated by the arrowsFor example it is understood that this orientation determines which of the duality morphisms in (213) to use

Cardy Algebras and Sewing Constraints I 885

For V isin C we also choose a basis b(iα)V of HomC(VUi ) and the dual basis bV(iβ)

of HomC(Ui V ) for i isin I such that b(iα)V bV(iβ) = δαβ idUi We use the graphical

notation

b(iα)V = α

Ui

V

bV(iα) = α

Ui

V

(236)

Given two modular tensor categories C and D by C D we mean the tensor productof additive categories over C [BK Def 1115] ie the category whose objects are directsums of pairs V times W of objects V isin C and W isin D and whose morphism spaces are

HomCD(V times W V prime times W prime) = HomC(V V prime)otimesC HomC(WW prime) (237)

for pairs and direct sums of these if the objects are direct sums of pairsIf we replace the braiding and the twist in C by the antibraiding cminus1 and the antitwist

θminus1 respectively we obtain another ribbon category structure on C In order to distin-guish these two distinct structures we denote (C c θ) and (C cminus1 θminus1) by C+ and Cminusrespectively As in the Introduction we will abbreviate

C2plusmn = C+ Cminus (238)

Note that a set of representatives of the simple objects in C2plusmn is given by Ui times U j fori j isin I

For the remainder of Sect 2 we fix a modular tensor category C

24 The functors T and R The tensor product bifunctor otimes can be naturally extendedto a functor T C2plusmn rarr C Namely T (oplusN

i=1Vi times Wi ) = oplusNi=1Vi otimes Wi for all Vi Wi isin C

and N isin N The functor T becomes a tensor functor as follows For φT0 1 rarr T (1 times 1)

take φT0 = id1 (or lminus1

1 in the non-strict case) Next notice that for U VW X isin C

T (U times V )otimes T (W times X)= (U otimes V )otimes (W otimes X)

T ((U times V )otimes (W times X))= (U otimes W )otimes (V otimes X)(239)

We define φT2 T (U times V )otimes T (W times X) rarr T ((U times V )otimes (W times X)) by

φT2 = idU otimes cminus1

W V otimes idX (240)

(In the non-strict case the appropriate associators have to be added) The above definitionof φT

2 can be naturally extended to a morphism φT2 T (M1 otimes M2) rarr T (M1)otimes T (M2)

for any pair of objects M1M2 in C2plusmn The following result can be checked by directcalculation [JS Prop 52]

Lemma 220 The triple (T φT2 φ

T0 ) gives a tensor functor

886 L Kong I Runkel

In particular (T φT2 φ

T0 ψ

T2 ψ

T0 ) where ψT

2 = (φT2 )

minus1 and ψT0 = (φT

0 )minus1 gives

a Frobenius functorDefine the functor R C rarr C2plusmn as follows for A isin C and f isin HomC(A B)

R(A) =oplusiisinI

(A otimes Uori )times Ui R( f ) =

oplusiisinI

( f otimes idUori)times idUi (241)

This functor was also considered in a slightly different context in [ENO2 Prop 23]The family of isomorphisms γ R

A = oplusiisinI DimCdim Ui

id(AotimesUori )timesUi

isin Aut(R(A)) defines a

natural isomorphism γ R R rarr ROur next aim is to show that R is left and right adjoint to T in other words R

and T form an ambidextrous adjunction (see eg [Ld] for a discussion of ambidextrousadjunctions) To this end we introduce two linear isomorphisms for A isin C and M isin C2plusmn

χ HomC(T (M) A) minusrarr HomC2plusmn(M R(A))

χ HomC(A T (M)) minusrarr HomC2plusmn(R(A)M)(242)

If we decompose M as M = oplusNn=1 Ml

n times Mrn then χ and χ are given by

χ Noplus

n=1

fn

Mln Mr

n

A

rarrNoplus

n=1

oplusiisinI

sumα

fn

Mln

Mrn

A

α

Uori

times α

Mrn

Ui

(243)

and

χ Noplus

n=1 gn

Mln Mr

n

A

rarrNoplus

n=1

oplusiisinI

sumα gn

Mln

Mrn

A

α

Uori

times α

Ui

Mrn

DimCdim Ui

(244)

Notice that χ and χ are independent of the choice of basis

Theorem 221 〈T R χ〉 and 〈R T χminus1〉 are adjunctions ie R is both left and rightadjoint of T

Proof Write M as M = oplusNn=1 Ml

n times Mrn The isomorphism χ amounts to the following

composition of natural isomorphisms

HomC(T (M) A) = oplusnHomC(Mln otimes Mr

n A)sim= oplusni HomC(Ml

n otimes Ui A)otimes HomC(Mrn Ui )

sim= oplusni HomC(Mln A otimes Uor

i )otimes HomC(Mrn Ui ) = HomC2plusmn(M R(A)) (245)

Cardy Algebras and Sewing Constraints I 887

Thus χ is natural Let (γ RA )

lowast HomC2plusmn(R(A)M) rarr HomC2plusmn(R(A)M) denote the

pull-back of γ RA The isomorphism χ is equal to the composition of (γ R

A )lowast and the

following sequence of natural isomorphisms

HomC(A T (M)) = oplusn HomC(AMln otimes Mr

n)

sim= oplusni HomC(AMln otimes Ui )otimes HomC(Ui Mr

n)

sim= oplusni HomC(A otimes Uori Ml

n)otimes HomC(Ui Mrn) = HomC2plusmn(R(A)M) (246)

We have proved that both χ and χ are natural isomorphisms There are four natural transformations associated to χ and χ namely

idC2plusmnδminusrarr RT

ρminusrarr idC2plusmn and idCδminusrarr T R

ρminusrarr idC (247)

defined by for A isin C M isin C2plusmn

δM = χ(idT (M)) ρA = χminus1(idR(A))

ρM = χ (idT (M)) δA = χminus1(idR(A))(248)

They can be expressed graphically as follows with M = oplusNn=1 Ml

n times Mrn

δM =oplusni

sumα

Mln

Mln Mr

n Uori

αtimes

Ui

Mrn

α

ρA =oplusiisinI

A

A

Uori Ui

(249)

ρM =oplusni

sumα

Mln

Mln

Mrn Uor

i

α times

Ui

Mrn

α DimCdim Ui

δA =oplusiisinI

A

A Uori Ui

dim UiDimC

Note that

ρM δM = Dim C middot idM and ρA δA = idA (250)

Lemma 222 The functors T and R as maps on the sets of morphisms have left inversesand thus are injective

Proof Let f A rarr B be a morphism in C We define a map Q R HomC2plusmn(R(A)

R(B)) rarr HomC(A B) by f prime rarr ρB T ( f prime) δA Then we have

Q R R( f ) = ρB T R( f ) δA = ρB δB f = f (251)

888 L Kong I Runkel

where we used naturality of δ and (250) in the second and third equalities respectivelySo Q R is a left inverse of R on morphisms Thus R is injective on morphisms Similarlylet g M rarr N be a morphism in C2plusmn We define a map QT HomC(T (M) T (N )) rarrHomC2plusmn(M N ) by gprime rarr (Dim C)minus1 middot ρN R(gprime) δM Then we have

QT T (g) = (Dim C)minus1 middot ρN RT (g) δM = (Dim C)minus1 middot ρN δN g = g (252)

So QT is a left inverse of T on morphisms Thus T is injective on morphisms Using (29) and (250) one can express the two inverse maps χminus1 χminus1 as follows

for f isin HomC2plusmn(M R(A)) and g isin HomC2plusmn(R(A)M)

χminus1( f ) = ρ T ( f ) χminus1(g) = T (g) δ (253)

By Proposition 25 and Lemma 27 R is both a lax and colax tensor functor In particularφR

0 1 times 1 rarr R(1) is given by

φR0 = χ (ψT

0 ) = R(ψT0 ) δ1 times 1 = id1times1 (254)

and φR2 R(A)otimes R(B) rarr R(A otimes B) by φR

2 = R(ρA otimes ρB) R(ψT2 ) δ which can

be expressed graphically as

φR2 =

oplusi jkisinI

sumα

A

A

Uori B

B

Uorj

Uork

α

times

Ui U j

α

Uk

(255)

Similarly ψ R0 R(1) rarr 1 times 1 is given by

ψ R0 = ρ1 R(φT

0 ) = Dim C id1times1 (256)

and ψ R2 R(A otimes B) rarr R(A) otimes R(B) by ψ R

2 = ρ R(φT2 ) R(δA otimes δB) which in

graphical notation reads

ψ R2 =

oplusi jkisinI

sumα

A

A Uori

B

B Uorj

Uork

α

times

Ui U j

α

Uk

dim Ui dim U j

dim Uk Dim C

(257)

If C has more than one simple object then R does not take the tensor unit of C to thetensor unit of C2plusmn and so is clearly not a tensor functor However we will show that R isstill a Frobenius functor This will imply that if A is a Frobenius algebra in C then

R(A) = (R(A)m R(A) ηR(A)R(A) εR(A)) (258)

Cardy Algebras and Sewing Constraints I 889

is a Frobenius algebra in C2plusmn where the structure morphisms were given in (217) and(218) In the case A = 1 it was proved in [Muuml1 Prop 41] (see also [Fr Lem 619]and [K1 Thm 52]) that (258) is a commutative simple symmetric normalised-specialFrobenius algebra in C2plusmn In fact given a Frobenius algebra A in C it is straightforwardto verify that the structure morphisms in (258) are precisely those of (A times 1)otimes R(1)cf Sect 22

Proposition 223 (R φR2 φ

R0 ψ

R2 ψ

R0 ) is a Frobenius functor

Proof Using the explicit graphical expression of φR2 φ

R0 ψ

R2 ψ

R0 it is easy to see that

the commutativity of the diagrams (25) and (26) are equivalent to the statement thatR(1) with structure morphisms as in (258) is a Frobenius algebra in C2plusmn The latterstatement is true by [Muuml1 Prop 41]

From Lemma 220 and Proposition 223 we see that T and R take Frobenius alge-bras to Frobenius algebras The following two propositions show how the properties ofFrobenius algebras are transported

Proposition 224 Let A be a Frobenius algebra in C2plusmn Then T (A) is a Frobenius algebrain C and

(i) A is symmetric iff T (A) is symmetric(ii) A is (normalised-)special iff T (A) is (normalised-)special

Proof For part (i) write A as a direct sum oplusNn=1 Al

n times Arn Then the maps IT (A) I prime

T (A) T (Aor) rarr T (A)or defined in (220) are given by

IT (A) = I primeT (A) = oplusN

n=1c(Aln)

or(Arn)

or (259)

Therefore by (221) T (A) = primeT (A) is equivalent to T (A) = T (prime

A) Since byLemma 222 T is injective on morphisms this proves part (i) Part (ii) can be checkedin the same way for example the condition mT (A) T (A) = ζ idT (A) is easily checkedto be equivalent to T (m A A) = ζ T (idA) Proposition 225 Let A be a Frobenius algebra A in C Then R(A) is a Frobeniusalgebra in C2plusmn and

(i) A is symmetric iff R(A) is symmetric(ii) A is (normalised-)special iff R(A) is (normalised-)special

Proof Recall that the structure morphisms of the Frobenius algebra R(A) are equal tothose of (A times 1) otimes R(1) Using this equality part (i) and (ii) follow because R(1) issymmetric and normalised-special For example

m R(A) R(A) = [(m A A)times id1] otimes idR(1) = R(m A A) (260)

so that m R(A) R(A) = ζ idR(A) is equivalent to R(m A A) = ζ R(idA) which byLemma 222 is equivalent to m A A = ζ idA

The functor R has one additional property not shared by T namely R takes absolutelysimple algebras to absolutely simple algebras We will see explicitly in Sect 33 thatthis is not true for T

890 L Kong I Runkel

Lemma 226 For a C-algebra A the map

R HomA|A(A A) rarr HomR(A)|R(A)(R(A) R(A)) (261)

given by f rarr R( f ) is well-defined and an isomorphism

Proof Since R is a lax tensor functor R(A) is naturally a R(A)-bimodule It is easy tosee that R in (261) is a well-defined map R(A) is also naturally a R(1)-bimodule whichcan be identified with the induced R(1)-bimodule structure on (A times 1)otimes R(1) wherethe left R(1) action on (Atimes1)otimes R(1) is given by (idAtimes1 otimesm R(1))(cminus1

Atimes1R(1)otimes idR(1))We have the following natural isomorphisms

HomR(1)|R(1)(R(A) R(A))sim=minusrarr HomC2plusmn(A times 1 R(A))

χminus1

minusminusrarr HomC(A A) (262)

which by (253) are given by for f isin HomR(1)|R(1)(R(A) R(A))

f rarr f prime = f (idAtimes1 otimes ηR(1)) rarr f primeprime = ρ T ( f prime) (263)

and its inverse is given by for g isin HomC(A A)

g rarr gprime = R(g) δ rarr gprimeprime = (idAtimes1 otimes m R(1)) (gprime otimes idR(1)) (264)

where gprimeprime is indeed a R(1)-bimodule map due to the commutativity of R(1) It is easy tocheck that gprimeprime = R(g) in (264) Therefore R gives an isomorphism from HomC(A A)to HomR(1)|R(1)(R(A) R(A)) Moreover one verifies that R(g) is an R(A)-bimodulemap iff g is an A-bimodule map In other words R g rarr R(g) gives an isomorphism

HomA|A(A A)sim=minusrarr HomR(A)|R(A)(R(A) R(A))

Corollary 227 Let A be a C-algebra

(i) A is absolutely simple iff R(A) is absolutely simple(ii) Let A be in addition Frobenius Then A is simple and special iff R(A) is simple and

special

Proof Part (i) immediately follows from Lemma 226 The statement of part (ii) withoutthe qualifier lsquosimplersquo is proved in Proposition 225 But as in the proof of (iii)rArr(i) inLemma 211 a special Frobenius algebra in a semi-simple category has a semi-simplecategory of bimodules and for a semi-simple C-linear category simple and absolutelysimple are equivalent Part (ii) then follows from part (i)

The following lemma will be needed in Sect 32 below to discuss the properties ofCardy algebras

Lemma 228 Let A be a Frobenius algebra in C Then (δA)lowast = ρA

Proof Recall from (250) that δA is a morphism A rarr T R(A) Since T and R are bothFrobenius functors T R(A) is a Frobenius algebra in C Substituting the definitions aftera short calculation one finds

Cardy Algebras and Sewing Constraints I 891

εT R(A) mT R(A) = Dim(C)oplusiisinI

A Uori Ui A U or

ı Uı

(265)

Substituting this in the definition of (δA)lowast gives again after a short calculation the mor-

phism ρA At an intermediate step one uses that the part of the morphism (δA)lowast which

is made up of Ui and Uı ribbons their duals and the basis morphisms λ(iı)0 andϒ(iı)0can be replaced by 1

dim Uimiddot dUi

3 Cardy Algebras

In this section we start by investigating the properties of Frobenius algebras which sat-isfy the so-called modular invariance condition We then give two definitions of a Cardyalgebra and prove their equivalence Finally in Sect 33 we study the properties of thesealgebras and state our main results

We fix a modular tensor category C Recall that C2plusmn is an abbreviation for C+ Cminus

31 Modular invariance In C2plusmn we define the object K and the morphism ω K rarr Kas

K =oplus

i jisinIUi times U j ω =

sumi jisinI

dim Ui dim U j

Dim C idUi timesU j (31)

They have the property (see eg [BK Cor 3111])

ω

K

(Uk times Ul )or

(Uk times Ul )or

Ui times U j

Ui times U j

= δik δ jlDim C

dim Ui dim U jbUi timesU j dUi timesU j (32)

892 L Kong I Runkel

Definition 31

(i) Let A B be objects of C2plusmn A morphism f A otimes B rarr B is called S-invariant iff

A

B

W

W

f =

A

B

W

W

f

K

ω(33)

holds for all for W isin C2plusmn(ii) A C2plusmn-algebra (Aclmcl ηcl) is called modular invariant iff θAcl = idAcl and mcl

is S-invariant

Lemma 32 The morphism f A otimes B rarr B is S-invariant if and only if

A

B

Ui times U j

Ui times U j

f = Dim Cdim Ui dim U j

sumα

A

B

B

f

Ui times U j

Ui times U j

α

α

(34)

holds for all i j isin I

Proof Condition (33) holds for all W iff it holds for all W = Ui times U j i j isin I so itis enough to show that the right-hand side of (33) with W = Ui times U j is equal to theright-hand side of (34) Recall the notation for basis morphisms in (236) Starting from(33) write

idBor otimes idUi timesU j =⎛⎝sum

klα

(b(ktimeslα)

B

)or (

bB(ktimeslα)

)or⎞⎠ otimes idUi timesU j (35)

and then apply (32) The graphical representation of the resulting morphism can bedeformed to give (34) Remark 33 As shown in [K3 Sect 61] the modular invariance condition of a C2plusmn-alge-bra exactly coincides with the modular invariance condition for torus 1-point correlationfunctions of a genus-01 closed CFT In particular the condition θAcl = idAcl is equiv-alent to invariance under the modular transformation T τ rarr τ + 1 and the condition(34) with f = mcl is equivalent to invariance under S τ rarr minus 1

τ Combining the

modular invariance condition with the genus zero properties of a genus-01 closed CFTresults in a modular invariant commutative symmetric Frobenius algebra in C2plusmn

Cardy Algebras and Sewing Constraints I 893

Let Acl be a modular invariant C2plusmn-algebra Evaluating (34) for f = mcl composingit with ηcl otimes idUi timesU j and taking the trace implies the following identity

Zi j = 1

DimCAclUi timesU j where

(36)

Zi j = dimC HomC2plusmn(UitimesU j Acl)

Decomposing Acl into simple objects this gives

Zi j =sum

klisinISik Zkl Sminus1

l j where Si j = si jradic

DimC (37)

which in CFT terms is of course nothing but the invariance of the torus partition functionunder the modular S-transformation

The following theorem gives a simple criterion for modular invariance

Theorem 34 Let Acl be a haploid commutative symmetric Frobenius algebra in C2plusmn

(i) If Acl is modular invariant then dim Acl = Dim C(ii) If dim Acl = Dim C then Acl is special and modular invariant

Proof Part (i) Since Acl is haploid for i = j = 0 Equation (36) reduces to 1 =dim AclDimCPart (ii) By the same reasoning as in the proof of (iii)rArr(i) in Lemma 211 one showsthat Acl is special Thus mcl cl = ζclidAcl for some ζcl = 0

By [KO Thm 45] the category (C2plusmn)locAcl

of local Acl-modules is again a mod-

ular tensor category and Dim (C2plusmn)locAcl

= (Dim C dim Acl)2 (see [Fr Prop 321 amp

Rem 323] for the same statement in the notation used here) Thus by assumption wehave Dim(C2plusmn)loc

Acl= 1 It then follows from [ENO1 Thm 23] that up to isomorphism

(C2plusmn)locAcl

has a unique simple object (namely the tensor unit) In other words every simplelocal Acl-module is isomorphic to Acl (seen as a left-module over itself)

We have the following isomorphisms between morphism spaces [FS Prop 47 amp 411]

HomAcl(Acl otimes (Ui times U j ) Acl) sim= HomC2plusmn(Ui times U j Acl)

HomAcl(Acl Acl otimes (Ui times U j )) sim= HomC2plusmn(AclUi times U j )

Using these to transport the bases (236) from the right to the left we obtain bases bα(i j)αof HomAcl(Acl otimes (Ui times U j ) Acl) and b(i j)

β β of HomAcl(Acl Acl otimes (Ui times U j )) Thesecan be expressed graphically as

894 L Kong I Runkel

bα(i j) =

Acl

Acl

Acl

Ui times U j

α

b(i j)β = DimC

dim Ui dim U j

1

ζcl

Acl

Ui times U jAcl

Acl

β

(38)

where the nonzero factor in b(i j)β is included for convenience Notice that bα(i j) b(i j)

β isa left Acl-module map Since Acl is simple as a left module over itself we have

bα(i j) b(i j)β = λαβ idAcl (39)

for some λαβ isin C By computing tr(bα(i j) b(i j)β ) it is easy to verify that λαβ = δαβ

We will now prove the following identity

1

ζcl

Acl

Acl

Ui times U j

Ui times U j

Acl

=sumα

DimCdim Ui dim U j

1

ζcl

Acl

Ui times U j

Ui times U j

Acl

Acl

Acl

α

α

(310)

One checks that the left-hand side of this equation is an idempotent which we denote byζminus1

cl PlAcl(Ui timesU j ) cf [Fr Sect 31] By [Fr Prop 41] the image Im(ζminus1

cl PlAcl(Ui timesU j ))

is a local Acl-module and hence isomorphic to AoplusNcl for some N isin Zge0

All left-module morphisms from Acl to Aclotimes(Ui timesU j ) are linear combinations of the

b(i j)β Furthermore one verifies that ζminus1

cl PlAcl(Ui timesU j )b(i j)

β = b(i j)β Therefore the b(i j)

β

describe precisely the image of the idempotent ie ζminus1cl Pl

Acl(Ui timesU j ) = sum

α b(i j)α bα(i j)

which is nothing but (310) Composing (310) with ζcl middot εcl otimes idUi timesU j from the left (iefrom the top) produces (34)

In addition since Acl is commutative symmetric Frobenius it satisfies θAcl = idAcl

[Fr Prop 225] Altogether this shows that Acl is modular invariant Remark 35 As we were writing this paper we heard that the results in Theorem 34were obtained independently by Kitaev and Muumlger [Ki]

Remark 36 Setting i = j = 0 in (36) gives the identity dim Acl = Z00 Dim C [KRProp 23] Combining this with Theorem 34 (ii) one may wonder if a general modularinvariant commutative symmetric Frobenius algebra Acl in C2plusmn is isomorphic to a directsum of simple such algebras However this is not so For example one can take the com-mutative symmetric Frobenius algebra Acl = C[x]x2 in the category of vector spacesequipped with the non-degenerate trace ε(ax + b) = a In this case the modular invari-ance condition holds automatically but Acl is clearly not a direct sum of two algebras

Cardy Algebras and Sewing Constraints I 895

For a general modular tensor category C the algebra C[x]x2 R(1) understood as analgebra in C2plusmn via the braided monoidal isomorphism Vect f (C) C2plusmn rarr C2plusmn providesanother counter-example

32 Two definitions Define a morphism PlA A rarr A for a Frobenius algebra A in C

or C2plusmn as follows [Fr Sect 24]

PlA =

A

A

A

m

(311)

If A is also commutative and obeys m A A = ζA idA we have PlA = ζA idA In partic-

ular this holds if A is commutative and special Using the fact that the Frobenius algebraR(1) is commutative and normalised-special one can check that Pl

R(A) R(A) rarr R(A)takes the following form

PlR(A) =

oplusiisinI

A

A

Uori

Uori

A

m

times

Ui

Ui

(312)

With these ingredients we can now give the first definition of a Cardy C|C2plusmn-algebrawhich was introduced in [K3 Def 514] cf Remark 315 below

Definition 37 (CardyC|C2plusmn-algebra I) A CardyC|C2plusmn-algebra is a triple (Aop|Acl ιcl-op)where (Aclmcl ηclcl εcl) is a modular invariant commutative symmetric Frobe-nius C2plusmn-algebra (Aopmop ηopop εop) is a symmetric Frobenius C-algebra andιcl-op Acl rarr R(Aop) an algebra homomorphism such that the following conditionsare satisfied

(i) Centre condition

Acl R(Aop)

R(Aop)

R(Aop)

ιcl-op

m R(Aop)

=

Acl

ιcl-op

R(Aop)

R(Aop)

R(Aop)

m R(Aop)

(313)

896 L Kong I Runkel

(ii) Cardy condition

ιcl-op ιlowastcl-op =

R(Aop)

R(Aop)

R(Aop)

(314)

Remark 38

(i) The name ldquoCardy C|C2plusmn-algebrardquo in Definition 37 was chosen because many ofthe important ingredients were first studied by Cardy the modular invariance ofthe closed theory [C1] the consistency of the annulus amplitude [C2] and thebulk-boundary OPE [CL] On the other hand the boundary-boundary OPE andthe OPE analogue of the centre condition were first considered in [Lw]

(ii) One can easily see that in the special case that C is the category Vect f (C) of finite-dimensional C-vector spaces a Cardy C|C2plusmn-algebra gives exactly the algebraicformulation of two-dimensional open-closed topological field theory over C (cfRemark 614 in [K3]) see [Lz Sect 48] [Mo Thm 11] [AN Thm 45] [LPCor 43] [MS Sect 22] When passing to a general modular tensor category Cthere are two important differences to the two-dimensional topological field the-ory Firstly the algebras Acl and Aop now live in different categories which inparticular affects the formulation of the centre condition and the Cardy conditionSecondly the modular invariance condition has to be imposed on Acl In the caseC = Vect f (C) modular invariance holds automatically

Definition 39 A homomorphism of Cardy C|C2plusmn-algebras (A(1)op |A(1)cl ι(1)cl-op) rarr (A(2)op

|A(2)cl ι(2)cl-op) is a pair ( fop fcl) of Frobenius algebra homomorphisms fop A(1)op rarr A(2)op

and fcl A(1)cl rarr A(2)cl such that the diagram

A(1)cl

fcl

ι(1)cl-op

A(2)cl

ι(2)cl-op

R(A(1)op )

R( fop) R(A(2)op )

(315)

commutes

Remark 310 Since a homomorphism of Frobenius algebras is invertible (cf Lemma218 (iv)) a homomorphism of Cardy algebras is always an isomorphism

For a homomorphism ( fop fcl) of Cardy C|C2plusmn-algebras using the commutativity of(315) and the fact that fcl and fop are both algebra and coalgebra homomorphisms itis easy to show that (315) commutes iff

R(A(1)op )R( fop)

(ι(1)cl-op)

lowast

R(A(2)op )

(ι(2)cl-op)

lowast

A(1)cl

fcl A(2)cl

(316)

Cardy Algebras and Sewing Constraints I 897

commutesLet (Aop|Acl ιcl-op) be a Cardy C|C2plusmn-algebra Define the morphism

ιcl-op = χminus1(ιcl-op) T (Acl) minusrarr Aop (317)

Decompose Acl as Acl = oplusNn=1Cl

n times Crn such that Cl

1 times Cr1 = ηcl(1 times 1) We use ι(n)cl-op

to denote the restriction of ιcl-op to Cln times Cr

n and ι(n)cl-op to denote the restriction of ιcl-op

to Cln otimes Cr

n We introduce the following graphical notation

ι(n)cl-op =

Cln Cr

n

Aop

(318)

By (243) ιcl-op can be expressed in terms of ιcl-op as follows

ιcl-op =Noplus

n=1

oplusiisinI

sumα

Cln

Crn

Aop

α

Uori

times α

Crn

Ui

(319)

Lemma 311 The centre condition (313) is equivalent to the following condition in C

Cln Cr

n

Aop

Aop

Aop

=

Cln Cr

n

Aop

Aop

Aop

for n = 1 N (320)

Proof First insert (319) and the definition (217) of m R(Aop) into (313) Then apply thecommutativity of R(1) to the left hand side of (313) The equivalence between (313)and (320) follows immediately Remark 312 The centre condition (320) is very natural from the open-closed confor-mal field theory point of view Correlators on the upper half plane are expressed in termsof conformal blocks on the full complex plane The objects Cl

n and Crn are associated to

the field insertion at a point z in the upper half plane and at the complex conjugate pointz in the lower half plane respectively The object Aop corresponds to a field inserted ata point r on the real axis The centre condition (320) simply says that the correlationfunctions in the disjoint domains |z| gt r gt 0 and r gt |z| gt 0 are analytic continuationsof each other see [K2 Prop 118]

898 L Kong I Runkel

Recall that we define ιlowastcl-op Aop rarr T (Acl) as in (230) We introduce the graphicalnotation

ιlowastcl-op =Noplus

n=1

Cln Cr

n

Aop

=Noplus

n=1

Cln Cr

n

Aop

Aop T (minus1Acl) (321)

where the second equality follows from (215) (221) and (259)

Lemma 313 The Cardy condition (314) is equivalent to the following identity in C

Noplusn=1

DimCdim Ui

sumα

Cln

Crn

Crn

α

α

Aop

Aop Uori

Uori

=Aop

Aop

Aop

Uori

Uori

op

mop for all i isin I (322)

Proof By (244) (312) and (319) it is easy to see that (322) is equivalent to thefollowing identity

ιcl-op χ(ιlowastcl-op) = PlR(Aop)

(323)

Therefore it is enough to show that

χ(ιlowastcl-op) = ιlowastcl-op (324)

We have

χminus1(ιlowastcl-op)(1)= T (ιlowastcl-op) δ (2)= T (ιcl-op)

lowast δlowastlowast (3)= (ρ T (ιcl-op)

)lowast (4)= (ιcl-op

)lowast (325)

In step (1) we use the expression (253) for χminus1 step (2) follows from Lemma 218 (v)and Lemma 219 Step (3) is Lemma 218 (i) and Lemma 228 and finally step (4)amounts to substituting (253) and (317) Acting with χ on both sides of the aboveequality produces (324)

Combining Lemmas 311 and 313 and Proposition 216 we obtain the followingequivalent definition of Cardy C|C2plusmn-algebra (recall the graphical notation (318) forιcl-op and (321) for ιlowastcl-op)

Cardy Algebras and Sewing Constraints I 899

Definition 314 (CardyC|C2plusmn-algebra II) A CardyC-algebra is a triple (Aop|Acl ιcl-op)where Acl is a commutative symmetric Frobenius C2plusmn-algebra satisfying property (34)with f = mcl Aop is a symmetric Frobenius C-algebra and ιcl-op T (Acl) rarr Aop isan algebra homomorphism satisfying the conditions (320) and (322)

Remark 315 Up to a choice of normalisation Definition 314 is the same as the orig-inal one in [K3 Def 613] The difference between the two definitions is the factorDim C dim Ui on the left hand side of (322) which in [K3 Def 613] is given byradic

Dim C dim Ui The two definitions are related by rescaling the coproduct cl andcounit εcl of Acl by 1

radicDim C and

radicDim C respectively We chose the convention in

(322) to remove all dimension factors from the expression (314) for the Cardy condition

33 Uniqueness and existence theorems In this subsection we investigate the structureof Cardy algebras We start with the following proposition which when combined withthe results of part II provides an alternative proof of [Fj Prop 422]

Proposition 316 Let (Aop|Acl ιcl-op) be a Cardy C|C2plusmn-algebra If Acl is simple anddim Aop = 0 then Aop is simple and special

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis therefore haploid Restricting the Cardy condition (322) to the case Ui = 1 andcomposing both sides with εop from the left we see that εop kills all terms associated toU j times 1 isin Acl in the sum except for a single 1 times 1 term Thus we obtain the followingidentity

β εop = dAop (mop otimes idAorop) (idAop otimes bAop) (326)

where β isin C Composing with ηop from the right in turn implies that βεop ηop =dim Aop which is nonzero by assumption Thus also β = 0 and εop is a nonzero multipleof the morphism on the right hand side of (326) By [FRS Lem 311] Aop is special

Since Aop is a special Frobenius algebra Aop is semi-simple as an Aop-bimodule(apply [FS Prop 524] to Aop tensored with its opposite algebra) Suppose Aop is not

simple so that we can write Aop = A(1)op oplus A(2)op for nonzero Aop-bimodules A(1)op and A(2)op We denote the canonical embeddings and projections associated to this decompositionas ι12 and π12 We have the identities

mop (ι1 otimes ι2) = 0 εop ηop =2sum

i=1

εop ιi πi ηop (327)

The first identity follows since π1 mop (ι1 otimes ι2) = 0 (as mop gives the left action of

Aop on Aop and hence it preserves A(2)op ) and similarly π2 mop (ι1 otimes ι2) = 0 Thesecond identity is just the completeness of ι12 π12

Since εop ηop = 0 without losing generality we can assume εop ι1 π1 ηop = 0Using that π2 is a bimodule map we compute

π2 [LHS of (322)]Ui =1 ι1 π1 ηop = π2 [RHS of (322)]Ui =1 ι1 π1 ηop

= PlA(2)op

π2 ι1 π1 ηop = 0 (328)

900 L Kong I Runkel

On the other hand using that Acl is haploid that ιcl-op is an algebra map and thatι lowastcl-op is a coalgebra map one can check that the left-hand side of (328) is equal toλ(εop ι1 π1 ηop) π2 ηop for some λ = 0 This implies that π2 ηop = 0 Thusηop = sum

i ιi πi ηop = ι1 π1 ηop Hence we have

0 = π2 ι2 = π2 mop (ηop otimes ι2) = π2 mop ((ι1 π1 ηop)otimes ι2) (329)

However the right-hand side is zero by (327) This is a contradiction and hence Aopmust be simple

To formulate the next theorem we need the notion of the full centre of an algebra [FjDef 49] Recall that an algebra A in a braided tensor category has a left centre and aright centre [VZO] both of which are sub-algebras of A Of these two we will onlyneed the left centre The following definition is [Fr Def 231] which in our setting isequivalent to that of [VZO]

Definition 317 Let A be a symmetric special Frobenius algebra such that m A A =ζA idA

(i) The left centre Cl(A) of A is the image of the idempotent ζminus1A Pl

A(ii) The full centre Z(A) is Cl(R(A))

That ζminus1A Pl

A is an idempotent follows from [FRS Lem 52] when keeping track ofthe factors ζA ([FRS] assumes normalised-special ie ζA = 1) Note that Cl(A) is againan object of C while Z(A) is an object of C2plusmn Let el

A Cl(A) rarr A be the embeddingof Cl(A) into A The left centre is in fact the maximal subobject of A such that

m A cAA (elA otimes idA) = m A (330)

see [Fr Lem 232] This observation explains the name left centre and also makes theconnection to [O Def 15]

The full centre is by definition the image of the idempotent ζminus1A Pl

R(A) R(A) rarrR(A) Since C2plusmn is abelian the idempotent splits and we obtain the embedding andrestriction morphisms

e Z(A) rarr R(A) and r R(A) Z(A) (331)

which obey r e = idZ(A) and e r = ζminus1A Pl

R(A) It follows from Proposition 225 and

[Fr Prop 237] that Z(A) is a commutative symmetric Frobenius algebra in C2plusmn withstructure morphisms5

m Z(A) = r m R(A) (e otimes e) ηZ(A) = r ηR(A)

Z(A) = ζA middot (r otimes r) R(A) e εZ(A) = ζminus1A middot εR(A) e

(332)

Moreover if A is simple then Z(A) is simple and if A is simple and dim A = 0 thenZ(A) is simple and special The normalisation of the counit is such that

εZ(A) ηZ(A) = ζminus2A dim A DimC (333)

5 The normalisation of product and unit is the standard one The factors in the coproduct and counit haveto be included in order for (A|Z(A) e) to be a Cardy algebra see Theorem 318 below The normalisation ofthe counit enters the Cardy condition (314) through the definition of ( middot )lowast

Cardy Algebras and Sewing Constraints I 901

Theorem 318 Let A be a special symmetric Frobenius C-algebra Then (A|Z(A) e)is a Cardy C|C2plusmn-algebra

The proof of this theorem makes use the following two lemmas

Lemma 319 e Z(A) rarr R(A) is an algebra map and elowast = ζA middot r

Proof It follows from [Fr Lem 229] (or by direct calculation using in particularm R(A) R(A) = ζAidR(A)) that

m R(A) (e otimes e) = ζminus1A middot Pl

R(A) m R(A) (e otimes e) (334)

Substituting e r = ζminus1A Pl

R(A) shows that e is compatible with multiplication For theunit one finds

e ηZ(A) = e r ηR(A) = ζminus1A Pl

R(A) ηR(A) = ηR(A) (335)

Thus e is an algebra map For the second statement one computes

elowast (1)= ζA

R(A)

r

R(A)

R(A)

Z(A)

R(A)

Z(A)

r

e

e

(2)=

R(A)

r

R(A)

R(A)

R(A)

Z(A)

(3)=

R(A)

r

R(A)

Z(A)

(4)= ζA middot r (336)

where in (1) the definitions (230) and (332) have been substituted step (2) is e r = ζminus1

A PlR(A) step (3) uses that R(A) is symmetric Frobenius and step (4) is again

e r = ζminus1A Pl

R(A) Lemma 320 Let A be a symmetric Frobenius algebra in C The morphism

PlR(A) m R(A)

(Pl

R(A) otimes PlR(A)

) R(A)otimes R(A) minusrarr R(A) (337)

is S-invariant

The proof of this lemma is a slightly lengthy explicit calculation and has been deferredto Appendix A2

902 L Kong I Runkel

Proof of Theorem 318 That e is an algebra map was proved in Lemma 319 The centrecondition (313) holds by property (330) of the left centre The Cardy condition (314)also is an immediate consequence of Lemma 319

ιcl-op ιlowastcl-op = e (ζA r) = PlR(A) (338)

The full centre Z(A) is a commutative symmetric Frobenius algebra It remains to provemodular invariance That θZ(A) = idZ(A) is implied by commutativity and symmetryof Z(A) [Fr Prop 225] The S-invariance condition (33) follows from Lemma 320In (337) substitute Pl

R(A) = ζA e r and then put the resulting morphism into (33)Compose the resulting equation with e otimes idW from the right (ie from the bottom) andsubstitute the definition (332) of m Z(A) This results in the statement that m Z(A) isS-invariant

The following theorem is analogous to [LR Prop 29] and [Fj Thm 426] whichroughly speaking answer the question under which circumstances the restriction of atwo-dimensional conformal field theory to the boundary already determines the entireconformal field theory The first work is set in Minkowski space and uses operator alge-bras and subfactors while the second work is set in Euclidean space and uses modulartensor categories

Theorem 321 Let (A|Acl ιcl-op) be a Cardy C|C2plusmn-algebra such that dim A = 0 andAcl is simple Then A is special and (A|Acl ιcl-op) sim= (A|Z(A) e) as Cardy algebras

Proof By Proposition 316 A is simple and special Since Acl is simple the algebramap ιcl-op Acl rarr R(A) is either zero or a monomorphism But ιcl-op ηcl = ηR(A)and so ιcl-op = 0 Thus ιcl-op is monic By Lemma 218 (ii) ζminus1

A ιlowastcl-op is epi The Cardycondition (314) implies

ιcl-op ζminus1A ιlowastcl-op = ζminus1

A PlR(A) = e r (339)

Composing this with e r from the left yields e r ιcl-op ζminus1A ιlowastcl-op = e r =

ιcl-op ζminus1A ιlowastcl-op Since ζminus1

A ιlowastcl-op is epi we have

e r ιcl-op = ιcl-op (340)

Actually (340) also follows from (313) and specialness of R(A) We will prove that

( fop fcl) (A|Acl ιcl-op) minusrarr (A|Z(A) e) where fop = idA fcl = r ιcl-op

(341)

is an isomorphism of Cardy algebrasfcl is an algebra map Compatibility with the units follows since ιcl-op is an algebramap

fcl ηcl = r ιcl-op ηcl = r ηR(A) = ηZ(A) (342)

Compatibility with the multiplication also follows since ιcl-op is an algebra map

m Z(A) ( fcl otimes fcl) = r m R(A) (e otimes e) (r otimes r) (ιcl-op otimes ιcl-op)

= r m R(A) (ιcl-op otimes ιcl-op) = r ιcl-op mcl = fcl mcl (343)

where in the second step we used (340)

Cardy Algebras and Sewing Constraints I 903

fcl is an isomorphism As above since fcl is an algebra map and since Acl is simple fcl

has to be monic By Lemma 319 rlowast = ζminus1A e Thus f lowast

cl = ιlowastcl-op rlowast = ζminus1A ιlowastcl-op e and

fcl f lowastcl = r ιcl-op ζminus1

A ιlowastcl-op e = r e r e = idZ(A) (344)

and so fcl is also epi and hence isofcl is a coalgebra map Since fcl is an algebra map so is f minus1

cl By (344) f minus1cl = f lowast

cland by Lemma 218 (iii) this implies that fcl is a also coalgebra mapThe diagram (315) commutes Commutativity of (315) is equivalent to e fcl = ιcl-opwhich holds by (340)

Let A be a special symmetric Frobenius algebra So far we have seen that (A|Z(A) e)is a Cardy algebra and that all Cardy algebras with Aop = A and simple Acl are of thisform It is now natural to ask if every simple Acl does occur as part of a Cardy algebraThe following theorem provides an affirmative answer Recall that for an A-left moduleM the object Mor otimesA M is an algebra (see eg [KR Lem 42])

Theorem 322 If Acl is a simple modular invariant commutative symmetric FrobeniusC2plusmn-algebra then there exist a simple special symmetric Frobenius C-algebra A and amorphism ιcl-op Acl rarr R(A) such that

(i) Acl sim= Z(A) as Frobenius algebras(ii) (A|Acl ιcl-op) is a Cardy C|C2plusmn-algebra

(iii) T (Acl) sim= oplusκisinJ Morκ otimesA Mκ as algebras where MκκisinJ is a set of representa-

tives of the isomorphism classes of simple A-left modules

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis haploid It then follows from Theorem 34 that Acl is special By Proposition 224T (Acl) is a special symmetric Frobenius algebra in C Thus T (Acl) = oplusi Ai where theAi are simple symmetric Frobenius algebras We will show that at least one of the Aiis special Since T (Acl) is special we have mT (Acl) T (Acl) = ζ idT (Acl) for someζ isin C

times Restricting this to the summand Ai shows mi i = ζ idAi FurthermoreεT (Acl) ηT (Acl) = ξ id1 for some ξ isin C

times But εT (Acl) ηT (Acl) = sumi εi ηi and so

at least one of the εi ηi has to be nonzero Therefore at least one of the Ai is speciallet A equiv Ai be this summand We denote the embedding A rarr T (Acl) by e0 and therestriction T (Acl) A by r0 Notice that r0 is an algebra homomorphism Define

ιcl-op = χ (r0) Acl minusrarr R(A) (345)

By Proposition 216 ιcl-op is an algebra homomorphism Next we verify the centre con-dition (313) or rather its equivalent form (320) By substituting the definitions onecan convince oneself that the commutativity mcl cAclAcl = mcl of Acl in C2plusmn implies thecondition mT (Acl) = mT (Acl) in C see [K2 Prop 36] Here T (Acl)otimes T (Acl) rarrT (Acl)otimes T (Acl) is given by

=oplusmn

(idCl

notimes cCl

m Crnotimes idCr

m

)

(cCl

m Clnotimes cminus1

Crn C

rm

)

(idCl

motimes cminus1

Cln C

rm

otimes idCrn

)

(346)

and we decomposed Acl as Acl = oplusnCln times Cr

n As a consequence we obtain the identity

r0 mT (Acl) (idAcl otimes e0) = r0 mT (Acl) (idAcl otimes e0) (347)

904 L Kong I Runkel

Using that r0 is an algebra map and that by definition ιcl-op = r0 we obtain (320)In order to show that (A|Acl ιcl-op) is a Cardy algebra it remains to show that the

Cardy condition (314) is satisfied We will demonstrate this via a detour by first provingthat Acl sim= Z(A) as Frobenius algebras

Recall the notations e and r given in (331) Using the centre condition (320) onecan check that Pl

R(A) ιcl-op = m R(A) R(A) ιcl-op By specialness of A we have

m A A = ζA idA and so together with e r = ζminus1A Pl

R(A) we get

e r ιcl-op = ιcl-op (348)

Next consider the morphism

fcl = r ιcl-op Acl minusrarr Z(A) (349)

By the same derivation as in (342) and (343) one sees that fcl is an algebra map Inparticular fcl ηcl = ηZ(A) = 0 and so fcl = 0 Since Acl is simple fcl has to be amonomorphism

By the same argument as used in the proof of Theorem 34 (ii) up to isomorphismAcl is the unique simple local Acl-(left-)module The algebra monomorphism fcl turnsZ(A) into an Acl-module Since Z(A) is commutative it is local as an Acl-module andso Z(A) sim= AoplusN

cl for some N ge 1 By construction A is a simple special symmetricFrobenius algebra Proposition 225 and Corollary 227 show that R(A) inherits all theseproperties and thus Z(A) is simple (see the comment below Eq (332)) By Theorem318 Z(A) is modular invariant and then by Theorem 34 (i) dim Z(A) = Dim C Thisimplies that N = 1 in Z(A) sim= AoplusN

cl and so fcl is in fact an isomorphismSince Acl and Z(A) are both haploid we have εZ(A) fcl = ξ εcl for some ξ isin C

timesThe counit uniquely determines the Frobenius structure on Acl and Z(A) (see eg [FRSLemma 37]) so that fcl is a coalgebra isomorphism iff ξ = 1 To compute ξ we composethe above identity with ηcl from the right Defining ζcl via εcl ηcl = ζminus1

cl Dim C middot id1 andusing (333) gives ξ = dim A ζclζ

2A By rescaling the comultiplication and the counit

of A and consequently changing ζA we can always achieve ξ = 1 This proves part (i)of the theorem

Equation (348) implies that ιcl-op = e fcl Since fcl is an isomorphism of Frobeniusalgebras by Lemmas 218 and 319 we have

ιcl-op ιlowastcl-op = e fcl f lowastcl elowast = ζA e r = Pl

R(A) (350)

Thus (A|Acl ιcl-op) is a Cardy algebra This proves part (ii) of the theoremPart (iii) can be seen as follows By [KR Prop 43] T Z(A) sim= oplusκisinJ Mor

κ otimesA Mκ asalgebras Together with the observation that T ( fcl) T (Acl) rarr T Z(A) is an isomor-phism of algebras this proves part (iii) Remark 323 Part (i) of Theorem 322 was announced by Muumlger [Muuml2] We provide anindependent proof in the setting of Cardy algebras

The above theorem together with Lemma 211 and Theorem 34 shows that a sim-ple commutative symmetric Frobenius C2plusmn-algebra Acl with dim Acl = Dim C is alwayspart of a Cardy algebra (Aop|Acl ιcl-op) for some simple special symmetric Frobeniusalgebra Aop in C However the above proof also illustrates that Aop is not unique Thisraises the question how two Cardy algebras with a given Acl can differ This question isanswered by [KR Thm 11] which in the present framework can be restated as follows

Cardy Algebras and Sewing Constraints I 905

Theorem 324 If (A(i)op |A(i)cl ι(i)cl-op) i = 1 2 are two Cardy C|C2plusmn-algebras such that

A(i)cl is simple and dim A(i)op = 0 for i = 1 2 then A(1)clsim= A(2)cl as algebras if and only

if A(1)op and A(2)op are Morita equivalent

Proof Theorem 11 in [KR] is stated for A(i)op being non-degenerate algebras and A(i)cl =Z(A(i)op ) for i = 1 2 By Proposition 316 A(i)op are simple and special for i = 1 2

Then by [KR Lem 21] A(i)op are non-degenerate algebras By Theorem 321 we have

A(i)clsim= Z(A(i)op ) as Frobenius algebras Finally by [KR Thm 11] Z(A(1)op ) sim= Z(A(2)op )

as algebras iff A(1)op and A(2)op are Morita equivalent Let Cmax(C2plusmn) be the set of equivalence classes [B] of simple modular invariant

commutative symmetric Frobenius algebras B in C2plusmn Two such algebras B and B prime areequivalent if B and B prime are isomorphic as algebras (but not necessarily as Frobenius alge-bras) Let Msimp(C) be the set of Morita classes of simple special symmetric Frobeniusalgebras in C Define the map z Msimp(C) rarr Cmax(C2plusmn) by z A rarr [Z(A)] whereA denotes the Morita class of A From Theorem 322 (i) and [KR Thm 11] we learn

Corollary 325 The map z Msimp(C) rarr Cmax(C2plusmn) is a bijection

A Appendix

A1 Proof of Lemma 27 We will show that if (F ψ F2 ψ

F0 ) is a colax tensor functor

from C1 to C2 then (G φG2 φ

G0 ) is a lax tensor functor from C2 to C1 Applying this

result to the opposed categories then gives the converse statementWe need to show that φG

0 and φG2 make the diagrams (21) and (22) commute We

first prove the commutativity of (21) Consider the following diagram

G(A)otimes (G(B)otimes G(C))

GψF2 δ

idG(A)otimesφG2 G(A)otimes G(B otimes C)

GψF2 δ

G(FG(A)otimes F(G(B)otimes G(C)))

G(F(idG(A))otimesF(φG2 ))

G(idFG(A)otimesψ F2 )

G(FG(A)otimes FG(B otimes C))

G(ρAotimesρBotimesC )

G(FG(A)otimes (FG(B)otimes FG(C)))

G(ρAotimes(ρBotimesρC )) G(A otimes (B otimes C))

(A1)

The top subdiagram is commutative because of the naturality of Gψ F2 δ The commu-

tativity of the bottom subdiagram follows from the following identities

(ρB otimes ρC ) ψ F2 = (ρB otimes ρC ) ψ F

2 ρF Fδ

= ρBotimesC FG(ρB otimes ρC ) FG(ψ F2 ) Fδ

= ρBotimesC F(φG2 ) (A2)

as a map F(G(B) otimes G(C)) rarr B otimes C The commutativity of (A1) implies that thecomposition of maps in the left column in (21) can be replaced by

G(ρA otimes (ρB otimes ρC )) G((idFG(A) otimes ψ F

2 ) ψ F2

) δ (A3)

906 L Kong I Runkel

Similarly we can show that the composition of maps in the right column in (21) can bereplaced by

G((ρA otimes ρB)otimes ρC ) G((ψ F

2 otimes idFG(C)) ψ F2

) δ (A4)

Using the commutativity of (23) it is easy to see that (21) with the left and right col-umns of (21) replaced by (A3) and (A4) respectively is commutative Hence (21) iscommutative

Now we prove the commutativity of the first diagram in (22)

φG2 (φG

0 otimes idG(A))

(1)= G(ρ12 otimes ρA) Gψ F2 δ

[(Gψ F

0 δ11)otimes idG(A)

]

(2)= G(ρ12 otimes ρA) Gψ F2 G F(Gψ0 otimes idG(A)) G F(δ11 otimes idG(A)) δ

(3)= G(ρ12 otimes ρA) G(FG(ψ F0 )otimes idFG(A)) Gψ F

2 G F(δ11 otimes idG(A)) δ(4)= G(id12 otimes ρA) G

([ψ F

0 ρF(11) (Fδ)11] otimes idFG(A)

) Gψ F

2 δ(5)= G(id12 otimes ρA) G(ψ F

0 otimes idFG(A)) Gψ F2 δ

(6)= G(id12 otimes ρA) G(lminus1FG(A)) G F(lG(A)) δ

(7)= G(lminus1A ) GρA δG lG(A)

(8)= G(lminus1A ) lG(A) (A5)

where in step (1) we substituted the definition of φG0 φ

G2 given in (210) in step (2)

we used the naturality of δ in step (3) we used the naturality of Gψ F2 in step (4) we

switched the position between Gρ12 and G FG(ψ F0 ) and the position between Gψ2 and

G F(δ12 otimes idG(A)) using the naturality of ρ and FψG2 respectively in step (5) we applied

the second identity in (28) in step (6) we used (24) in step (7) we used the naturalityof lminus1 and δ in step (8) we used the first identity in (28)

The proof of the commutativity of the second diagram in (22) is similar Thus wehave shown that G is a lax tensor functor

A2 Proof of Lemma 320 To prepare the proof recall that for a given object B isin Cthe modular group P SL(2Z) acts on the space oplusi HomC(B otimes Ui Ui ) see eg [BKSect 31] and [K3 Eq (455)] We will only need the action of S and Sminus1 Let f isinoplusi HomC(B otimes Ui Ui ) Then

S oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A6)

Cardy Algebras and Sewing Constraints I 907

Sminus1 oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A7)

By Lemma 32 to establish that (337) is S-invariant it is enough to prove the identity(34) when f is given by (337) Using (A6) and (A7) we can see that Eq (34) simplysays that oplusi j [RHS of (34)] is invariant under the action of S times S Consider the elementg of oplus jkisinIHomC2plusmn(R(A)otimes (Uor

j times Uk)Uorj times Uk) given by

g =oplusjkisinI

sumα

R(A)

R(A)

R(A)

R(A)

Uorj times Uk

Uorj times Uk

α

α

(A8)

By the above arguments proving S-invariance of (337) is equivalent to proving invari-ance of g under the action of S times S

For i isin I we denote by gi the component of g in(oplus jisinIHomC(A otimes Uor

i otimes Uorj U

orj )

)otimes (opluskisinIHomC(Ui otimes UkUk))

We view the second Hom-space in the above tensor product as a Hom-space in C+ insteadof Cminus It is enough to show that gi is invariant under the action of S times Sminus1 Note thatthe action of Sminus1 in C+ is equivalent to that of S in Cminus

The morphism gi can be canonically identified with a bilinear pairing

( middot middot )i ⎛⎝oplus

jisinIHomC(Uor

j A otimes Uori otimes Uor

j )

⎞⎠

times(oplus

kisinIHomC(UkUi otimes Uk)

)minusrarr C (A9)

as follows For h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) we set

(h1 h2)i = (dim U j dim Uk)minus1 trUor

j timesUk[gi (h1 times h2)] (A10)

908 L Kong I Runkel

When substituting the explicit form of the product m R(A) of R(A) = (A times 1) otimes R(1)after a short calculation one finds

(h1 h2)i =sumα

1

dim U j

A

A

A

A

A A

A α

α

h1

h2

U jUk

Uorj

Ui

Uk (A11)

Here the top morphism PlR(A) has been simplified with the help of the identity

PR(A) m R(A) (PR(A) otimes PR(A))

=(oplus

iisinI((m A A)otimes idUor

i)times idUi

) m R(A) (PR(A) otimes PR(A)) (A12)

which can be checked by direct calculation along the same lines as in the proof of [FrLem 310]

The action of the modular transformation S on oplusiisinI(B otimes Ui Ui ) for B isin C nat-urally induces an action on oplusiisinI(Ui B otimes Ui ) [K3 Prop 514] which we denote bySlowast In the present case we get an action of Slowast on oplus jisinIHomC(Uor

j A otimes Uori otimes Uor

j ) and

opluskisinIHomC(UkUi otimes Uk) Then to show gi is invariant under the action of S times Sminus1

amounts to showing that

(h1 h2)i = ((Sminus1)lowast h1 Slowast h2)i (A13)

Cardy Algebras and Sewing Constraints I 909

for all h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) We have

((Sminus1)lowast h1 Slowast h2)i =sum

mnα

dim Un

DimC

A

A

A

A

A A

A

Uk

α

α

h1

h2

Ui

U j

UmUn

(A14)

Now drag the upper vertex indexed by α in the above graph along its Uorm -leg until it

meets the lower vertex also indexed by α then sum over α and m This gives

((Sminus1)lowast h1 Slowast h2)i =sum

n

dim Un

DimC

A

A

A

A

A A

A

Uk

Un

h1

h2

Ui

U j

(A15)

910 L Kong I Runkel

If we just look at the neighbourhood of the Un-loop in the above graph we see thefollowing subgraphs

sumn

dim Un

DimC

A Aor

Uorj U j

Uk Uork

Un=

sumα

1

dim U j

A Aor Uk Uork

Uorj U j

αα

(A16)

where we have applied [BK Cor 3111] Substituting this subgraph back to the originalgraph in (A15) we obtain

((Sminus1)lowast h1 Slowast h2)i =sumα

1

dim U j

A

A

A

A

A A

A

A

A

Uorj

α

α

h1

h2m A

Ui

U jUk

Uk

(A17)

The graph in (A17) is equal to that in (A11) In order to see this we first drag theldquobubblerdquo (m A A) along A lines and through the m A vertex (because m A A is abimodule map) until it reaches the lower-left leg of the upper vertex indexed by α Thendrag the m A vertex along the (red) dotted line in the above graph Finally we applythe associativity of A (A12) and [Fr Lem 311] Then we see that the graph in (A17)exactly matches with the one in (A11)

Acknowledgements We would like to thank the organisers of the Oberwolfach Arbeitsgemeinschaft ldquoAlge-braic structures in conformal field theoriesrdquo (April 2007) where this work was started for an inspiring meetingWe would further like to thank the Hausdorff Institute for Mathematics in Bonn and the organisers of the stim-ulating meeting ldquoGeometry and Physicsrdquo (May 2008) We are indebted to Alexei Davydov Jens FjelstadJuumlrgen Fuchs Yi-Zhi Huang Alexei Kitaev Urs Schreiber Christoph Schweigert Stephan Stolz and PeterTeichner for helpful discussions andor comments on a draft of this paper The research of IR was partiallysupported by the EPSRC First Grant EPE0050471 the PPARC rolling grant PPC5071451 and the MarieCurie network lsquoSuperstring Theoryrsquo (MRTN-CT-2004-512194)

Cardy Algebras and Sewing Constraints I 911

References

[AN] Alexeevski A Natanzon SM Noncommutative two-dimensional topological field theories andhurwitz numbers for real algebraic curves Sel Math New Ser 12 307ndash377 (2006)

[Bi] Bichon J Cosovereign hopf algebras J Pure Appl Alg 157 121ndash133 (2001)[BK] Bakalov B Kirillov AA Lectures on Tensor Categories and Modular Functors Providence

RI Amer Math Soc 2001[C1] Cardy JL Operator content of two-dimensional conformal invariant theories Nucl Phys

B 270 186ndash204 (1986)[C2] Cardy JL Boundary conditions fusion rules and the verlinde formula Nucl Phys B 324

581ndash596 (1989)[CL] Cardy JL Lewellen DC Bulk and boundary operators in conformal field theory Phys Lett

B 259 274ndash278 (1991)[DMZ] Dong C-Y Mason G Zhu Y-C Discrete series of the Virasoro algebra and the moonshine

module In Algebraic Groups and Their Generalizations Quantum and infinite-dimensional Meth-ods Proc Symp Pure Math 56 Part 2 Providence RI Amer MathSoc 1994 pp 295ndash316

[DP] Day B Pastro C Note on frobenius monoidal functors New York J Math 14 733ndash742 (2008)[ENO1] Etingof PI Nikshych D Ostrik V On fusion categories Ann Math 162 581ndash642 (2005)[ENO2] Etingof PI Nikshych D Ostrik V An analogue of radfordrsquos s4 formula for finite tensor

categories Int Math Research Notices 54 2915ndash2933 (2004)[Fe] Felder G Froumlhlich J Fuchs J Schweigert C Correlation functions and boundary conditions in

rational conformal field theory and three-dimensional topology Comp Math 131 189ndash237 (2002)[FHL] Frenkel IB Huang Y-Z Lepowsky J On axiomatic approaches to vertex operator algebras

and modules Mem Amer Math Soc 104 (1993)[Fj] Fjelstad J Fuchs J Runkel I Schweigert C Uniqueness of openclosed rational cft with

given algebra of open states Adv Theor Math Phys 12 1283ndash1375 (2008)[FK] Froumlhlich J King C The chern-simons theory and knot polynomials Commun Math

Phys 126 167ndash199 (1989)[Fr] Froumlhlich J Fuchs J Runkel I Schweigert C Correspondences of ribbon categories Adv

Math 199 192ndash329 (2006)[FRS] Fuchs J Runkel I Schweigert C Tft construction of rcft correlators I partition functions Nucl

Phys B 646 353ndash497 (2002)[FS] Fuchs J Schweigert C Category theory for conformal boundary conditions Fields Inst Com-

mun 39 25ndash71 (2003)[H1] Huang Y-Z Two-dimensional conformal geometry and vertex operator algebras Progress in

Mathematics Vol 148 Boston Birkhaumluser 1997[H2] Huang Y-Z Rigidity and modularity of vertex tensor categories Comm Contemp Math 10

871ndash911 (2008)[HK1] Huang Y-Z Kong L Open-string vertex algebra category and operads Commun Math

Phys 250 433ndash471 (2004)[HK2] Huang Y-Z Kong L Full field algebras Commun Math Phys 272 345ndash396 (2007)[HK3] Huang Y-Z Kong L Modular invariance for conformal full field algebras httparxivorgabs

math0609570v2[mathQA] 2006[HL] Huang Y-Z Lepowsky J Tensor products of modules for a vertex operator algebra and vertex

tensor categories In Lie Theory and Geometry in honor of Bertram Kostant ed R BrylinskiJ-L Brylinski V Guillemin V Kac Boston Birkhaumluser 1994 pp 349ndash383

[JS] Joyal A Street R Braided tensor categories Adv Math 102 20ndash78 (1993)[K1] Kong L Full field algebras operads and tensor categories Adv Math 213 271ndash340 (2007)[K2] Kong L Open-closed field algebras Commun Math Phys 280 207ndash261 (2008)[K3] Kong L Cardy condition for open-closed field algebras Commun Math Phys 283 25ndash92 (2008)[Ki] Kitaev A Private communication[KO] Kirillov AA Ostrik V On q-analog of mckay correspondence and ade classification of sl(2)

conformal field theories Adv Math 171 183ndash227 (2002)[KR] Kong L Runkel I Morita classes of algebras in modular tensor categories Adv Math 219

1548ndash1576 (2008)[Ld] Lauda AD Frobenius algebras and ambidextrous adjunctions Theo Appl Cat 16 84ndash

122 (2006)[Li1] Li H-S Regular representations of vertex operator algebras Commun Contemp Math 4

639ndash683 (2002)[Li2] Li H-S Regular representations and huang-lepowsky tensor functors for vertex operator

algebras J Alge 255 423ndash462 (2002)

Cardy Algebras and Sewing Constraints I 879

Definition 29 A Frobenius algebra A = (Am η ε) is an algebra and a coalge-bra such that the coproduct is an intertwiner of A-bimodules

(idA otimes m) (otimes idA) = otimes m = (m otimes idA) (idA otimes)

We will use the following graphical representation for the morphisms of a Frobeniusalgebra

m =A A

A

η =A

=A A

A

ε =A

(212)

A Frobenius algebra A in a k-linear tensor category for k a field is called special iffm = ζ idA and ε η = ξ id1 for nonzero constants ζ ξ isin k If ζ = 1 we call A nor-malised-special A Frobenius algebra homomorphism between two Frobenius algebrasis both an algebra homomorphism and a coalgebra homomorphism

A (strictly) sovereign tensor category is a tensor category equipped with a left and aright duality which agree on objects and morphisms (see eg [BiFS] for more details)We will write the dualities as

Uor U

= dU Uor otimes U rarr 1U Uor

= dU U otimes Uor rarr 1

U Uor

= bU 1 rarr U otimes UorUor U

= bU 1 rarr Uor otimes U

(213)

In terms of these we define the left and right dimension of an object U as

diml U = dU bU dimr U = dU bU (214)

both of which are elements of Hom(1 1)Let now C be a sovereign tensor category For a Frobenius algebra A in C we define

two morphisms

A =

A

Aor

primeA =

A

Aor

(215)

Definition 210 A Frobenius algebra A is symmetric iff A = primeA

The following lemma shows that under certain conditions we do not need to distin-guish the various notions of simplicity in Definition 28

Lemma 211 Let A be a commutative symmetric Frobenius algebra in a C-linear semi-simple sovereign braided tensor category C and suppose that diml A = 0 Then thefollowing are equivalent

880 L Kong I Runkel

(i) A is simple(ii) A is absolutely simple

(iii) A is haploid

Proof (ii)hArr(iii) A is haploid iff it is absolutely simple as a left module over itself[FS Eq (417)] Furthermore for a commutative algebra we have HomA(A A) =HomA|A(A A) and so A is haploid iff it is absolutely simple(i)rArr(ii) If A is simple then every nonzero element of HomA|A(A A) is invertibleHence this space forms a division algebra over C and is therefore isomorphic to C(iii)rArr(i) Since C is semi-simple and A is haploid also Hom(A 1) is one-dimensionalThe counit ε is a nonzero element in this space and so gives a basis This implies firstlythat ε η = 0 and secondly that there is a constant β isin C such that

β middot ε = dA (idAor otimes m) (bA otimes idA) (216)

Composing with η from the right yields β εη = diml A The right-hand side is nonzeroand so β = 0 By [FRS Lem 311] A is special We have already proved (ii)hArr(iii)and so A is absolutely simple A special Frobenius algebra in a semi-simple categoryhas a semi-simple category of bimodules (apply [FS Prop 524] to the algebra tensoredwith its opposite algebra) For semi-simple C-linear categories simple and absolutelysimple are equivalent2 Thus A is simple Remark 212 For a Frobenius algebra A the morphisms (215) are invertible and henceA sim= Aor In this case one has diml A = dimr A [FS Rem 363] and so we could havestated the above lemma equivalently with the condition dimr A = 0

Let F C1 rarr C2 be a lax tensor functor between two tensor categories C1 C2 and

let (Am A ηA) be an algebra in C1 Define morphisms F(A)otimes F(A)m F(A)minusminusminusrarr F(A) and

12ηF(A)minusminusminusrarr F(A) as

m F(A) = F(m A) φF2 ηF(A) = F(ηA) φF

0 (217)

Then (F(A)m F(A) ηF(A)) is an algebra in C2 [JS Prop 55] If f A rarr B is analgebra homomorphism between two algebras A B isin C1 then F( f ) F(A) rarr F(B)is also an algebra homomorphism If (MmM ) is a left (or right) A-module in C1 then(F(M) F(mM )φF

2 ) is a left (or right) F(A)-module if M has a A-bimodule structurethen F(M) naturally has a F(A)-bimodule structure

Similarly if (AA εA) is a coalgebra in C1 and F C1 rarr C2 is a colax tensor func-

tor then F(A) with coproduct F(A)F(A)minusminusminusrarr F(A) otimes F(A) and counit F(A)

εF(A)minusminusminusrarr 12given by

F(A) = ψ F2 F(A) εF(A) = ψ F

0 F(εA) (218)

is a coalgebra in C2 If f A rarr B is a coalgebra homomorphism between two coalgebrasA B isin C1 then F( f ) F(A) rarr F(B) is also a coalgebra homomorphism

2 To see this note that if U is simple then the C-vector space Hom(UU ) is a division algebra and henceHom(UU ) = C idU Conversely if U is not simple then U = U1 oplus U2 and Hom(UU ) contains at leasttwo linearly independent elements namely idU1 and idU2

Cardy Algebras and Sewing Constraints I 881

Proposition 2133 If F C1 rarr C2 is a Frobenius functor and (Am A ηAA εA) aFrobenius algebra in C1 then (F(A)m F(A) ηF(A)F(A) εF(A)) is a Frobenius alge-bra in C2

Proof One Frobenius property (m F(A)otimes idF(A)) (idF(A)otimesF(A)) = F(A) m F(A)follows from the commutativity of the following diagram (we spell out the associativityisomorphisms)

F(A)otimes F(A)idF(A)otimesF(A)

φF2

F(A)otimes F(A otimes A)idF(A)otimesψF

2

φF2

F(A)otimes (F(A)otimes F(A))

αF(A)F(A)F(A)

F(A otimes A)

F(idAotimesA)

F(m A)

F(A otimes (A otimes A))

F(αAAA)

(F(A)otimes F(A))otimes F(A)

φF2 otimesidF(A)

F((A otimes A)otimes A)

F(m AotimesidA)

ψF2 F(A otimes A)otimes F(A)

F(m A)otimesidF(A)

F(A)

F(A) F(A otimes A)ψF

2 F(A)otimes F(A)

(219)

The commutativity of the upper-left subdiagram follows from the naturalness of φF2 that

of the upper-right subdiagram follows from (25) that of the lower-left subdiagram fol-lows from the Frobenius properties of A and that of the lower-right subdiagram followsfrom the naturalness of ψ F

2 The proof of the other Frobenius property is similar Proposition 214 If F C1 rarr C2 is a tensor functor and A a Frobenius algebra in C1then

(i) F(A) has a natural structure of Frobenius algebra as given in Proposition 213(ii) If A is (normalised-)special so is F(A)

Proof Part (i) follows from Propositions 25 and 213 Part (ii) is a straightforwardverification of the definition using ψ F

2 = (φF2 )

minus1 and ψ F0 = (φF

0 )minus1

Let C1 C2 be sovereign tensor categories and F C1 rarr C2 a Frobenius functor Wedefine two morphisms IF(Aor) I prime

F(Aor) F(Aor) rarr F(A)or for a Frobenius algebra A inC1 as follows

IF(Aor) = ((ψ F0 F(dA) φF

2 )otimes idF(A)or) (idF(Aor) otimes bF(A))

I primeF(Aor) = (idF(A)or otimes (ψ F

0 F(dA) φF2 )) (bF(A) otimes idF(Aor))

(220)

It is easy to see that these are isomorphisms

Lemma 215 If F C1 rarr C2 is a Frobenius functor and A a Frobenius algebra in C1then

F(A) = IF(Aor) F(A) primeF(A) = I prime

F(Aor) F(primeA) (221)

3 After the preprint of the present paper appeared we noticed that this proposition is also proved in[DP Cor 5]

882 L Kong I Runkel

Proof We only prove the first equality the second one can be seen in the same way Bydefinition we have

IF(Aor) F(A)

= ((ψ F0 F(dA) φF

2 )otimes idF(A)or) (idF(Aor) otimes bF(A)) F(A)

=[(ψ F

0 F(dA) φF2 ) (F(A)otimes idF(A))

]otimes idF(A)or

(idF(A) otimes bF(A))

For the term inside the square brackets we find

ψ F0 F(dA) φF

2 (F(A)otimes idF(A)) = ψ F0 F(dA) F(A otimes idA) φF

2

= ψ F0 F(dA (A otimes idA)) φF

2 = ψ F0 F(εA m A) φF

2 (222)

On the other hand by definitionF(A) = [((ψ F0 F(εA) (F(m A)φF

2 ))otimes idF(A)or] (idF(A) otimes bF(A)) This demonstrates the first equality in (221) Proposition 216 Let F C1 rarr C2 be a tensor functor G C2 rarr C1 a functor〈FG χ〉 an adjunction A a C1-algebra and B a C2-algebra Then f A rarr G(B)is an algebra homomorphism if and only if f = χminus1( f ) F(A) rarr B is an algebrahomomorphism

Proof We need to show that

mG(B) ( f otimes f ) = f m A and f ηA = ηG(B) (223)

is equivalent to

m B ( f otimes f ) = f m F(A) and f ηF(A) = ηBs (224)

We first prove that the first identity in (223) is equivalent to the first identity in (224)For the left-hand side of the first identity in (223) we have the following equalities

mG(B) ( f otimes f )(1)= G(m B) φG

2 ( f otimes f )(2)= G(m B) G(ρ otimes ρ) G(ψ F

2 ) δ ( f otimes f )(3)= G(m B) G(ρ otimes ρ) G(ψ F

2 ) G F( f otimes f ) δ(4)= G(m B) G(ρ otimes ρ) G(F( f )otimes F( f )) G(ψ F

2 ) δ(5)= G

(m B (ρ otimes ρ) (F( f )otimes F( f )) ψ F

2

) δ

(6)= χ(

m B (ρ otimes ρ) (F( f )otimes F( f )) ψ F2

) (225)

where (1) is the definition of mG(B) in (217) (2) is the second identity in (210) (3)and (4) are naturality of δ and ψ F

2 respectively step (5) is functoriality of G and finallystep (6) is (29) For the right hand side of the first identity in (223) we get

f m A(1)= Gρ δG ( f m A)

(2)= Gρ G F( f m A) δ(3)= G(ρ F( f m A)) δ(4)= χ(ρ F( f ) F(m A)) (226)

Cardy Algebras and Sewing Constraints I 883

where (1) is the adjunction property (28) (2) is naturality of δ (3) functoriality of Gand (4) amounts to (29) and functoriality of F

On the other hand we see that the first equality in (224) is equivalent to

m B (ρ otimes ρ) (F( f )otimes F( f )) = ρ F( f ) F(m A) φF2 (227)

Using that φF2 is invertible with inverse (φF

2 )minus1 = ψ F

2 and that χ is an isomorphismit follows that the statement that (225) is equal to (226) is equivalent to the identity(227)

Now we prove that the second identity in (223) is equivalent to the second identity in(224) Using (217) and (210) we can write ηG(B) = G(ηB)φF

0 = G(ηB)G(ψ F0 )δ1

Together with (29) this shows that the second identity in (223) is equivalent to

f ηA = χ(ηB ψ F0 ) (228)

On the other hand the second identity in (224) is equivalent to

ρ F( f ) F(ηA) φF0 = ηB (229)

which by φF0 = (ψ F

0 )minus1 and (29) is further equivalent to (228)

Definition 217 Let (Am A ηAA εA) and (Bm B ηBB εB) be two Frobeniusalgebras in a tensor category C For f A rarr B we define f lowast B rarr A by

f lowast = ((εB m B)otimes idA) (idB otimes f otimes idA) (idB otimes (A ηA)) (230)

The following lemma is immediate from the definition of (middot)lowast and the properties ofFrobenius algebras We omit the proof

Lemma 218 Let C be a tensor category let A BC be Frobenius algebras in C andlet f A rarr B and g B rarr C be morphisms

(i) (g f )lowast = f lowast glowast(ii) f is a monomorphism iff f lowast is an epimorphism

(iii) f is an algebra map iff f lowast is a coalgebra map(iv) If f is a homomorphism of Frobenius algebras then f lowast f = idA and f f lowast =

idB(v) If C is sovereign and if A and B are symmetric then f lowastlowast = f

Let C and D be tensor categories and let F C rarr D be a Frobenius functor GivenFrobenius algebras A B in C and a morphism f A rarr B the next lemma shows how(middot)lowast behaves under F

Lemma 219 F( f lowast) = F( f )lowast

Proof The definition of the structure morphisms of the Frobenius algebra F(A) is givenin (217) and (218) Substituting these definitions gives

F( f )lowast =[(ψ F

0 F(εB) F(m B) φF2

)otimes idF(A)

] [

idF(B) otimes F( f )otimes idF(A)]

[idF(B) otimes

(ψ F

2 F(A) F(ηA) φF0

)]

= (ψ F0 otimes idF(A)) [

F (εB m B (idB otimes f ))otimes idF(A)]

(φF2 otimes idF(A)) (idF(B) otimes ψ F

2 )

[idF(B) otimes F(A ηA)

] (idF(B) otimes φF0 ) (231)

884 L Kong I Runkel

In the middle line of the last expression we can use the defining property (25) of F namely we substitute (φF

2 otimes idF(A)) (idF(B) otimes ψ F2 ) = ψ F

2 φF2 Then ψ F

2 can bemoved to the left and φF

2 to the right until they can be omitted against ψ F0 and φF

0 respectively using (22) and (24) This results in

F( f )lowast = F ((εB m B (idB otimes f ))otimes idA) F (idB otimes (A ηA)) (232)

which is nothing but F( f lowast)

23 Modular tensor categories Let C be a modular tensor category [TBK] ie anabelian semi-simple finite C-linear ribbon category with simple tensor unit 1 and a non-degeneracy condition on the braiding (to be stated in a moment) We denote the set ofequivalence classes of simple objects in C by I elements in I by i j k isin I and theirrepresentatives by Ui U j Uk We also set U0 = 1 and for an index k isin I we define kby Uk

sim= Uork

Since the tensor unit is simple we shall for modular tensor categories identifyHom(1 1) sim= C (cf footnote 2) Define numbers si j isin C by4

si j = UiU j (233)

They obey si j = s ji and s0i = dim Ui see eg [BK Sect 31] (In a ribbon cate-gory the left and right dimension (214) of Ui coincide and are denoted by dim Ui )The non-degeneracy condition on the braiding of a modular tensor category is that the|I|times|I|-matrix s should be invertible In fact [BK Thm 317]

sumkisinI

sik sk j = Dim C δij (234)

where Dim C = sumiisinI(dim Ui )

2 One can show (even in the weaker context of fusioncategories over C) that Dim C ge 1 [ENO1 Thm 23] In particular Dim C = 0 We fixonce and for all a square root

radicDim C of Dim C

Let us fix a basis λα(i j)kN k

i jα=1 in HomC(Ui otimesU j Uk) and the dual basis ϒ(i j)k

α N ki j

α=1

in HomC(UkUi otimesU j ) The duality of the bases means that λα(i j)k ϒ(i j)kβ = δαβ idUk

We also fix λ(0i)i = λ(i0)i = idUi We denote the basis vectors graphically as follows

λα(i j)k = α

Uk

Ui U j

ϒ(i j)kα = α

Uk

Ui U j

(235)

4 In the graphical notation used below we have given an orientation to the ribbons indicated by the arrowsFor example it is understood that this orientation determines which of the duality morphisms in (213) to use

Cardy Algebras and Sewing Constraints I 885

For V isin C we also choose a basis b(iα)V of HomC(VUi ) and the dual basis bV(iβ)

of HomC(Ui V ) for i isin I such that b(iα)V bV(iβ) = δαβ idUi We use the graphical

notation

b(iα)V = α

Ui

V

bV(iα) = α

Ui

V

(236)

Given two modular tensor categories C and D by C D we mean the tensor productof additive categories over C [BK Def 1115] ie the category whose objects are directsums of pairs V times W of objects V isin C and W isin D and whose morphism spaces are

HomCD(V times W V prime times W prime) = HomC(V V prime)otimesC HomC(WW prime) (237)

for pairs and direct sums of these if the objects are direct sums of pairsIf we replace the braiding and the twist in C by the antibraiding cminus1 and the antitwist

θminus1 respectively we obtain another ribbon category structure on C In order to distin-guish these two distinct structures we denote (C c θ) and (C cminus1 θminus1) by C+ and Cminusrespectively As in the Introduction we will abbreviate

C2plusmn = C+ Cminus (238)

Note that a set of representatives of the simple objects in C2plusmn is given by Ui times U j fori j isin I

For the remainder of Sect 2 we fix a modular tensor category C

24 The functors T and R The tensor product bifunctor otimes can be naturally extendedto a functor T C2plusmn rarr C Namely T (oplusN

i=1Vi times Wi ) = oplusNi=1Vi otimes Wi for all Vi Wi isin C

and N isin N The functor T becomes a tensor functor as follows For φT0 1 rarr T (1 times 1)

take φT0 = id1 (or lminus1

1 in the non-strict case) Next notice that for U VW X isin C

T (U times V )otimes T (W times X)= (U otimes V )otimes (W otimes X)

T ((U times V )otimes (W times X))= (U otimes W )otimes (V otimes X)(239)

We define φT2 T (U times V )otimes T (W times X) rarr T ((U times V )otimes (W times X)) by

φT2 = idU otimes cminus1

W V otimes idX (240)

(In the non-strict case the appropriate associators have to be added) The above definitionof φT

2 can be naturally extended to a morphism φT2 T (M1 otimes M2) rarr T (M1)otimes T (M2)

for any pair of objects M1M2 in C2plusmn The following result can be checked by directcalculation [JS Prop 52]

Lemma 220 The triple (T φT2 φ

T0 ) gives a tensor functor

886 L Kong I Runkel

In particular (T φT2 φ

T0 ψ

T2 ψ

T0 ) where ψT

2 = (φT2 )

minus1 and ψT0 = (φT

0 )minus1 gives

a Frobenius functorDefine the functor R C rarr C2plusmn as follows for A isin C and f isin HomC(A B)

R(A) =oplusiisinI

(A otimes Uori )times Ui R( f ) =

oplusiisinI

( f otimes idUori)times idUi (241)

This functor was also considered in a slightly different context in [ENO2 Prop 23]The family of isomorphisms γ R

A = oplusiisinI DimCdim Ui

id(AotimesUori )timesUi

isin Aut(R(A)) defines a

natural isomorphism γ R R rarr ROur next aim is to show that R is left and right adjoint to T in other words R

and T form an ambidextrous adjunction (see eg [Ld] for a discussion of ambidextrousadjunctions) To this end we introduce two linear isomorphisms for A isin C and M isin C2plusmn

χ HomC(T (M) A) minusrarr HomC2plusmn(M R(A))

χ HomC(A T (M)) minusrarr HomC2plusmn(R(A)M)(242)

If we decompose M as M = oplusNn=1 Ml

n times Mrn then χ and χ are given by

χ Noplus

n=1

fn

Mln Mr

n

A

rarrNoplus

n=1

oplusiisinI

sumα

fn

Mln

Mrn

A

α

Uori

times α

Mrn

Ui

(243)

and

χ Noplus

n=1 gn

Mln Mr

n

A

rarrNoplus

n=1

oplusiisinI

sumα gn

Mln

Mrn

A

α

Uori

times α

Ui

Mrn

DimCdim Ui

(244)

Notice that χ and χ are independent of the choice of basis

Theorem 221 〈T R χ〉 and 〈R T χminus1〉 are adjunctions ie R is both left and rightadjoint of T

Proof Write M as M = oplusNn=1 Ml

n times Mrn The isomorphism χ amounts to the following

composition of natural isomorphisms

HomC(T (M) A) = oplusnHomC(Mln otimes Mr

n A)sim= oplusni HomC(Ml

n otimes Ui A)otimes HomC(Mrn Ui )

sim= oplusni HomC(Mln A otimes Uor

i )otimes HomC(Mrn Ui ) = HomC2plusmn(M R(A)) (245)

Cardy Algebras and Sewing Constraints I 887

Thus χ is natural Let (γ RA )

lowast HomC2plusmn(R(A)M) rarr HomC2plusmn(R(A)M) denote the

pull-back of γ RA The isomorphism χ is equal to the composition of (γ R

A )lowast and the

following sequence of natural isomorphisms

HomC(A T (M)) = oplusn HomC(AMln otimes Mr

n)

sim= oplusni HomC(AMln otimes Ui )otimes HomC(Ui Mr

n)

sim= oplusni HomC(A otimes Uori Ml

n)otimes HomC(Ui Mrn) = HomC2plusmn(R(A)M) (246)

We have proved that both χ and χ are natural isomorphisms There are four natural transformations associated to χ and χ namely

idC2plusmnδminusrarr RT

ρminusrarr idC2plusmn and idCδminusrarr T R

ρminusrarr idC (247)

defined by for A isin C M isin C2plusmn

δM = χ(idT (M)) ρA = χminus1(idR(A))

ρM = χ (idT (M)) δA = χminus1(idR(A))(248)

They can be expressed graphically as follows with M = oplusNn=1 Ml

n times Mrn

δM =oplusni

sumα

Mln

Mln Mr

n Uori

αtimes

Ui

Mrn

α

ρA =oplusiisinI

A

A

Uori Ui

(249)

ρM =oplusni

sumα

Mln

Mln

Mrn Uor

i

α times

Ui

Mrn

α DimCdim Ui

δA =oplusiisinI

A

A Uori Ui

dim UiDimC

Note that

ρM δM = Dim C middot idM and ρA δA = idA (250)

Lemma 222 The functors T and R as maps on the sets of morphisms have left inversesand thus are injective

Proof Let f A rarr B be a morphism in C We define a map Q R HomC2plusmn(R(A)

R(B)) rarr HomC(A B) by f prime rarr ρB T ( f prime) δA Then we have

Q R R( f ) = ρB T R( f ) δA = ρB δB f = f (251)

888 L Kong I Runkel

where we used naturality of δ and (250) in the second and third equalities respectivelySo Q R is a left inverse of R on morphisms Thus R is injective on morphisms Similarlylet g M rarr N be a morphism in C2plusmn We define a map QT HomC(T (M) T (N )) rarrHomC2plusmn(M N ) by gprime rarr (Dim C)minus1 middot ρN R(gprime) δM Then we have

QT T (g) = (Dim C)minus1 middot ρN RT (g) δM = (Dim C)minus1 middot ρN δN g = g (252)

So QT is a left inverse of T on morphisms Thus T is injective on morphisms Using (29) and (250) one can express the two inverse maps χminus1 χminus1 as follows

for f isin HomC2plusmn(M R(A)) and g isin HomC2plusmn(R(A)M)

χminus1( f ) = ρ T ( f ) χminus1(g) = T (g) δ (253)

By Proposition 25 and Lemma 27 R is both a lax and colax tensor functor In particularφR

0 1 times 1 rarr R(1) is given by

φR0 = χ (ψT

0 ) = R(ψT0 ) δ1 times 1 = id1times1 (254)

and φR2 R(A)otimes R(B) rarr R(A otimes B) by φR

2 = R(ρA otimes ρB) R(ψT2 ) δ which can

be expressed graphically as

φR2 =

oplusi jkisinI

sumα

A

A

Uori B

B

Uorj

Uork

α

times

Ui U j

α

Uk

(255)

Similarly ψ R0 R(1) rarr 1 times 1 is given by

ψ R0 = ρ1 R(φT

0 ) = Dim C id1times1 (256)

and ψ R2 R(A otimes B) rarr R(A) otimes R(B) by ψ R

2 = ρ R(φT2 ) R(δA otimes δB) which in

graphical notation reads

ψ R2 =

oplusi jkisinI

sumα

A

A Uori

B

B Uorj

Uork

α

times

Ui U j

α

Uk

dim Ui dim U j

dim Uk Dim C

(257)

If C has more than one simple object then R does not take the tensor unit of C to thetensor unit of C2plusmn and so is clearly not a tensor functor However we will show that R isstill a Frobenius functor This will imply that if A is a Frobenius algebra in C then

R(A) = (R(A)m R(A) ηR(A)R(A) εR(A)) (258)

Cardy Algebras and Sewing Constraints I 889

is a Frobenius algebra in C2plusmn where the structure morphisms were given in (217) and(218) In the case A = 1 it was proved in [Muuml1 Prop 41] (see also [Fr Lem 619]and [K1 Thm 52]) that (258) is a commutative simple symmetric normalised-specialFrobenius algebra in C2plusmn In fact given a Frobenius algebra A in C it is straightforwardto verify that the structure morphisms in (258) are precisely those of (A times 1)otimes R(1)cf Sect 22

Proposition 223 (R φR2 φ

R0 ψ

R2 ψ

R0 ) is a Frobenius functor

Proof Using the explicit graphical expression of φR2 φ

R0 ψ

R2 ψ

R0 it is easy to see that

the commutativity of the diagrams (25) and (26) are equivalent to the statement thatR(1) with structure morphisms as in (258) is a Frobenius algebra in C2plusmn The latterstatement is true by [Muuml1 Prop 41]

From Lemma 220 and Proposition 223 we see that T and R take Frobenius alge-bras to Frobenius algebras The following two propositions show how the properties ofFrobenius algebras are transported

Proposition 224 Let A be a Frobenius algebra in C2plusmn Then T (A) is a Frobenius algebrain C and

(i) A is symmetric iff T (A) is symmetric(ii) A is (normalised-)special iff T (A) is (normalised-)special

Proof For part (i) write A as a direct sum oplusNn=1 Al

n times Arn Then the maps IT (A) I prime

T (A) T (Aor) rarr T (A)or defined in (220) are given by

IT (A) = I primeT (A) = oplusN

n=1c(Aln)

or(Arn)

or (259)

Therefore by (221) T (A) = primeT (A) is equivalent to T (A) = T (prime

A) Since byLemma 222 T is injective on morphisms this proves part (i) Part (ii) can be checkedin the same way for example the condition mT (A) T (A) = ζ idT (A) is easily checkedto be equivalent to T (m A A) = ζ T (idA) Proposition 225 Let A be a Frobenius algebra A in C Then R(A) is a Frobeniusalgebra in C2plusmn and

(i) A is symmetric iff R(A) is symmetric(ii) A is (normalised-)special iff R(A) is (normalised-)special

Proof Recall that the structure morphisms of the Frobenius algebra R(A) are equal tothose of (A times 1) otimes R(1) Using this equality part (i) and (ii) follow because R(1) issymmetric and normalised-special For example

m R(A) R(A) = [(m A A)times id1] otimes idR(1) = R(m A A) (260)

so that m R(A) R(A) = ζ idR(A) is equivalent to R(m A A) = ζ R(idA) which byLemma 222 is equivalent to m A A = ζ idA

The functor R has one additional property not shared by T namely R takes absolutelysimple algebras to absolutely simple algebras We will see explicitly in Sect 33 thatthis is not true for T

890 L Kong I Runkel

Lemma 226 For a C-algebra A the map

R HomA|A(A A) rarr HomR(A)|R(A)(R(A) R(A)) (261)

given by f rarr R( f ) is well-defined and an isomorphism

Proof Since R is a lax tensor functor R(A) is naturally a R(A)-bimodule It is easy tosee that R in (261) is a well-defined map R(A) is also naturally a R(1)-bimodule whichcan be identified with the induced R(1)-bimodule structure on (A times 1)otimes R(1) wherethe left R(1) action on (Atimes1)otimes R(1) is given by (idAtimes1 otimesm R(1))(cminus1

Atimes1R(1)otimes idR(1))We have the following natural isomorphisms

HomR(1)|R(1)(R(A) R(A))sim=minusrarr HomC2plusmn(A times 1 R(A))

χminus1

minusminusrarr HomC(A A) (262)

which by (253) are given by for f isin HomR(1)|R(1)(R(A) R(A))

f rarr f prime = f (idAtimes1 otimes ηR(1)) rarr f primeprime = ρ T ( f prime) (263)

and its inverse is given by for g isin HomC(A A)

g rarr gprime = R(g) δ rarr gprimeprime = (idAtimes1 otimes m R(1)) (gprime otimes idR(1)) (264)

where gprimeprime is indeed a R(1)-bimodule map due to the commutativity of R(1) It is easy tocheck that gprimeprime = R(g) in (264) Therefore R gives an isomorphism from HomC(A A)to HomR(1)|R(1)(R(A) R(A)) Moreover one verifies that R(g) is an R(A)-bimodulemap iff g is an A-bimodule map In other words R g rarr R(g) gives an isomorphism

HomA|A(A A)sim=minusrarr HomR(A)|R(A)(R(A) R(A))

Corollary 227 Let A be a C-algebra

(i) A is absolutely simple iff R(A) is absolutely simple(ii) Let A be in addition Frobenius Then A is simple and special iff R(A) is simple and

special

Proof Part (i) immediately follows from Lemma 226 The statement of part (ii) withoutthe qualifier lsquosimplersquo is proved in Proposition 225 But as in the proof of (iii)rArr(i) inLemma 211 a special Frobenius algebra in a semi-simple category has a semi-simplecategory of bimodules and for a semi-simple C-linear category simple and absolutelysimple are equivalent Part (ii) then follows from part (i)

The following lemma will be needed in Sect 32 below to discuss the properties ofCardy algebras

Lemma 228 Let A be a Frobenius algebra in C Then (δA)lowast = ρA

Proof Recall from (250) that δA is a morphism A rarr T R(A) Since T and R are bothFrobenius functors T R(A) is a Frobenius algebra in C Substituting the definitions aftera short calculation one finds

Cardy Algebras and Sewing Constraints I 891

εT R(A) mT R(A) = Dim(C)oplusiisinI

A Uori Ui A U or

ı Uı

(265)

Substituting this in the definition of (δA)lowast gives again after a short calculation the mor-

phism ρA At an intermediate step one uses that the part of the morphism (δA)lowast which

is made up of Ui and Uı ribbons their duals and the basis morphisms λ(iı)0 andϒ(iı)0can be replaced by 1

dim Uimiddot dUi

3 Cardy Algebras

In this section we start by investigating the properties of Frobenius algebras which sat-isfy the so-called modular invariance condition We then give two definitions of a Cardyalgebra and prove their equivalence Finally in Sect 33 we study the properties of thesealgebras and state our main results

We fix a modular tensor category C Recall that C2plusmn is an abbreviation for C+ Cminus

31 Modular invariance In C2plusmn we define the object K and the morphism ω K rarr Kas

K =oplus

i jisinIUi times U j ω =

sumi jisinI

dim Ui dim U j

Dim C idUi timesU j (31)

They have the property (see eg [BK Cor 3111])

ω

K

(Uk times Ul )or

(Uk times Ul )or

Ui times U j

Ui times U j

= δik δ jlDim C

dim Ui dim U jbUi timesU j dUi timesU j (32)

892 L Kong I Runkel

Definition 31

(i) Let A B be objects of C2plusmn A morphism f A otimes B rarr B is called S-invariant iff

A

B

W

W

f =

A

B

W

W

f

K

ω(33)

holds for all for W isin C2plusmn(ii) A C2plusmn-algebra (Aclmcl ηcl) is called modular invariant iff θAcl = idAcl and mcl

is S-invariant

Lemma 32 The morphism f A otimes B rarr B is S-invariant if and only if

A

B

Ui times U j

Ui times U j

f = Dim Cdim Ui dim U j

sumα

A

B

B

f

Ui times U j

Ui times U j

α

α

(34)

holds for all i j isin I

Proof Condition (33) holds for all W iff it holds for all W = Ui times U j i j isin I so itis enough to show that the right-hand side of (33) with W = Ui times U j is equal to theright-hand side of (34) Recall the notation for basis morphisms in (236) Starting from(33) write

idBor otimes idUi timesU j =⎛⎝sum

klα

(b(ktimeslα)

B

)or (

bB(ktimeslα)

)or⎞⎠ otimes idUi timesU j (35)

and then apply (32) The graphical representation of the resulting morphism can bedeformed to give (34) Remark 33 As shown in [K3 Sect 61] the modular invariance condition of a C2plusmn-alge-bra exactly coincides with the modular invariance condition for torus 1-point correlationfunctions of a genus-01 closed CFT In particular the condition θAcl = idAcl is equiv-alent to invariance under the modular transformation T τ rarr τ + 1 and the condition(34) with f = mcl is equivalent to invariance under S τ rarr minus 1

τ Combining the

modular invariance condition with the genus zero properties of a genus-01 closed CFTresults in a modular invariant commutative symmetric Frobenius algebra in C2plusmn

Cardy Algebras and Sewing Constraints I 893

Let Acl be a modular invariant C2plusmn-algebra Evaluating (34) for f = mcl composingit with ηcl otimes idUi timesU j and taking the trace implies the following identity

Zi j = 1

DimCAclUi timesU j where

(36)

Zi j = dimC HomC2plusmn(UitimesU j Acl)

Decomposing Acl into simple objects this gives

Zi j =sum

klisinISik Zkl Sminus1

l j where Si j = si jradic

DimC (37)

which in CFT terms is of course nothing but the invariance of the torus partition functionunder the modular S-transformation

The following theorem gives a simple criterion for modular invariance

Theorem 34 Let Acl be a haploid commutative symmetric Frobenius algebra in C2plusmn

(i) If Acl is modular invariant then dim Acl = Dim C(ii) If dim Acl = Dim C then Acl is special and modular invariant

Proof Part (i) Since Acl is haploid for i = j = 0 Equation (36) reduces to 1 =dim AclDimCPart (ii) By the same reasoning as in the proof of (iii)rArr(i) in Lemma 211 one showsthat Acl is special Thus mcl cl = ζclidAcl for some ζcl = 0

By [KO Thm 45] the category (C2plusmn)locAcl

of local Acl-modules is again a mod-

ular tensor category and Dim (C2plusmn)locAcl

= (Dim C dim Acl)2 (see [Fr Prop 321 amp

Rem 323] for the same statement in the notation used here) Thus by assumption wehave Dim(C2plusmn)loc

Acl= 1 It then follows from [ENO1 Thm 23] that up to isomorphism

(C2plusmn)locAcl

has a unique simple object (namely the tensor unit) In other words every simplelocal Acl-module is isomorphic to Acl (seen as a left-module over itself)

We have the following isomorphisms between morphism spaces [FS Prop 47 amp 411]

HomAcl(Acl otimes (Ui times U j ) Acl) sim= HomC2plusmn(Ui times U j Acl)

HomAcl(Acl Acl otimes (Ui times U j )) sim= HomC2plusmn(AclUi times U j )

Using these to transport the bases (236) from the right to the left we obtain bases bα(i j)αof HomAcl(Acl otimes (Ui times U j ) Acl) and b(i j)

β β of HomAcl(Acl Acl otimes (Ui times U j )) Thesecan be expressed graphically as

894 L Kong I Runkel

bα(i j) =

Acl

Acl

Acl

Ui times U j

α

b(i j)β = DimC

dim Ui dim U j

1

ζcl

Acl

Ui times U jAcl

Acl

β

(38)

where the nonzero factor in b(i j)β is included for convenience Notice that bα(i j) b(i j)

β isa left Acl-module map Since Acl is simple as a left module over itself we have

bα(i j) b(i j)β = λαβ idAcl (39)

for some λαβ isin C By computing tr(bα(i j) b(i j)β ) it is easy to verify that λαβ = δαβ

We will now prove the following identity

1

ζcl

Acl

Acl

Ui times U j

Ui times U j

Acl

=sumα

DimCdim Ui dim U j

1

ζcl

Acl

Ui times U j

Ui times U j

Acl

Acl

Acl

α

α

(310)

One checks that the left-hand side of this equation is an idempotent which we denote byζminus1

cl PlAcl(Ui timesU j ) cf [Fr Sect 31] By [Fr Prop 41] the image Im(ζminus1

cl PlAcl(Ui timesU j ))

is a local Acl-module and hence isomorphic to AoplusNcl for some N isin Zge0

All left-module morphisms from Acl to Aclotimes(Ui timesU j ) are linear combinations of the

b(i j)β Furthermore one verifies that ζminus1

cl PlAcl(Ui timesU j )b(i j)

β = b(i j)β Therefore the b(i j)

β

describe precisely the image of the idempotent ie ζminus1cl Pl

Acl(Ui timesU j ) = sum

α b(i j)α bα(i j)

which is nothing but (310) Composing (310) with ζcl middot εcl otimes idUi timesU j from the left (iefrom the top) produces (34)

In addition since Acl is commutative symmetric Frobenius it satisfies θAcl = idAcl

[Fr Prop 225] Altogether this shows that Acl is modular invariant Remark 35 As we were writing this paper we heard that the results in Theorem 34were obtained independently by Kitaev and Muumlger [Ki]

Remark 36 Setting i = j = 0 in (36) gives the identity dim Acl = Z00 Dim C [KRProp 23] Combining this with Theorem 34 (ii) one may wonder if a general modularinvariant commutative symmetric Frobenius algebra Acl in C2plusmn is isomorphic to a directsum of simple such algebras However this is not so For example one can take the com-mutative symmetric Frobenius algebra Acl = C[x]x2 in the category of vector spacesequipped with the non-degenerate trace ε(ax + b) = a In this case the modular invari-ance condition holds automatically but Acl is clearly not a direct sum of two algebras

Cardy Algebras and Sewing Constraints I 895

For a general modular tensor category C the algebra C[x]x2 R(1) understood as analgebra in C2plusmn via the braided monoidal isomorphism Vect f (C) C2plusmn rarr C2plusmn providesanother counter-example

32 Two definitions Define a morphism PlA A rarr A for a Frobenius algebra A in C

or C2plusmn as follows [Fr Sect 24]

PlA =

A

A

A

m

(311)

If A is also commutative and obeys m A A = ζA idA we have PlA = ζA idA In partic-

ular this holds if A is commutative and special Using the fact that the Frobenius algebraR(1) is commutative and normalised-special one can check that Pl

R(A) R(A) rarr R(A)takes the following form

PlR(A) =

oplusiisinI

A

A

Uori

Uori

A

m

times

Ui

Ui

(312)

With these ingredients we can now give the first definition of a Cardy C|C2plusmn-algebrawhich was introduced in [K3 Def 514] cf Remark 315 below

Definition 37 (CardyC|C2plusmn-algebra I) A CardyC|C2plusmn-algebra is a triple (Aop|Acl ιcl-op)where (Aclmcl ηclcl εcl) is a modular invariant commutative symmetric Frobe-nius C2plusmn-algebra (Aopmop ηopop εop) is a symmetric Frobenius C-algebra andιcl-op Acl rarr R(Aop) an algebra homomorphism such that the following conditionsare satisfied

(i) Centre condition

Acl R(Aop)

R(Aop)

R(Aop)

ιcl-op

m R(Aop)

=

Acl

ιcl-op

R(Aop)

R(Aop)

R(Aop)

m R(Aop)

(313)

896 L Kong I Runkel

(ii) Cardy condition

ιcl-op ιlowastcl-op =

R(Aop)

R(Aop)

R(Aop)

(314)

Remark 38

(i) The name ldquoCardy C|C2plusmn-algebrardquo in Definition 37 was chosen because many ofthe important ingredients were first studied by Cardy the modular invariance ofthe closed theory [C1] the consistency of the annulus amplitude [C2] and thebulk-boundary OPE [CL] On the other hand the boundary-boundary OPE andthe OPE analogue of the centre condition were first considered in [Lw]

(ii) One can easily see that in the special case that C is the category Vect f (C) of finite-dimensional C-vector spaces a Cardy C|C2plusmn-algebra gives exactly the algebraicformulation of two-dimensional open-closed topological field theory over C (cfRemark 614 in [K3]) see [Lz Sect 48] [Mo Thm 11] [AN Thm 45] [LPCor 43] [MS Sect 22] When passing to a general modular tensor category Cthere are two important differences to the two-dimensional topological field the-ory Firstly the algebras Acl and Aop now live in different categories which inparticular affects the formulation of the centre condition and the Cardy conditionSecondly the modular invariance condition has to be imposed on Acl In the caseC = Vect f (C) modular invariance holds automatically

Definition 39 A homomorphism of Cardy C|C2plusmn-algebras (A(1)op |A(1)cl ι(1)cl-op) rarr (A(2)op

|A(2)cl ι(2)cl-op) is a pair ( fop fcl) of Frobenius algebra homomorphisms fop A(1)op rarr A(2)op

and fcl A(1)cl rarr A(2)cl such that the diagram

A(1)cl

fcl

ι(1)cl-op

A(2)cl

ι(2)cl-op

R(A(1)op )

R( fop) R(A(2)op )

(315)

commutes

Remark 310 Since a homomorphism of Frobenius algebras is invertible (cf Lemma218 (iv)) a homomorphism of Cardy algebras is always an isomorphism

For a homomorphism ( fop fcl) of Cardy C|C2plusmn-algebras using the commutativity of(315) and the fact that fcl and fop are both algebra and coalgebra homomorphisms itis easy to show that (315) commutes iff

R(A(1)op )R( fop)

(ι(1)cl-op)

lowast

R(A(2)op )

(ι(2)cl-op)

lowast

A(1)cl

fcl A(2)cl

(316)

Cardy Algebras and Sewing Constraints I 897

commutesLet (Aop|Acl ιcl-op) be a Cardy C|C2plusmn-algebra Define the morphism

ιcl-op = χminus1(ιcl-op) T (Acl) minusrarr Aop (317)

Decompose Acl as Acl = oplusNn=1Cl

n times Crn such that Cl

1 times Cr1 = ηcl(1 times 1) We use ι(n)cl-op

to denote the restriction of ιcl-op to Cln times Cr

n and ι(n)cl-op to denote the restriction of ιcl-op

to Cln otimes Cr

n We introduce the following graphical notation

ι(n)cl-op =

Cln Cr

n

Aop

(318)

By (243) ιcl-op can be expressed in terms of ιcl-op as follows

ιcl-op =Noplus

n=1

oplusiisinI

sumα

Cln

Crn

Aop

α

Uori

times α

Crn

Ui

(319)

Lemma 311 The centre condition (313) is equivalent to the following condition in C

Cln Cr

n

Aop

Aop

Aop

=

Cln Cr

n

Aop

Aop

Aop

for n = 1 N (320)

Proof First insert (319) and the definition (217) of m R(Aop) into (313) Then apply thecommutativity of R(1) to the left hand side of (313) The equivalence between (313)and (320) follows immediately Remark 312 The centre condition (320) is very natural from the open-closed confor-mal field theory point of view Correlators on the upper half plane are expressed in termsof conformal blocks on the full complex plane The objects Cl

n and Crn are associated to

the field insertion at a point z in the upper half plane and at the complex conjugate pointz in the lower half plane respectively The object Aop corresponds to a field inserted ata point r on the real axis The centre condition (320) simply says that the correlationfunctions in the disjoint domains |z| gt r gt 0 and r gt |z| gt 0 are analytic continuationsof each other see [K2 Prop 118]

898 L Kong I Runkel

Recall that we define ιlowastcl-op Aop rarr T (Acl) as in (230) We introduce the graphicalnotation

ιlowastcl-op =Noplus

n=1

Cln Cr

n

Aop

=Noplus

n=1

Cln Cr

n

Aop

Aop T (minus1Acl) (321)

where the second equality follows from (215) (221) and (259)

Lemma 313 The Cardy condition (314) is equivalent to the following identity in C

Noplusn=1

DimCdim Ui

sumα

Cln

Crn

Crn

α

α

Aop

Aop Uori

Uori

=Aop

Aop

Aop

Uori

Uori

op

mop for all i isin I (322)

Proof By (244) (312) and (319) it is easy to see that (322) is equivalent to thefollowing identity

ιcl-op χ(ιlowastcl-op) = PlR(Aop)

(323)

Therefore it is enough to show that

χ(ιlowastcl-op) = ιlowastcl-op (324)

We have

χminus1(ιlowastcl-op)(1)= T (ιlowastcl-op) δ (2)= T (ιcl-op)

lowast δlowastlowast (3)= (ρ T (ιcl-op)

)lowast (4)= (ιcl-op

)lowast (325)

In step (1) we use the expression (253) for χminus1 step (2) follows from Lemma 218 (v)and Lemma 219 Step (3) is Lemma 218 (i) and Lemma 228 and finally step (4)amounts to substituting (253) and (317) Acting with χ on both sides of the aboveequality produces (324)

Combining Lemmas 311 and 313 and Proposition 216 we obtain the followingequivalent definition of Cardy C|C2plusmn-algebra (recall the graphical notation (318) forιcl-op and (321) for ιlowastcl-op)

Cardy Algebras and Sewing Constraints I 899

Definition 314 (CardyC|C2plusmn-algebra II) A CardyC-algebra is a triple (Aop|Acl ιcl-op)where Acl is a commutative symmetric Frobenius C2plusmn-algebra satisfying property (34)with f = mcl Aop is a symmetric Frobenius C-algebra and ιcl-op T (Acl) rarr Aop isan algebra homomorphism satisfying the conditions (320) and (322)

Remark 315 Up to a choice of normalisation Definition 314 is the same as the orig-inal one in [K3 Def 613] The difference between the two definitions is the factorDim C dim Ui on the left hand side of (322) which in [K3 Def 613] is given byradic

Dim C dim Ui The two definitions are related by rescaling the coproduct cl andcounit εcl of Acl by 1

radicDim C and

radicDim C respectively We chose the convention in

(322) to remove all dimension factors from the expression (314) for the Cardy condition

33 Uniqueness and existence theorems In this subsection we investigate the structureof Cardy algebras We start with the following proposition which when combined withthe results of part II provides an alternative proof of [Fj Prop 422]

Proposition 316 Let (Aop|Acl ιcl-op) be a Cardy C|C2plusmn-algebra If Acl is simple anddim Aop = 0 then Aop is simple and special

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis therefore haploid Restricting the Cardy condition (322) to the case Ui = 1 andcomposing both sides with εop from the left we see that εop kills all terms associated toU j times 1 isin Acl in the sum except for a single 1 times 1 term Thus we obtain the followingidentity

β εop = dAop (mop otimes idAorop) (idAop otimes bAop) (326)

where β isin C Composing with ηop from the right in turn implies that βεop ηop =dim Aop which is nonzero by assumption Thus also β = 0 and εop is a nonzero multipleof the morphism on the right hand side of (326) By [FRS Lem 311] Aop is special

Since Aop is a special Frobenius algebra Aop is semi-simple as an Aop-bimodule(apply [FS Prop 524] to Aop tensored with its opposite algebra) Suppose Aop is not

simple so that we can write Aop = A(1)op oplus A(2)op for nonzero Aop-bimodules A(1)op and A(2)op We denote the canonical embeddings and projections associated to this decompositionas ι12 and π12 We have the identities

mop (ι1 otimes ι2) = 0 εop ηop =2sum

i=1

εop ιi πi ηop (327)

The first identity follows since π1 mop (ι1 otimes ι2) = 0 (as mop gives the left action of

Aop on Aop and hence it preserves A(2)op ) and similarly π2 mop (ι1 otimes ι2) = 0 Thesecond identity is just the completeness of ι12 π12

Since εop ηop = 0 without losing generality we can assume εop ι1 π1 ηop = 0Using that π2 is a bimodule map we compute

π2 [LHS of (322)]Ui =1 ι1 π1 ηop = π2 [RHS of (322)]Ui =1 ι1 π1 ηop

= PlA(2)op

π2 ι1 π1 ηop = 0 (328)

900 L Kong I Runkel

On the other hand using that Acl is haploid that ιcl-op is an algebra map and thatι lowastcl-op is a coalgebra map one can check that the left-hand side of (328) is equal toλ(εop ι1 π1 ηop) π2 ηop for some λ = 0 This implies that π2 ηop = 0 Thusηop = sum

i ιi πi ηop = ι1 π1 ηop Hence we have

0 = π2 ι2 = π2 mop (ηop otimes ι2) = π2 mop ((ι1 π1 ηop)otimes ι2) (329)

However the right-hand side is zero by (327) This is a contradiction and hence Aopmust be simple

To formulate the next theorem we need the notion of the full centre of an algebra [FjDef 49] Recall that an algebra A in a braided tensor category has a left centre and aright centre [VZO] both of which are sub-algebras of A Of these two we will onlyneed the left centre The following definition is [Fr Def 231] which in our setting isequivalent to that of [VZO]

Definition 317 Let A be a symmetric special Frobenius algebra such that m A A =ζA idA

(i) The left centre Cl(A) of A is the image of the idempotent ζminus1A Pl

A(ii) The full centre Z(A) is Cl(R(A))

That ζminus1A Pl

A is an idempotent follows from [FRS Lem 52] when keeping track ofthe factors ζA ([FRS] assumes normalised-special ie ζA = 1) Note that Cl(A) is againan object of C while Z(A) is an object of C2plusmn Let el

A Cl(A) rarr A be the embeddingof Cl(A) into A The left centre is in fact the maximal subobject of A such that

m A cAA (elA otimes idA) = m A (330)

see [Fr Lem 232] This observation explains the name left centre and also makes theconnection to [O Def 15]

The full centre is by definition the image of the idempotent ζminus1A Pl

R(A) R(A) rarrR(A) Since C2plusmn is abelian the idempotent splits and we obtain the embedding andrestriction morphisms

e Z(A) rarr R(A) and r R(A) Z(A) (331)

which obey r e = idZ(A) and e r = ζminus1A Pl

R(A) It follows from Proposition 225 and

[Fr Prop 237] that Z(A) is a commutative symmetric Frobenius algebra in C2plusmn withstructure morphisms5

m Z(A) = r m R(A) (e otimes e) ηZ(A) = r ηR(A)

Z(A) = ζA middot (r otimes r) R(A) e εZ(A) = ζminus1A middot εR(A) e

(332)

Moreover if A is simple then Z(A) is simple and if A is simple and dim A = 0 thenZ(A) is simple and special The normalisation of the counit is such that

εZ(A) ηZ(A) = ζminus2A dim A DimC (333)

5 The normalisation of product and unit is the standard one The factors in the coproduct and counit haveto be included in order for (A|Z(A) e) to be a Cardy algebra see Theorem 318 below The normalisation ofthe counit enters the Cardy condition (314) through the definition of ( middot )lowast

Cardy Algebras and Sewing Constraints I 901

Theorem 318 Let A be a special symmetric Frobenius C-algebra Then (A|Z(A) e)is a Cardy C|C2plusmn-algebra

The proof of this theorem makes use the following two lemmas

Lemma 319 e Z(A) rarr R(A) is an algebra map and elowast = ζA middot r

Proof It follows from [Fr Lem 229] (or by direct calculation using in particularm R(A) R(A) = ζAidR(A)) that

m R(A) (e otimes e) = ζminus1A middot Pl

R(A) m R(A) (e otimes e) (334)

Substituting e r = ζminus1A Pl

R(A) shows that e is compatible with multiplication For theunit one finds

e ηZ(A) = e r ηR(A) = ζminus1A Pl

R(A) ηR(A) = ηR(A) (335)

Thus e is an algebra map For the second statement one computes

elowast (1)= ζA

R(A)

r

R(A)

R(A)

Z(A)

R(A)

Z(A)

r

e

e

(2)=

R(A)

r

R(A)

R(A)

R(A)

Z(A)

(3)=

R(A)

r

R(A)

Z(A)

(4)= ζA middot r (336)

where in (1) the definitions (230) and (332) have been substituted step (2) is e r = ζminus1

A PlR(A) step (3) uses that R(A) is symmetric Frobenius and step (4) is again

e r = ζminus1A Pl

R(A) Lemma 320 Let A be a symmetric Frobenius algebra in C The morphism

PlR(A) m R(A)

(Pl

R(A) otimes PlR(A)

) R(A)otimes R(A) minusrarr R(A) (337)

is S-invariant

The proof of this lemma is a slightly lengthy explicit calculation and has been deferredto Appendix A2

902 L Kong I Runkel

Proof of Theorem 318 That e is an algebra map was proved in Lemma 319 The centrecondition (313) holds by property (330) of the left centre The Cardy condition (314)also is an immediate consequence of Lemma 319

ιcl-op ιlowastcl-op = e (ζA r) = PlR(A) (338)

The full centre Z(A) is a commutative symmetric Frobenius algebra It remains to provemodular invariance That θZ(A) = idZ(A) is implied by commutativity and symmetryof Z(A) [Fr Prop 225] The S-invariance condition (33) follows from Lemma 320In (337) substitute Pl

R(A) = ζA e r and then put the resulting morphism into (33)Compose the resulting equation with e otimes idW from the right (ie from the bottom) andsubstitute the definition (332) of m Z(A) This results in the statement that m Z(A) isS-invariant

The following theorem is analogous to [LR Prop 29] and [Fj Thm 426] whichroughly speaking answer the question under which circumstances the restriction of atwo-dimensional conformal field theory to the boundary already determines the entireconformal field theory The first work is set in Minkowski space and uses operator alge-bras and subfactors while the second work is set in Euclidean space and uses modulartensor categories

Theorem 321 Let (A|Acl ιcl-op) be a Cardy C|C2plusmn-algebra such that dim A = 0 andAcl is simple Then A is special and (A|Acl ιcl-op) sim= (A|Z(A) e) as Cardy algebras

Proof By Proposition 316 A is simple and special Since Acl is simple the algebramap ιcl-op Acl rarr R(A) is either zero or a monomorphism But ιcl-op ηcl = ηR(A)and so ιcl-op = 0 Thus ιcl-op is monic By Lemma 218 (ii) ζminus1

A ιlowastcl-op is epi The Cardycondition (314) implies

ιcl-op ζminus1A ιlowastcl-op = ζminus1

A PlR(A) = e r (339)

Composing this with e r from the left yields e r ιcl-op ζminus1A ιlowastcl-op = e r =

ιcl-op ζminus1A ιlowastcl-op Since ζminus1

A ιlowastcl-op is epi we have

e r ιcl-op = ιcl-op (340)

Actually (340) also follows from (313) and specialness of R(A) We will prove that

( fop fcl) (A|Acl ιcl-op) minusrarr (A|Z(A) e) where fop = idA fcl = r ιcl-op

(341)

is an isomorphism of Cardy algebrasfcl is an algebra map Compatibility with the units follows since ιcl-op is an algebramap

fcl ηcl = r ιcl-op ηcl = r ηR(A) = ηZ(A) (342)

Compatibility with the multiplication also follows since ιcl-op is an algebra map

m Z(A) ( fcl otimes fcl) = r m R(A) (e otimes e) (r otimes r) (ιcl-op otimes ιcl-op)

= r m R(A) (ιcl-op otimes ιcl-op) = r ιcl-op mcl = fcl mcl (343)

where in the second step we used (340)

Cardy Algebras and Sewing Constraints I 903

fcl is an isomorphism As above since fcl is an algebra map and since Acl is simple fcl

has to be monic By Lemma 319 rlowast = ζminus1A e Thus f lowast

cl = ιlowastcl-op rlowast = ζminus1A ιlowastcl-op e and

fcl f lowastcl = r ιcl-op ζminus1

A ιlowastcl-op e = r e r e = idZ(A) (344)

and so fcl is also epi and hence isofcl is a coalgebra map Since fcl is an algebra map so is f minus1

cl By (344) f minus1cl = f lowast

cland by Lemma 218 (iii) this implies that fcl is a also coalgebra mapThe diagram (315) commutes Commutativity of (315) is equivalent to e fcl = ιcl-opwhich holds by (340)

Let A be a special symmetric Frobenius algebra So far we have seen that (A|Z(A) e)is a Cardy algebra and that all Cardy algebras with Aop = A and simple Acl are of thisform It is now natural to ask if every simple Acl does occur as part of a Cardy algebraThe following theorem provides an affirmative answer Recall that for an A-left moduleM the object Mor otimesA M is an algebra (see eg [KR Lem 42])

Theorem 322 If Acl is a simple modular invariant commutative symmetric FrobeniusC2plusmn-algebra then there exist a simple special symmetric Frobenius C-algebra A and amorphism ιcl-op Acl rarr R(A) such that

(i) Acl sim= Z(A) as Frobenius algebras(ii) (A|Acl ιcl-op) is a Cardy C|C2plusmn-algebra

(iii) T (Acl) sim= oplusκisinJ Morκ otimesA Mκ as algebras where MκκisinJ is a set of representa-

tives of the isomorphism classes of simple A-left modules

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis haploid It then follows from Theorem 34 that Acl is special By Proposition 224T (Acl) is a special symmetric Frobenius algebra in C Thus T (Acl) = oplusi Ai where theAi are simple symmetric Frobenius algebras We will show that at least one of the Aiis special Since T (Acl) is special we have mT (Acl) T (Acl) = ζ idT (Acl) for someζ isin C

times Restricting this to the summand Ai shows mi i = ζ idAi FurthermoreεT (Acl) ηT (Acl) = ξ id1 for some ξ isin C

times But εT (Acl) ηT (Acl) = sumi εi ηi and so

at least one of the εi ηi has to be nonzero Therefore at least one of the Ai is speciallet A equiv Ai be this summand We denote the embedding A rarr T (Acl) by e0 and therestriction T (Acl) A by r0 Notice that r0 is an algebra homomorphism Define

ιcl-op = χ (r0) Acl minusrarr R(A) (345)

By Proposition 216 ιcl-op is an algebra homomorphism Next we verify the centre con-dition (313) or rather its equivalent form (320) By substituting the definitions onecan convince oneself that the commutativity mcl cAclAcl = mcl of Acl in C2plusmn implies thecondition mT (Acl) = mT (Acl) in C see [K2 Prop 36] Here T (Acl)otimes T (Acl) rarrT (Acl)otimes T (Acl) is given by

=oplusmn

(idCl

notimes cCl

m Crnotimes idCr

m

)

(cCl

m Clnotimes cminus1

Crn C

rm

)

(idCl

motimes cminus1

Cln C

rm

otimes idCrn

)

(346)

and we decomposed Acl as Acl = oplusnCln times Cr

n As a consequence we obtain the identity

r0 mT (Acl) (idAcl otimes e0) = r0 mT (Acl) (idAcl otimes e0) (347)

904 L Kong I Runkel

Using that r0 is an algebra map and that by definition ιcl-op = r0 we obtain (320)In order to show that (A|Acl ιcl-op) is a Cardy algebra it remains to show that the

Cardy condition (314) is satisfied We will demonstrate this via a detour by first provingthat Acl sim= Z(A) as Frobenius algebras

Recall the notations e and r given in (331) Using the centre condition (320) onecan check that Pl

R(A) ιcl-op = m R(A) R(A) ιcl-op By specialness of A we have

m A A = ζA idA and so together with e r = ζminus1A Pl

R(A) we get

e r ιcl-op = ιcl-op (348)

Next consider the morphism

fcl = r ιcl-op Acl minusrarr Z(A) (349)

By the same derivation as in (342) and (343) one sees that fcl is an algebra map Inparticular fcl ηcl = ηZ(A) = 0 and so fcl = 0 Since Acl is simple fcl has to be amonomorphism

By the same argument as used in the proof of Theorem 34 (ii) up to isomorphismAcl is the unique simple local Acl-(left-)module The algebra monomorphism fcl turnsZ(A) into an Acl-module Since Z(A) is commutative it is local as an Acl-module andso Z(A) sim= AoplusN

cl for some N ge 1 By construction A is a simple special symmetricFrobenius algebra Proposition 225 and Corollary 227 show that R(A) inherits all theseproperties and thus Z(A) is simple (see the comment below Eq (332)) By Theorem318 Z(A) is modular invariant and then by Theorem 34 (i) dim Z(A) = Dim C Thisimplies that N = 1 in Z(A) sim= AoplusN

cl and so fcl is in fact an isomorphismSince Acl and Z(A) are both haploid we have εZ(A) fcl = ξ εcl for some ξ isin C

timesThe counit uniquely determines the Frobenius structure on Acl and Z(A) (see eg [FRSLemma 37]) so that fcl is a coalgebra isomorphism iff ξ = 1 To compute ξ we composethe above identity with ηcl from the right Defining ζcl via εcl ηcl = ζminus1

cl Dim C middot id1 andusing (333) gives ξ = dim A ζclζ

2A By rescaling the comultiplication and the counit

of A and consequently changing ζA we can always achieve ξ = 1 This proves part (i)of the theorem

Equation (348) implies that ιcl-op = e fcl Since fcl is an isomorphism of Frobeniusalgebras by Lemmas 218 and 319 we have

ιcl-op ιlowastcl-op = e fcl f lowastcl elowast = ζA e r = Pl

R(A) (350)

Thus (A|Acl ιcl-op) is a Cardy algebra This proves part (ii) of the theoremPart (iii) can be seen as follows By [KR Prop 43] T Z(A) sim= oplusκisinJ Mor

κ otimesA Mκ asalgebras Together with the observation that T ( fcl) T (Acl) rarr T Z(A) is an isomor-phism of algebras this proves part (iii) Remark 323 Part (i) of Theorem 322 was announced by Muumlger [Muuml2] We provide anindependent proof in the setting of Cardy algebras

The above theorem together with Lemma 211 and Theorem 34 shows that a sim-ple commutative symmetric Frobenius C2plusmn-algebra Acl with dim Acl = Dim C is alwayspart of a Cardy algebra (Aop|Acl ιcl-op) for some simple special symmetric Frobeniusalgebra Aop in C However the above proof also illustrates that Aop is not unique Thisraises the question how two Cardy algebras with a given Acl can differ This question isanswered by [KR Thm 11] which in the present framework can be restated as follows

Cardy Algebras and Sewing Constraints I 905

Theorem 324 If (A(i)op |A(i)cl ι(i)cl-op) i = 1 2 are two Cardy C|C2plusmn-algebras such that

A(i)cl is simple and dim A(i)op = 0 for i = 1 2 then A(1)clsim= A(2)cl as algebras if and only

if A(1)op and A(2)op are Morita equivalent

Proof Theorem 11 in [KR] is stated for A(i)op being non-degenerate algebras and A(i)cl =Z(A(i)op ) for i = 1 2 By Proposition 316 A(i)op are simple and special for i = 1 2

Then by [KR Lem 21] A(i)op are non-degenerate algebras By Theorem 321 we have

A(i)clsim= Z(A(i)op ) as Frobenius algebras Finally by [KR Thm 11] Z(A(1)op ) sim= Z(A(2)op )

as algebras iff A(1)op and A(2)op are Morita equivalent Let Cmax(C2plusmn) be the set of equivalence classes [B] of simple modular invariant

commutative symmetric Frobenius algebras B in C2plusmn Two such algebras B and B prime areequivalent if B and B prime are isomorphic as algebras (but not necessarily as Frobenius alge-bras) Let Msimp(C) be the set of Morita classes of simple special symmetric Frobeniusalgebras in C Define the map z Msimp(C) rarr Cmax(C2plusmn) by z A rarr [Z(A)] whereA denotes the Morita class of A From Theorem 322 (i) and [KR Thm 11] we learn

Corollary 325 The map z Msimp(C) rarr Cmax(C2plusmn) is a bijection

A Appendix

A1 Proof of Lemma 27 We will show that if (F ψ F2 ψ

F0 ) is a colax tensor functor

from C1 to C2 then (G φG2 φ

G0 ) is a lax tensor functor from C2 to C1 Applying this

result to the opposed categories then gives the converse statementWe need to show that φG

0 and φG2 make the diagrams (21) and (22) commute We

first prove the commutativity of (21) Consider the following diagram

G(A)otimes (G(B)otimes G(C))

GψF2 δ

idG(A)otimesφG2 G(A)otimes G(B otimes C)

GψF2 δ

G(FG(A)otimes F(G(B)otimes G(C)))

G(F(idG(A))otimesF(φG2 ))

G(idFG(A)otimesψ F2 )

G(FG(A)otimes FG(B otimes C))

G(ρAotimesρBotimesC )

G(FG(A)otimes (FG(B)otimes FG(C)))

G(ρAotimes(ρBotimesρC )) G(A otimes (B otimes C))

(A1)

The top subdiagram is commutative because of the naturality of Gψ F2 δ The commu-

tativity of the bottom subdiagram follows from the following identities

(ρB otimes ρC ) ψ F2 = (ρB otimes ρC ) ψ F

2 ρF Fδ

= ρBotimesC FG(ρB otimes ρC ) FG(ψ F2 ) Fδ

= ρBotimesC F(φG2 ) (A2)

as a map F(G(B) otimes G(C)) rarr B otimes C The commutativity of (A1) implies that thecomposition of maps in the left column in (21) can be replaced by

G(ρA otimes (ρB otimes ρC )) G((idFG(A) otimes ψ F

2 ) ψ F2

) δ (A3)

906 L Kong I Runkel

Similarly we can show that the composition of maps in the right column in (21) can bereplaced by

G((ρA otimes ρB)otimes ρC ) G((ψ F

2 otimes idFG(C)) ψ F2

) δ (A4)

Using the commutativity of (23) it is easy to see that (21) with the left and right col-umns of (21) replaced by (A3) and (A4) respectively is commutative Hence (21) iscommutative

Now we prove the commutativity of the first diagram in (22)

φG2 (φG

0 otimes idG(A))

(1)= G(ρ12 otimes ρA) Gψ F2 δ

[(Gψ F

0 δ11)otimes idG(A)

]

(2)= G(ρ12 otimes ρA) Gψ F2 G F(Gψ0 otimes idG(A)) G F(δ11 otimes idG(A)) δ

(3)= G(ρ12 otimes ρA) G(FG(ψ F0 )otimes idFG(A)) Gψ F

2 G F(δ11 otimes idG(A)) δ(4)= G(id12 otimes ρA) G

([ψ F

0 ρF(11) (Fδ)11] otimes idFG(A)

) Gψ F

2 δ(5)= G(id12 otimes ρA) G(ψ F

0 otimes idFG(A)) Gψ F2 δ

(6)= G(id12 otimes ρA) G(lminus1FG(A)) G F(lG(A)) δ

(7)= G(lminus1A ) GρA δG lG(A)

(8)= G(lminus1A ) lG(A) (A5)

where in step (1) we substituted the definition of φG0 φ

G2 given in (210) in step (2)

we used the naturality of δ in step (3) we used the naturality of Gψ F2 in step (4) we

switched the position between Gρ12 and G FG(ψ F0 ) and the position between Gψ2 and

G F(δ12 otimes idG(A)) using the naturality of ρ and FψG2 respectively in step (5) we applied

the second identity in (28) in step (6) we used (24) in step (7) we used the naturalityof lminus1 and δ in step (8) we used the first identity in (28)

The proof of the commutativity of the second diagram in (22) is similar Thus wehave shown that G is a lax tensor functor

A2 Proof of Lemma 320 To prepare the proof recall that for a given object B isin Cthe modular group P SL(2Z) acts on the space oplusi HomC(B otimes Ui Ui ) see eg [BKSect 31] and [K3 Eq (455)] We will only need the action of S and Sminus1 Let f isinoplusi HomC(B otimes Ui Ui ) Then

S oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A6)

Cardy Algebras and Sewing Constraints I 907

Sminus1 oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A7)

By Lemma 32 to establish that (337) is S-invariant it is enough to prove the identity(34) when f is given by (337) Using (A6) and (A7) we can see that Eq (34) simplysays that oplusi j [RHS of (34)] is invariant under the action of S times S Consider the elementg of oplus jkisinIHomC2plusmn(R(A)otimes (Uor

j times Uk)Uorj times Uk) given by

g =oplusjkisinI

sumα

R(A)

R(A)

R(A)

R(A)

Uorj times Uk

Uorj times Uk

α

α

(A8)

By the above arguments proving S-invariance of (337) is equivalent to proving invari-ance of g under the action of S times S

For i isin I we denote by gi the component of g in(oplus jisinIHomC(A otimes Uor

i otimes Uorj U

orj )

)otimes (opluskisinIHomC(Ui otimes UkUk))

We view the second Hom-space in the above tensor product as a Hom-space in C+ insteadof Cminus It is enough to show that gi is invariant under the action of S times Sminus1 Note thatthe action of Sminus1 in C+ is equivalent to that of S in Cminus

The morphism gi can be canonically identified with a bilinear pairing

( middot middot )i ⎛⎝oplus

jisinIHomC(Uor

j A otimes Uori otimes Uor

j )

⎞⎠

times(oplus

kisinIHomC(UkUi otimes Uk)

)minusrarr C (A9)

as follows For h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) we set

(h1 h2)i = (dim U j dim Uk)minus1 trUor

j timesUk[gi (h1 times h2)] (A10)

908 L Kong I Runkel

When substituting the explicit form of the product m R(A) of R(A) = (A times 1) otimes R(1)after a short calculation one finds

(h1 h2)i =sumα

1

dim U j

A

A

A

A

A A

A α

α

h1

h2

U jUk

Uorj

Ui

Uk (A11)

Here the top morphism PlR(A) has been simplified with the help of the identity

PR(A) m R(A) (PR(A) otimes PR(A))

=(oplus

iisinI((m A A)otimes idUor

i)times idUi

) m R(A) (PR(A) otimes PR(A)) (A12)

which can be checked by direct calculation along the same lines as in the proof of [FrLem 310]

The action of the modular transformation S on oplusiisinI(B otimes Ui Ui ) for B isin C nat-urally induces an action on oplusiisinI(Ui B otimes Ui ) [K3 Prop 514] which we denote bySlowast In the present case we get an action of Slowast on oplus jisinIHomC(Uor

j A otimes Uori otimes Uor

j ) and

opluskisinIHomC(UkUi otimes Uk) Then to show gi is invariant under the action of S times Sminus1

amounts to showing that

(h1 h2)i = ((Sminus1)lowast h1 Slowast h2)i (A13)

Cardy Algebras and Sewing Constraints I 909

for all h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) We have

((Sminus1)lowast h1 Slowast h2)i =sum

mnα

dim Un

DimC

A

A

A

A

A A

A

Uk

α

α

h1

h2

Ui

U j

UmUn

(A14)

Now drag the upper vertex indexed by α in the above graph along its Uorm -leg until it

meets the lower vertex also indexed by α then sum over α and m This gives

((Sminus1)lowast h1 Slowast h2)i =sum

n

dim Un

DimC

A

A

A

A

A A

A

Uk

Un

h1

h2

Ui

U j

(A15)

910 L Kong I Runkel

If we just look at the neighbourhood of the Un-loop in the above graph we see thefollowing subgraphs

sumn

dim Un

DimC

A Aor

Uorj U j

Uk Uork

Un=

sumα

1

dim U j

A Aor Uk Uork

Uorj U j

αα

(A16)

where we have applied [BK Cor 3111] Substituting this subgraph back to the originalgraph in (A15) we obtain

((Sminus1)lowast h1 Slowast h2)i =sumα

1

dim U j

A

A

A

A

A A

A

A

A

Uorj

α

α

h1

h2m A

Ui

U jUk

Uk

(A17)

The graph in (A17) is equal to that in (A11) In order to see this we first drag theldquobubblerdquo (m A A) along A lines and through the m A vertex (because m A A is abimodule map) until it reaches the lower-left leg of the upper vertex indexed by α Thendrag the m A vertex along the (red) dotted line in the above graph Finally we applythe associativity of A (A12) and [Fr Lem 311] Then we see that the graph in (A17)exactly matches with the one in (A11)

Acknowledgements We would like to thank the organisers of the Oberwolfach Arbeitsgemeinschaft ldquoAlge-braic structures in conformal field theoriesrdquo (April 2007) where this work was started for an inspiring meetingWe would further like to thank the Hausdorff Institute for Mathematics in Bonn and the organisers of the stim-ulating meeting ldquoGeometry and Physicsrdquo (May 2008) We are indebted to Alexei Davydov Jens FjelstadJuumlrgen Fuchs Yi-Zhi Huang Alexei Kitaev Urs Schreiber Christoph Schweigert Stephan Stolz and PeterTeichner for helpful discussions andor comments on a draft of this paper The research of IR was partiallysupported by the EPSRC First Grant EPE0050471 the PPARC rolling grant PPC5071451 and the MarieCurie network lsquoSuperstring Theoryrsquo (MRTN-CT-2004-512194)

Cardy Algebras and Sewing Constraints I 911

References

[AN] Alexeevski A Natanzon SM Noncommutative two-dimensional topological field theories andhurwitz numbers for real algebraic curves Sel Math New Ser 12 307ndash377 (2006)

[Bi] Bichon J Cosovereign hopf algebras J Pure Appl Alg 157 121ndash133 (2001)[BK] Bakalov B Kirillov AA Lectures on Tensor Categories and Modular Functors Providence

RI Amer Math Soc 2001[C1] Cardy JL Operator content of two-dimensional conformal invariant theories Nucl Phys

B 270 186ndash204 (1986)[C2] Cardy JL Boundary conditions fusion rules and the verlinde formula Nucl Phys B 324

581ndash596 (1989)[CL] Cardy JL Lewellen DC Bulk and boundary operators in conformal field theory Phys Lett

B 259 274ndash278 (1991)[DMZ] Dong C-Y Mason G Zhu Y-C Discrete series of the Virasoro algebra and the moonshine

module In Algebraic Groups and Their Generalizations Quantum and infinite-dimensional Meth-ods Proc Symp Pure Math 56 Part 2 Providence RI Amer MathSoc 1994 pp 295ndash316

[DP] Day B Pastro C Note on frobenius monoidal functors New York J Math 14 733ndash742 (2008)[ENO1] Etingof PI Nikshych D Ostrik V On fusion categories Ann Math 162 581ndash642 (2005)[ENO2] Etingof PI Nikshych D Ostrik V An analogue of radfordrsquos s4 formula for finite tensor

categories Int Math Research Notices 54 2915ndash2933 (2004)[Fe] Felder G Froumlhlich J Fuchs J Schweigert C Correlation functions and boundary conditions in

rational conformal field theory and three-dimensional topology Comp Math 131 189ndash237 (2002)[FHL] Frenkel IB Huang Y-Z Lepowsky J On axiomatic approaches to vertex operator algebras

and modules Mem Amer Math Soc 104 (1993)[Fj] Fjelstad J Fuchs J Runkel I Schweigert C Uniqueness of openclosed rational cft with

given algebra of open states Adv Theor Math Phys 12 1283ndash1375 (2008)[FK] Froumlhlich J King C The chern-simons theory and knot polynomials Commun Math

Phys 126 167ndash199 (1989)[Fr] Froumlhlich J Fuchs J Runkel I Schweigert C Correspondences of ribbon categories Adv

Math 199 192ndash329 (2006)[FRS] Fuchs J Runkel I Schweigert C Tft construction of rcft correlators I partition functions Nucl

Phys B 646 353ndash497 (2002)[FS] Fuchs J Schweigert C Category theory for conformal boundary conditions Fields Inst Com-

mun 39 25ndash71 (2003)[H1] Huang Y-Z Two-dimensional conformal geometry and vertex operator algebras Progress in

Mathematics Vol 148 Boston Birkhaumluser 1997[H2] Huang Y-Z Rigidity and modularity of vertex tensor categories Comm Contemp Math 10

871ndash911 (2008)[HK1] Huang Y-Z Kong L Open-string vertex algebra category and operads Commun Math

Phys 250 433ndash471 (2004)[HK2] Huang Y-Z Kong L Full field algebras Commun Math Phys 272 345ndash396 (2007)[HK3] Huang Y-Z Kong L Modular invariance for conformal full field algebras httparxivorgabs

math0609570v2[mathQA] 2006[HL] Huang Y-Z Lepowsky J Tensor products of modules for a vertex operator algebra and vertex

tensor categories In Lie Theory and Geometry in honor of Bertram Kostant ed R BrylinskiJ-L Brylinski V Guillemin V Kac Boston Birkhaumluser 1994 pp 349ndash383

[JS] Joyal A Street R Braided tensor categories Adv Math 102 20ndash78 (1993)[K1] Kong L Full field algebras operads and tensor categories Adv Math 213 271ndash340 (2007)[K2] Kong L Open-closed field algebras Commun Math Phys 280 207ndash261 (2008)[K3] Kong L Cardy condition for open-closed field algebras Commun Math Phys 283 25ndash92 (2008)[Ki] Kitaev A Private communication[KO] Kirillov AA Ostrik V On q-analog of mckay correspondence and ade classification of sl(2)

conformal field theories Adv Math 171 183ndash227 (2002)[KR] Kong L Runkel I Morita classes of algebras in modular tensor categories Adv Math 219

1548ndash1576 (2008)[Ld] Lauda AD Frobenius algebras and ambidextrous adjunctions Theo Appl Cat 16 84ndash

122 (2006)[Li1] Li H-S Regular representations of vertex operator algebras Commun Contemp Math 4

639ndash683 (2002)[Li2] Li H-S Regular representations and huang-lepowsky tensor functors for vertex operator

algebras J Alge 255 423ndash462 (2002)

880 L Kong I Runkel

(i) A is simple(ii) A is absolutely simple

(iii) A is haploid

Proof (ii)hArr(iii) A is haploid iff it is absolutely simple as a left module over itself[FS Eq (417)] Furthermore for a commutative algebra we have HomA(A A) =HomA|A(A A) and so A is haploid iff it is absolutely simple(i)rArr(ii) If A is simple then every nonzero element of HomA|A(A A) is invertibleHence this space forms a division algebra over C and is therefore isomorphic to C(iii)rArr(i) Since C is semi-simple and A is haploid also Hom(A 1) is one-dimensionalThe counit ε is a nonzero element in this space and so gives a basis This implies firstlythat ε η = 0 and secondly that there is a constant β isin C such that

β middot ε = dA (idAor otimes m) (bA otimes idA) (216)

Composing with η from the right yields β εη = diml A The right-hand side is nonzeroand so β = 0 By [FRS Lem 311] A is special We have already proved (ii)hArr(iii)and so A is absolutely simple A special Frobenius algebra in a semi-simple categoryhas a semi-simple category of bimodules (apply [FS Prop 524] to the algebra tensoredwith its opposite algebra) For semi-simple C-linear categories simple and absolutelysimple are equivalent2 Thus A is simple Remark 212 For a Frobenius algebra A the morphisms (215) are invertible and henceA sim= Aor In this case one has diml A = dimr A [FS Rem 363] and so we could havestated the above lemma equivalently with the condition dimr A = 0

Let F C1 rarr C2 be a lax tensor functor between two tensor categories C1 C2 and

let (Am A ηA) be an algebra in C1 Define morphisms F(A)otimes F(A)m F(A)minusminusminusrarr F(A) and

12ηF(A)minusminusminusrarr F(A) as

m F(A) = F(m A) φF2 ηF(A) = F(ηA) φF

0 (217)

Then (F(A)m F(A) ηF(A)) is an algebra in C2 [JS Prop 55] If f A rarr B is analgebra homomorphism between two algebras A B isin C1 then F( f ) F(A) rarr F(B)is also an algebra homomorphism If (MmM ) is a left (or right) A-module in C1 then(F(M) F(mM )φF

2 ) is a left (or right) F(A)-module if M has a A-bimodule structurethen F(M) naturally has a F(A)-bimodule structure

Similarly if (AA εA) is a coalgebra in C1 and F C1 rarr C2 is a colax tensor func-

tor then F(A) with coproduct F(A)F(A)minusminusminusrarr F(A) otimes F(A) and counit F(A)

εF(A)minusminusminusrarr 12given by

F(A) = ψ F2 F(A) εF(A) = ψ F

0 F(εA) (218)

is a coalgebra in C2 If f A rarr B is a coalgebra homomorphism between two coalgebrasA B isin C1 then F( f ) F(A) rarr F(B) is also a coalgebra homomorphism

2 To see this note that if U is simple then the C-vector space Hom(UU ) is a division algebra and henceHom(UU ) = C idU Conversely if U is not simple then U = U1 oplus U2 and Hom(UU ) contains at leasttwo linearly independent elements namely idU1 and idU2

Cardy Algebras and Sewing Constraints I 881

Proposition 2133 If F C1 rarr C2 is a Frobenius functor and (Am A ηAA εA) aFrobenius algebra in C1 then (F(A)m F(A) ηF(A)F(A) εF(A)) is a Frobenius alge-bra in C2

Proof One Frobenius property (m F(A)otimes idF(A)) (idF(A)otimesF(A)) = F(A) m F(A)follows from the commutativity of the following diagram (we spell out the associativityisomorphisms)

F(A)otimes F(A)idF(A)otimesF(A)

φF2

F(A)otimes F(A otimes A)idF(A)otimesψF

2

φF2

F(A)otimes (F(A)otimes F(A))

αF(A)F(A)F(A)

F(A otimes A)

F(idAotimesA)

F(m A)

F(A otimes (A otimes A))

F(αAAA)

(F(A)otimes F(A))otimes F(A)

φF2 otimesidF(A)

F((A otimes A)otimes A)

F(m AotimesidA)

ψF2 F(A otimes A)otimes F(A)

F(m A)otimesidF(A)

F(A)

F(A) F(A otimes A)ψF

2 F(A)otimes F(A)

(219)

The commutativity of the upper-left subdiagram follows from the naturalness of φF2 that

of the upper-right subdiagram follows from (25) that of the lower-left subdiagram fol-lows from the Frobenius properties of A and that of the lower-right subdiagram followsfrom the naturalness of ψ F

2 The proof of the other Frobenius property is similar Proposition 214 If F C1 rarr C2 is a tensor functor and A a Frobenius algebra in C1then

(i) F(A) has a natural structure of Frobenius algebra as given in Proposition 213(ii) If A is (normalised-)special so is F(A)

Proof Part (i) follows from Propositions 25 and 213 Part (ii) is a straightforwardverification of the definition using ψ F

2 = (φF2 )

minus1 and ψ F0 = (φF

0 )minus1

Let C1 C2 be sovereign tensor categories and F C1 rarr C2 a Frobenius functor Wedefine two morphisms IF(Aor) I prime

F(Aor) F(Aor) rarr F(A)or for a Frobenius algebra A inC1 as follows

IF(Aor) = ((ψ F0 F(dA) φF

2 )otimes idF(A)or) (idF(Aor) otimes bF(A))

I primeF(Aor) = (idF(A)or otimes (ψ F

0 F(dA) φF2 )) (bF(A) otimes idF(Aor))

(220)

It is easy to see that these are isomorphisms

Lemma 215 If F C1 rarr C2 is a Frobenius functor and A a Frobenius algebra in C1then

F(A) = IF(Aor) F(A) primeF(A) = I prime

F(Aor) F(primeA) (221)

3 After the preprint of the present paper appeared we noticed that this proposition is also proved in[DP Cor 5]

882 L Kong I Runkel

Proof We only prove the first equality the second one can be seen in the same way Bydefinition we have

IF(Aor) F(A)

= ((ψ F0 F(dA) φF

2 )otimes idF(A)or) (idF(Aor) otimes bF(A)) F(A)

=[(ψ F

0 F(dA) φF2 ) (F(A)otimes idF(A))

]otimes idF(A)or

(idF(A) otimes bF(A))

For the term inside the square brackets we find

ψ F0 F(dA) φF

2 (F(A)otimes idF(A)) = ψ F0 F(dA) F(A otimes idA) φF

2

= ψ F0 F(dA (A otimes idA)) φF

2 = ψ F0 F(εA m A) φF

2 (222)

On the other hand by definitionF(A) = [((ψ F0 F(εA) (F(m A)φF

2 ))otimes idF(A)or] (idF(A) otimes bF(A)) This demonstrates the first equality in (221) Proposition 216 Let F C1 rarr C2 be a tensor functor G C2 rarr C1 a functor〈FG χ〉 an adjunction A a C1-algebra and B a C2-algebra Then f A rarr G(B)is an algebra homomorphism if and only if f = χminus1( f ) F(A) rarr B is an algebrahomomorphism

Proof We need to show that

mG(B) ( f otimes f ) = f m A and f ηA = ηG(B) (223)

is equivalent to

m B ( f otimes f ) = f m F(A) and f ηF(A) = ηBs (224)

We first prove that the first identity in (223) is equivalent to the first identity in (224)For the left-hand side of the first identity in (223) we have the following equalities

mG(B) ( f otimes f )(1)= G(m B) φG

2 ( f otimes f )(2)= G(m B) G(ρ otimes ρ) G(ψ F

2 ) δ ( f otimes f )(3)= G(m B) G(ρ otimes ρ) G(ψ F

2 ) G F( f otimes f ) δ(4)= G(m B) G(ρ otimes ρ) G(F( f )otimes F( f )) G(ψ F

2 ) δ(5)= G

(m B (ρ otimes ρ) (F( f )otimes F( f )) ψ F

2

) δ

(6)= χ(

m B (ρ otimes ρ) (F( f )otimes F( f )) ψ F2

) (225)

where (1) is the definition of mG(B) in (217) (2) is the second identity in (210) (3)and (4) are naturality of δ and ψ F

2 respectively step (5) is functoriality of G and finallystep (6) is (29) For the right hand side of the first identity in (223) we get

f m A(1)= Gρ δG ( f m A)

(2)= Gρ G F( f m A) δ(3)= G(ρ F( f m A)) δ(4)= χ(ρ F( f ) F(m A)) (226)

Cardy Algebras and Sewing Constraints I 883

where (1) is the adjunction property (28) (2) is naturality of δ (3) functoriality of Gand (4) amounts to (29) and functoriality of F

On the other hand we see that the first equality in (224) is equivalent to

m B (ρ otimes ρ) (F( f )otimes F( f )) = ρ F( f ) F(m A) φF2 (227)

Using that φF2 is invertible with inverse (φF

2 )minus1 = ψ F

2 and that χ is an isomorphismit follows that the statement that (225) is equal to (226) is equivalent to the identity(227)

Now we prove that the second identity in (223) is equivalent to the second identity in(224) Using (217) and (210) we can write ηG(B) = G(ηB)φF

0 = G(ηB)G(ψ F0 )δ1

Together with (29) this shows that the second identity in (223) is equivalent to

f ηA = χ(ηB ψ F0 ) (228)

On the other hand the second identity in (224) is equivalent to

ρ F( f ) F(ηA) φF0 = ηB (229)

which by φF0 = (ψ F

0 )minus1 and (29) is further equivalent to (228)

Definition 217 Let (Am A ηAA εA) and (Bm B ηBB εB) be two Frobeniusalgebras in a tensor category C For f A rarr B we define f lowast B rarr A by

f lowast = ((εB m B)otimes idA) (idB otimes f otimes idA) (idB otimes (A ηA)) (230)

The following lemma is immediate from the definition of (middot)lowast and the properties ofFrobenius algebras We omit the proof

Lemma 218 Let C be a tensor category let A BC be Frobenius algebras in C andlet f A rarr B and g B rarr C be morphisms

(i) (g f )lowast = f lowast glowast(ii) f is a monomorphism iff f lowast is an epimorphism

(iii) f is an algebra map iff f lowast is a coalgebra map(iv) If f is a homomorphism of Frobenius algebras then f lowast f = idA and f f lowast =

idB(v) If C is sovereign and if A and B are symmetric then f lowastlowast = f

Let C and D be tensor categories and let F C rarr D be a Frobenius functor GivenFrobenius algebras A B in C and a morphism f A rarr B the next lemma shows how(middot)lowast behaves under F

Lemma 219 F( f lowast) = F( f )lowast

Proof The definition of the structure morphisms of the Frobenius algebra F(A) is givenin (217) and (218) Substituting these definitions gives

F( f )lowast =[(ψ F

0 F(εB) F(m B) φF2

)otimes idF(A)

] [

idF(B) otimes F( f )otimes idF(A)]

[idF(B) otimes

(ψ F

2 F(A) F(ηA) φF0

)]

= (ψ F0 otimes idF(A)) [

F (εB m B (idB otimes f ))otimes idF(A)]

(φF2 otimes idF(A)) (idF(B) otimes ψ F

2 )

[idF(B) otimes F(A ηA)

] (idF(B) otimes φF0 ) (231)

884 L Kong I Runkel

In the middle line of the last expression we can use the defining property (25) of F namely we substitute (φF

2 otimes idF(A)) (idF(B) otimes ψ F2 ) = ψ F

2 φF2 Then ψ F

2 can bemoved to the left and φF

2 to the right until they can be omitted against ψ F0 and φF

0 respectively using (22) and (24) This results in

F( f )lowast = F ((εB m B (idB otimes f ))otimes idA) F (idB otimes (A ηA)) (232)

which is nothing but F( f lowast)

23 Modular tensor categories Let C be a modular tensor category [TBK] ie anabelian semi-simple finite C-linear ribbon category with simple tensor unit 1 and a non-degeneracy condition on the braiding (to be stated in a moment) We denote the set ofequivalence classes of simple objects in C by I elements in I by i j k isin I and theirrepresentatives by Ui U j Uk We also set U0 = 1 and for an index k isin I we define kby Uk

sim= Uork

Since the tensor unit is simple we shall for modular tensor categories identifyHom(1 1) sim= C (cf footnote 2) Define numbers si j isin C by4

si j = UiU j (233)

They obey si j = s ji and s0i = dim Ui see eg [BK Sect 31] (In a ribbon cate-gory the left and right dimension (214) of Ui coincide and are denoted by dim Ui )The non-degeneracy condition on the braiding of a modular tensor category is that the|I|times|I|-matrix s should be invertible In fact [BK Thm 317]

sumkisinI

sik sk j = Dim C δij (234)

where Dim C = sumiisinI(dim Ui )

2 One can show (even in the weaker context of fusioncategories over C) that Dim C ge 1 [ENO1 Thm 23] In particular Dim C = 0 We fixonce and for all a square root

radicDim C of Dim C

Let us fix a basis λα(i j)kN k

i jα=1 in HomC(Ui otimesU j Uk) and the dual basis ϒ(i j)k

α N ki j

α=1

in HomC(UkUi otimesU j ) The duality of the bases means that λα(i j)k ϒ(i j)kβ = δαβ idUk

We also fix λ(0i)i = λ(i0)i = idUi We denote the basis vectors graphically as follows

λα(i j)k = α

Uk

Ui U j

ϒ(i j)kα = α

Uk

Ui U j

(235)

4 In the graphical notation used below we have given an orientation to the ribbons indicated by the arrowsFor example it is understood that this orientation determines which of the duality morphisms in (213) to use

Cardy Algebras and Sewing Constraints I 885

For V isin C we also choose a basis b(iα)V of HomC(VUi ) and the dual basis bV(iβ)

of HomC(Ui V ) for i isin I such that b(iα)V bV(iβ) = δαβ idUi We use the graphical

notation

b(iα)V = α

Ui

V

bV(iα) = α

Ui

V

(236)

Given two modular tensor categories C and D by C D we mean the tensor productof additive categories over C [BK Def 1115] ie the category whose objects are directsums of pairs V times W of objects V isin C and W isin D and whose morphism spaces are

HomCD(V times W V prime times W prime) = HomC(V V prime)otimesC HomC(WW prime) (237)

for pairs and direct sums of these if the objects are direct sums of pairsIf we replace the braiding and the twist in C by the antibraiding cminus1 and the antitwist

θminus1 respectively we obtain another ribbon category structure on C In order to distin-guish these two distinct structures we denote (C c θ) and (C cminus1 θminus1) by C+ and Cminusrespectively As in the Introduction we will abbreviate

C2plusmn = C+ Cminus (238)

Note that a set of representatives of the simple objects in C2plusmn is given by Ui times U j fori j isin I

For the remainder of Sect 2 we fix a modular tensor category C

24 The functors T and R The tensor product bifunctor otimes can be naturally extendedto a functor T C2plusmn rarr C Namely T (oplusN

i=1Vi times Wi ) = oplusNi=1Vi otimes Wi for all Vi Wi isin C

and N isin N The functor T becomes a tensor functor as follows For φT0 1 rarr T (1 times 1)

take φT0 = id1 (or lminus1

1 in the non-strict case) Next notice that for U VW X isin C

T (U times V )otimes T (W times X)= (U otimes V )otimes (W otimes X)

T ((U times V )otimes (W times X))= (U otimes W )otimes (V otimes X)(239)

We define φT2 T (U times V )otimes T (W times X) rarr T ((U times V )otimes (W times X)) by

φT2 = idU otimes cminus1

W V otimes idX (240)

(In the non-strict case the appropriate associators have to be added) The above definitionof φT

2 can be naturally extended to a morphism φT2 T (M1 otimes M2) rarr T (M1)otimes T (M2)

for any pair of objects M1M2 in C2plusmn The following result can be checked by directcalculation [JS Prop 52]

Lemma 220 The triple (T φT2 φ

T0 ) gives a tensor functor

886 L Kong I Runkel

In particular (T φT2 φ

T0 ψ

T2 ψ

T0 ) where ψT

2 = (φT2 )

minus1 and ψT0 = (φT

0 )minus1 gives

a Frobenius functorDefine the functor R C rarr C2plusmn as follows for A isin C and f isin HomC(A B)

R(A) =oplusiisinI

(A otimes Uori )times Ui R( f ) =

oplusiisinI

( f otimes idUori)times idUi (241)

This functor was also considered in a slightly different context in [ENO2 Prop 23]The family of isomorphisms γ R

A = oplusiisinI DimCdim Ui

id(AotimesUori )timesUi

isin Aut(R(A)) defines a

natural isomorphism γ R R rarr ROur next aim is to show that R is left and right adjoint to T in other words R

and T form an ambidextrous adjunction (see eg [Ld] for a discussion of ambidextrousadjunctions) To this end we introduce two linear isomorphisms for A isin C and M isin C2plusmn

χ HomC(T (M) A) minusrarr HomC2plusmn(M R(A))

χ HomC(A T (M)) minusrarr HomC2plusmn(R(A)M)(242)

If we decompose M as M = oplusNn=1 Ml

n times Mrn then χ and χ are given by

χ Noplus

n=1

fn

Mln Mr

n

A

rarrNoplus

n=1

oplusiisinI

sumα

fn

Mln

Mrn

A

α

Uori

times α

Mrn

Ui

(243)

and

χ Noplus

n=1 gn

Mln Mr

n

A

rarrNoplus

n=1

oplusiisinI

sumα gn

Mln

Mrn

A

α

Uori

times α

Ui

Mrn

DimCdim Ui

(244)

Notice that χ and χ are independent of the choice of basis

Theorem 221 〈T R χ〉 and 〈R T χminus1〉 are adjunctions ie R is both left and rightadjoint of T

Proof Write M as M = oplusNn=1 Ml

n times Mrn The isomorphism χ amounts to the following

composition of natural isomorphisms

HomC(T (M) A) = oplusnHomC(Mln otimes Mr

n A)sim= oplusni HomC(Ml

n otimes Ui A)otimes HomC(Mrn Ui )

sim= oplusni HomC(Mln A otimes Uor

i )otimes HomC(Mrn Ui ) = HomC2plusmn(M R(A)) (245)

Cardy Algebras and Sewing Constraints I 887

Thus χ is natural Let (γ RA )

lowast HomC2plusmn(R(A)M) rarr HomC2plusmn(R(A)M) denote the

pull-back of γ RA The isomorphism χ is equal to the composition of (γ R

A )lowast and the

following sequence of natural isomorphisms

HomC(A T (M)) = oplusn HomC(AMln otimes Mr

n)

sim= oplusni HomC(AMln otimes Ui )otimes HomC(Ui Mr

n)

sim= oplusni HomC(A otimes Uori Ml

n)otimes HomC(Ui Mrn) = HomC2plusmn(R(A)M) (246)

We have proved that both χ and χ are natural isomorphisms There are four natural transformations associated to χ and χ namely

idC2plusmnδminusrarr RT

ρminusrarr idC2plusmn and idCδminusrarr T R

ρminusrarr idC (247)

defined by for A isin C M isin C2plusmn

δM = χ(idT (M)) ρA = χminus1(idR(A))

ρM = χ (idT (M)) δA = χminus1(idR(A))(248)

They can be expressed graphically as follows with M = oplusNn=1 Ml

n times Mrn

δM =oplusni

sumα

Mln

Mln Mr

n Uori

αtimes

Ui

Mrn

α

ρA =oplusiisinI

A

A

Uori Ui

(249)

ρM =oplusni

sumα

Mln

Mln

Mrn Uor

i

α times

Ui

Mrn

α DimCdim Ui

δA =oplusiisinI

A

A Uori Ui

dim UiDimC

Note that

ρM δM = Dim C middot idM and ρA δA = idA (250)

Lemma 222 The functors T and R as maps on the sets of morphisms have left inversesand thus are injective

Proof Let f A rarr B be a morphism in C We define a map Q R HomC2plusmn(R(A)

R(B)) rarr HomC(A B) by f prime rarr ρB T ( f prime) δA Then we have

Q R R( f ) = ρB T R( f ) δA = ρB δB f = f (251)

888 L Kong I Runkel

where we used naturality of δ and (250) in the second and third equalities respectivelySo Q R is a left inverse of R on morphisms Thus R is injective on morphisms Similarlylet g M rarr N be a morphism in C2plusmn We define a map QT HomC(T (M) T (N )) rarrHomC2plusmn(M N ) by gprime rarr (Dim C)minus1 middot ρN R(gprime) δM Then we have

QT T (g) = (Dim C)minus1 middot ρN RT (g) δM = (Dim C)minus1 middot ρN δN g = g (252)

So QT is a left inverse of T on morphisms Thus T is injective on morphisms Using (29) and (250) one can express the two inverse maps χminus1 χminus1 as follows

for f isin HomC2plusmn(M R(A)) and g isin HomC2plusmn(R(A)M)

χminus1( f ) = ρ T ( f ) χminus1(g) = T (g) δ (253)

By Proposition 25 and Lemma 27 R is both a lax and colax tensor functor In particularφR

0 1 times 1 rarr R(1) is given by

φR0 = χ (ψT

0 ) = R(ψT0 ) δ1 times 1 = id1times1 (254)

and φR2 R(A)otimes R(B) rarr R(A otimes B) by φR

2 = R(ρA otimes ρB) R(ψT2 ) δ which can

be expressed graphically as

φR2 =

oplusi jkisinI

sumα

A

A

Uori B

B

Uorj

Uork

α

times

Ui U j

α

Uk

(255)

Similarly ψ R0 R(1) rarr 1 times 1 is given by

ψ R0 = ρ1 R(φT

0 ) = Dim C id1times1 (256)

and ψ R2 R(A otimes B) rarr R(A) otimes R(B) by ψ R

2 = ρ R(φT2 ) R(δA otimes δB) which in

graphical notation reads

ψ R2 =

oplusi jkisinI

sumα

A

A Uori

B

B Uorj

Uork

α

times

Ui U j

α

Uk

dim Ui dim U j

dim Uk Dim C

(257)

If C has more than one simple object then R does not take the tensor unit of C to thetensor unit of C2plusmn and so is clearly not a tensor functor However we will show that R isstill a Frobenius functor This will imply that if A is a Frobenius algebra in C then

R(A) = (R(A)m R(A) ηR(A)R(A) εR(A)) (258)

Cardy Algebras and Sewing Constraints I 889

is a Frobenius algebra in C2plusmn where the structure morphisms were given in (217) and(218) In the case A = 1 it was proved in [Muuml1 Prop 41] (see also [Fr Lem 619]and [K1 Thm 52]) that (258) is a commutative simple symmetric normalised-specialFrobenius algebra in C2plusmn In fact given a Frobenius algebra A in C it is straightforwardto verify that the structure morphisms in (258) are precisely those of (A times 1)otimes R(1)cf Sect 22

Proposition 223 (R φR2 φ

R0 ψ

R2 ψ

R0 ) is a Frobenius functor

Proof Using the explicit graphical expression of φR2 φ

R0 ψ

R2 ψ

R0 it is easy to see that

the commutativity of the diagrams (25) and (26) are equivalent to the statement thatR(1) with structure morphisms as in (258) is a Frobenius algebra in C2plusmn The latterstatement is true by [Muuml1 Prop 41]

From Lemma 220 and Proposition 223 we see that T and R take Frobenius alge-bras to Frobenius algebras The following two propositions show how the properties ofFrobenius algebras are transported

Proposition 224 Let A be a Frobenius algebra in C2plusmn Then T (A) is a Frobenius algebrain C and

(i) A is symmetric iff T (A) is symmetric(ii) A is (normalised-)special iff T (A) is (normalised-)special

Proof For part (i) write A as a direct sum oplusNn=1 Al

n times Arn Then the maps IT (A) I prime

T (A) T (Aor) rarr T (A)or defined in (220) are given by

IT (A) = I primeT (A) = oplusN

n=1c(Aln)

or(Arn)

or (259)

Therefore by (221) T (A) = primeT (A) is equivalent to T (A) = T (prime

A) Since byLemma 222 T is injective on morphisms this proves part (i) Part (ii) can be checkedin the same way for example the condition mT (A) T (A) = ζ idT (A) is easily checkedto be equivalent to T (m A A) = ζ T (idA) Proposition 225 Let A be a Frobenius algebra A in C Then R(A) is a Frobeniusalgebra in C2plusmn and

(i) A is symmetric iff R(A) is symmetric(ii) A is (normalised-)special iff R(A) is (normalised-)special

Proof Recall that the structure morphisms of the Frobenius algebra R(A) are equal tothose of (A times 1) otimes R(1) Using this equality part (i) and (ii) follow because R(1) issymmetric and normalised-special For example

m R(A) R(A) = [(m A A)times id1] otimes idR(1) = R(m A A) (260)

so that m R(A) R(A) = ζ idR(A) is equivalent to R(m A A) = ζ R(idA) which byLemma 222 is equivalent to m A A = ζ idA

The functor R has one additional property not shared by T namely R takes absolutelysimple algebras to absolutely simple algebras We will see explicitly in Sect 33 thatthis is not true for T

890 L Kong I Runkel

Lemma 226 For a C-algebra A the map

R HomA|A(A A) rarr HomR(A)|R(A)(R(A) R(A)) (261)

given by f rarr R( f ) is well-defined and an isomorphism

Proof Since R is a lax tensor functor R(A) is naturally a R(A)-bimodule It is easy tosee that R in (261) is a well-defined map R(A) is also naturally a R(1)-bimodule whichcan be identified with the induced R(1)-bimodule structure on (A times 1)otimes R(1) wherethe left R(1) action on (Atimes1)otimes R(1) is given by (idAtimes1 otimesm R(1))(cminus1

Atimes1R(1)otimes idR(1))We have the following natural isomorphisms

HomR(1)|R(1)(R(A) R(A))sim=minusrarr HomC2plusmn(A times 1 R(A))

χminus1

minusminusrarr HomC(A A) (262)

which by (253) are given by for f isin HomR(1)|R(1)(R(A) R(A))

f rarr f prime = f (idAtimes1 otimes ηR(1)) rarr f primeprime = ρ T ( f prime) (263)

and its inverse is given by for g isin HomC(A A)

g rarr gprime = R(g) δ rarr gprimeprime = (idAtimes1 otimes m R(1)) (gprime otimes idR(1)) (264)

where gprimeprime is indeed a R(1)-bimodule map due to the commutativity of R(1) It is easy tocheck that gprimeprime = R(g) in (264) Therefore R gives an isomorphism from HomC(A A)to HomR(1)|R(1)(R(A) R(A)) Moreover one verifies that R(g) is an R(A)-bimodulemap iff g is an A-bimodule map In other words R g rarr R(g) gives an isomorphism

HomA|A(A A)sim=minusrarr HomR(A)|R(A)(R(A) R(A))

Corollary 227 Let A be a C-algebra

(i) A is absolutely simple iff R(A) is absolutely simple(ii) Let A be in addition Frobenius Then A is simple and special iff R(A) is simple and

special

Proof Part (i) immediately follows from Lemma 226 The statement of part (ii) withoutthe qualifier lsquosimplersquo is proved in Proposition 225 But as in the proof of (iii)rArr(i) inLemma 211 a special Frobenius algebra in a semi-simple category has a semi-simplecategory of bimodules and for a semi-simple C-linear category simple and absolutelysimple are equivalent Part (ii) then follows from part (i)

The following lemma will be needed in Sect 32 below to discuss the properties ofCardy algebras

Lemma 228 Let A be a Frobenius algebra in C Then (δA)lowast = ρA

Proof Recall from (250) that δA is a morphism A rarr T R(A) Since T and R are bothFrobenius functors T R(A) is a Frobenius algebra in C Substituting the definitions aftera short calculation one finds

Cardy Algebras and Sewing Constraints I 891

εT R(A) mT R(A) = Dim(C)oplusiisinI

A Uori Ui A U or

ı Uı

(265)

Substituting this in the definition of (δA)lowast gives again after a short calculation the mor-

phism ρA At an intermediate step one uses that the part of the morphism (δA)lowast which

is made up of Ui and Uı ribbons their duals and the basis morphisms λ(iı)0 andϒ(iı)0can be replaced by 1

dim Uimiddot dUi

3 Cardy Algebras

In this section we start by investigating the properties of Frobenius algebras which sat-isfy the so-called modular invariance condition We then give two definitions of a Cardyalgebra and prove their equivalence Finally in Sect 33 we study the properties of thesealgebras and state our main results

We fix a modular tensor category C Recall that C2plusmn is an abbreviation for C+ Cminus

31 Modular invariance In C2plusmn we define the object K and the morphism ω K rarr Kas

K =oplus

i jisinIUi times U j ω =

sumi jisinI

dim Ui dim U j

Dim C idUi timesU j (31)

They have the property (see eg [BK Cor 3111])

ω

K

(Uk times Ul )or

(Uk times Ul )or

Ui times U j

Ui times U j

= δik δ jlDim C

dim Ui dim U jbUi timesU j dUi timesU j (32)

892 L Kong I Runkel

Definition 31

(i) Let A B be objects of C2plusmn A morphism f A otimes B rarr B is called S-invariant iff

A

B

W

W

f =

A

B

W

W

f

K

ω(33)

holds for all for W isin C2plusmn(ii) A C2plusmn-algebra (Aclmcl ηcl) is called modular invariant iff θAcl = idAcl and mcl

is S-invariant

Lemma 32 The morphism f A otimes B rarr B is S-invariant if and only if

A

B

Ui times U j

Ui times U j

f = Dim Cdim Ui dim U j

sumα

A

B

B

f

Ui times U j

Ui times U j

α

α

(34)

holds for all i j isin I

Proof Condition (33) holds for all W iff it holds for all W = Ui times U j i j isin I so itis enough to show that the right-hand side of (33) with W = Ui times U j is equal to theright-hand side of (34) Recall the notation for basis morphisms in (236) Starting from(33) write

idBor otimes idUi timesU j =⎛⎝sum

klα

(b(ktimeslα)

B

)or (

bB(ktimeslα)

)or⎞⎠ otimes idUi timesU j (35)

and then apply (32) The graphical representation of the resulting morphism can bedeformed to give (34) Remark 33 As shown in [K3 Sect 61] the modular invariance condition of a C2plusmn-alge-bra exactly coincides with the modular invariance condition for torus 1-point correlationfunctions of a genus-01 closed CFT In particular the condition θAcl = idAcl is equiv-alent to invariance under the modular transformation T τ rarr τ + 1 and the condition(34) with f = mcl is equivalent to invariance under S τ rarr minus 1

τ Combining the

modular invariance condition with the genus zero properties of a genus-01 closed CFTresults in a modular invariant commutative symmetric Frobenius algebra in C2plusmn

Cardy Algebras and Sewing Constraints I 893

Let Acl be a modular invariant C2plusmn-algebra Evaluating (34) for f = mcl composingit with ηcl otimes idUi timesU j and taking the trace implies the following identity

Zi j = 1

DimCAclUi timesU j where

(36)

Zi j = dimC HomC2plusmn(UitimesU j Acl)

Decomposing Acl into simple objects this gives

Zi j =sum

klisinISik Zkl Sminus1

l j where Si j = si jradic

DimC (37)

which in CFT terms is of course nothing but the invariance of the torus partition functionunder the modular S-transformation

The following theorem gives a simple criterion for modular invariance

Theorem 34 Let Acl be a haploid commutative symmetric Frobenius algebra in C2plusmn

(i) If Acl is modular invariant then dim Acl = Dim C(ii) If dim Acl = Dim C then Acl is special and modular invariant

Proof Part (i) Since Acl is haploid for i = j = 0 Equation (36) reduces to 1 =dim AclDimCPart (ii) By the same reasoning as in the proof of (iii)rArr(i) in Lemma 211 one showsthat Acl is special Thus mcl cl = ζclidAcl for some ζcl = 0

By [KO Thm 45] the category (C2plusmn)locAcl

of local Acl-modules is again a mod-

ular tensor category and Dim (C2plusmn)locAcl

= (Dim C dim Acl)2 (see [Fr Prop 321 amp

Rem 323] for the same statement in the notation used here) Thus by assumption wehave Dim(C2plusmn)loc

Acl= 1 It then follows from [ENO1 Thm 23] that up to isomorphism

(C2plusmn)locAcl

has a unique simple object (namely the tensor unit) In other words every simplelocal Acl-module is isomorphic to Acl (seen as a left-module over itself)

We have the following isomorphisms between morphism spaces [FS Prop 47 amp 411]

HomAcl(Acl otimes (Ui times U j ) Acl) sim= HomC2plusmn(Ui times U j Acl)

HomAcl(Acl Acl otimes (Ui times U j )) sim= HomC2plusmn(AclUi times U j )

Using these to transport the bases (236) from the right to the left we obtain bases bα(i j)αof HomAcl(Acl otimes (Ui times U j ) Acl) and b(i j)

β β of HomAcl(Acl Acl otimes (Ui times U j )) Thesecan be expressed graphically as

894 L Kong I Runkel

bα(i j) =

Acl

Acl

Acl

Ui times U j

α

b(i j)β = DimC

dim Ui dim U j

1

ζcl

Acl

Ui times U jAcl

Acl

β

(38)

where the nonzero factor in b(i j)β is included for convenience Notice that bα(i j) b(i j)

β isa left Acl-module map Since Acl is simple as a left module over itself we have

bα(i j) b(i j)β = λαβ idAcl (39)

for some λαβ isin C By computing tr(bα(i j) b(i j)β ) it is easy to verify that λαβ = δαβ

We will now prove the following identity

1

ζcl

Acl

Acl

Ui times U j

Ui times U j

Acl

=sumα

DimCdim Ui dim U j

1

ζcl

Acl

Ui times U j

Ui times U j

Acl

Acl

Acl

α

α

(310)

One checks that the left-hand side of this equation is an idempotent which we denote byζminus1

cl PlAcl(Ui timesU j ) cf [Fr Sect 31] By [Fr Prop 41] the image Im(ζminus1

cl PlAcl(Ui timesU j ))

is a local Acl-module and hence isomorphic to AoplusNcl for some N isin Zge0

All left-module morphisms from Acl to Aclotimes(Ui timesU j ) are linear combinations of the

b(i j)β Furthermore one verifies that ζminus1

cl PlAcl(Ui timesU j )b(i j)

β = b(i j)β Therefore the b(i j)

β

describe precisely the image of the idempotent ie ζminus1cl Pl

Acl(Ui timesU j ) = sum

α b(i j)α bα(i j)

which is nothing but (310) Composing (310) with ζcl middot εcl otimes idUi timesU j from the left (iefrom the top) produces (34)

In addition since Acl is commutative symmetric Frobenius it satisfies θAcl = idAcl

[Fr Prop 225] Altogether this shows that Acl is modular invariant Remark 35 As we were writing this paper we heard that the results in Theorem 34were obtained independently by Kitaev and Muumlger [Ki]

Remark 36 Setting i = j = 0 in (36) gives the identity dim Acl = Z00 Dim C [KRProp 23] Combining this with Theorem 34 (ii) one may wonder if a general modularinvariant commutative symmetric Frobenius algebra Acl in C2plusmn is isomorphic to a directsum of simple such algebras However this is not so For example one can take the com-mutative symmetric Frobenius algebra Acl = C[x]x2 in the category of vector spacesequipped with the non-degenerate trace ε(ax + b) = a In this case the modular invari-ance condition holds automatically but Acl is clearly not a direct sum of two algebras

Cardy Algebras and Sewing Constraints I 895

For a general modular tensor category C the algebra C[x]x2 R(1) understood as analgebra in C2plusmn via the braided monoidal isomorphism Vect f (C) C2plusmn rarr C2plusmn providesanother counter-example

32 Two definitions Define a morphism PlA A rarr A for a Frobenius algebra A in C

or C2plusmn as follows [Fr Sect 24]

PlA =

A

A

A

m

(311)

If A is also commutative and obeys m A A = ζA idA we have PlA = ζA idA In partic-

ular this holds if A is commutative and special Using the fact that the Frobenius algebraR(1) is commutative and normalised-special one can check that Pl

R(A) R(A) rarr R(A)takes the following form

PlR(A) =

oplusiisinI

A

A

Uori

Uori

A

m

times

Ui

Ui

(312)

With these ingredients we can now give the first definition of a Cardy C|C2plusmn-algebrawhich was introduced in [K3 Def 514] cf Remark 315 below

Definition 37 (CardyC|C2plusmn-algebra I) A CardyC|C2plusmn-algebra is a triple (Aop|Acl ιcl-op)where (Aclmcl ηclcl εcl) is a modular invariant commutative symmetric Frobe-nius C2plusmn-algebra (Aopmop ηopop εop) is a symmetric Frobenius C-algebra andιcl-op Acl rarr R(Aop) an algebra homomorphism such that the following conditionsare satisfied

(i) Centre condition

Acl R(Aop)

R(Aop)

R(Aop)

ιcl-op

m R(Aop)

=

Acl

ιcl-op

R(Aop)

R(Aop)

R(Aop)

m R(Aop)

(313)

896 L Kong I Runkel

(ii) Cardy condition

ιcl-op ιlowastcl-op =

R(Aop)

R(Aop)

R(Aop)

(314)

Remark 38

(i) The name ldquoCardy C|C2plusmn-algebrardquo in Definition 37 was chosen because many ofthe important ingredients were first studied by Cardy the modular invariance ofthe closed theory [C1] the consistency of the annulus amplitude [C2] and thebulk-boundary OPE [CL] On the other hand the boundary-boundary OPE andthe OPE analogue of the centre condition were first considered in [Lw]

(ii) One can easily see that in the special case that C is the category Vect f (C) of finite-dimensional C-vector spaces a Cardy C|C2plusmn-algebra gives exactly the algebraicformulation of two-dimensional open-closed topological field theory over C (cfRemark 614 in [K3]) see [Lz Sect 48] [Mo Thm 11] [AN Thm 45] [LPCor 43] [MS Sect 22] When passing to a general modular tensor category Cthere are two important differences to the two-dimensional topological field the-ory Firstly the algebras Acl and Aop now live in different categories which inparticular affects the formulation of the centre condition and the Cardy conditionSecondly the modular invariance condition has to be imposed on Acl In the caseC = Vect f (C) modular invariance holds automatically

Definition 39 A homomorphism of Cardy C|C2plusmn-algebras (A(1)op |A(1)cl ι(1)cl-op) rarr (A(2)op

|A(2)cl ι(2)cl-op) is a pair ( fop fcl) of Frobenius algebra homomorphisms fop A(1)op rarr A(2)op

and fcl A(1)cl rarr A(2)cl such that the diagram

A(1)cl

fcl

ι(1)cl-op

A(2)cl

ι(2)cl-op

R(A(1)op )

R( fop) R(A(2)op )

(315)

commutes

Remark 310 Since a homomorphism of Frobenius algebras is invertible (cf Lemma218 (iv)) a homomorphism of Cardy algebras is always an isomorphism

For a homomorphism ( fop fcl) of Cardy C|C2plusmn-algebras using the commutativity of(315) and the fact that fcl and fop are both algebra and coalgebra homomorphisms itis easy to show that (315) commutes iff

R(A(1)op )R( fop)

(ι(1)cl-op)

lowast

R(A(2)op )

(ι(2)cl-op)

lowast

A(1)cl

fcl A(2)cl

(316)

Cardy Algebras and Sewing Constraints I 897

commutesLet (Aop|Acl ιcl-op) be a Cardy C|C2plusmn-algebra Define the morphism

ιcl-op = χminus1(ιcl-op) T (Acl) minusrarr Aop (317)

Decompose Acl as Acl = oplusNn=1Cl

n times Crn such that Cl

1 times Cr1 = ηcl(1 times 1) We use ι(n)cl-op

to denote the restriction of ιcl-op to Cln times Cr

n and ι(n)cl-op to denote the restriction of ιcl-op

to Cln otimes Cr

n We introduce the following graphical notation

ι(n)cl-op =

Cln Cr

n

Aop

(318)

By (243) ιcl-op can be expressed in terms of ιcl-op as follows

ιcl-op =Noplus

n=1

oplusiisinI

sumα

Cln

Crn

Aop

α

Uori

times α

Crn

Ui

(319)

Lemma 311 The centre condition (313) is equivalent to the following condition in C

Cln Cr

n

Aop

Aop

Aop

=

Cln Cr

n

Aop

Aop

Aop

for n = 1 N (320)

Proof First insert (319) and the definition (217) of m R(Aop) into (313) Then apply thecommutativity of R(1) to the left hand side of (313) The equivalence between (313)and (320) follows immediately Remark 312 The centre condition (320) is very natural from the open-closed confor-mal field theory point of view Correlators on the upper half plane are expressed in termsof conformal blocks on the full complex plane The objects Cl

n and Crn are associated to

the field insertion at a point z in the upper half plane and at the complex conjugate pointz in the lower half plane respectively The object Aop corresponds to a field inserted ata point r on the real axis The centre condition (320) simply says that the correlationfunctions in the disjoint domains |z| gt r gt 0 and r gt |z| gt 0 are analytic continuationsof each other see [K2 Prop 118]

898 L Kong I Runkel

Recall that we define ιlowastcl-op Aop rarr T (Acl) as in (230) We introduce the graphicalnotation

ιlowastcl-op =Noplus

n=1

Cln Cr

n

Aop

=Noplus

n=1

Cln Cr

n

Aop

Aop T (minus1Acl) (321)

where the second equality follows from (215) (221) and (259)

Lemma 313 The Cardy condition (314) is equivalent to the following identity in C

Noplusn=1

DimCdim Ui

sumα

Cln

Crn

Crn

α

α

Aop

Aop Uori

Uori

=Aop

Aop

Aop

Uori

Uori

op

mop for all i isin I (322)

Proof By (244) (312) and (319) it is easy to see that (322) is equivalent to thefollowing identity

ιcl-op χ(ιlowastcl-op) = PlR(Aop)

(323)

Therefore it is enough to show that

χ(ιlowastcl-op) = ιlowastcl-op (324)

We have

χminus1(ιlowastcl-op)(1)= T (ιlowastcl-op) δ (2)= T (ιcl-op)

lowast δlowastlowast (3)= (ρ T (ιcl-op)

)lowast (4)= (ιcl-op

)lowast (325)

In step (1) we use the expression (253) for χminus1 step (2) follows from Lemma 218 (v)and Lemma 219 Step (3) is Lemma 218 (i) and Lemma 228 and finally step (4)amounts to substituting (253) and (317) Acting with χ on both sides of the aboveequality produces (324)

Combining Lemmas 311 and 313 and Proposition 216 we obtain the followingequivalent definition of Cardy C|C2plusmn-algebra (recall the graphical notation (318) forιcl-op and (321) for ιlowastcl-op)

Cardy Algebras and Sewing Constraints I 899

Definition 314 (CardyC|C2plusmn-algebra II) A CardyC-algebra is a triple (Aop|Acl ιcl-op)where Acl is a commutative symmetric Frobenius C2plusmn-algebra satisfying property (34)with f = mcl Aop is a symmetric Frobenius C-algebra and ιcl-op T (Acl) rarr Aop isan algebra homomorphism satisfying the conditions (320) and (322)

Remark 315 Up to a choice of normalisation Definition 314 is the same as the orig-inal one in [K3 Def 613] The difference between the two definitions is the factorDim C dim Ui on the left hand side of (322) which in [K3 Def 613] is given byradic

Dim C dim Ui The two definitions are related by rescaling the coproduct cl andcounit εcl of Acl by 1

radicDim C and

radicDim C respectively We chose the convention in

(322) to remove all dimension factors from the expression (314) for the Cardy condition

33 Uniqueness and existence theorems In this subsection we investigate the structureof Cardy algebras We start with the following proposition which when combined withthe results of part II provides an alternative proof of [Fj Prop 422]

Proposition 316 Let (Aop|Acl ιcl-op) be a Cardy C|C2plusmn-algebra If Acl is simple anddim Aop = 0 then Aop is simple and special

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis therefore haploid Restricting the Cardy condition (322) to the case Ui = 1 andcomposing both sides with εop from the left we see that εop kills all terms associated toU j times 1 isin Acl in the sum except for a single 1 times 1 term Thus we obtain the followingidentity

β εop = dAop (mop otimes idAorop) (idAop otimes bAop) (326)

where β isin C Composing with ηop from the right in turn implies that βεop ηop =dim Aop which is nonzero by assumption Thus also β = 0 and εop is a nonzero multipleof the morphism on the right hand side of (326) By [FRS Lem 311] Aop is special

Since Aop is a special Frobenius algebra Aop is semi-simple as an Aop-bimodule(apply [FS Prop 524] to Aop tensored with its opposite algebra) Suppose Aop is not

simple so that we can write Aop = A(1)op oplus A(2)op for nonzero Aop-bimodules A(1)op and A(2)op We denote the canonical embeddings and projections associated to this decompositionas ι12 and π12 We have the identities

mop (ι1 otimes ι2) = 0 εop ηop =2sum

i=1

εop ιi πi ηop (327)

The first identity follows since π1 mop (ι1 otimes ι2) = 0 (as mop gives the left action of

Aop on Aop and hence it preserves A(2)op ) and similarly π2 mop (ι1 otimes ι2) = 0 Thesecond identity is just the completeness of ι12 π12

Since εop ηop = 0 without losing generality we can assume εop ι1 π1 ηop = 0Using that π2 is a bimodule map we compute

π2 [LHS of (322)]Ui =1 ι1 π1 ηop = π2 [RHS of (322)]Ui =1 ι1 π1 ηop

= PlA(2)op

π2 ι1 π1 ηop = 0 (328)

900 L Kong I Runkel

On the other hand using that Acl is haploid that ιcl-op is an algebra map and thatι lowastcl-op is a coalgebra map one can check that the left-hand side of (328) is equal toλ(εop ι1 π1 ηop) π2 ηop for some λ = 0 This implies that π2 ηop = 0 Thusηop = sum

i ιi πi ηop = ι1 π1 ηop Hence we have

0 = π2 ι2 = π2 mop (ηop otimes ι2) = π2 mop ((ι1 π1 ηop)otimes ι2) (329)

However the right-hand side is zero by (327) This is a contradiction and hence Aopmust be simple

To formulate the next theorem we need the notion of the full centre of an algebra [FjDef 49] Recall that an algebra A in a braided tensor category has a left centre and aright centre [VZO] both of which are sub-algebras of A Of these two we will onlyneed the left centre The following definition is [Fr Def 231] which in our setting isequivalent to that of [VZO]

Definition 317 Let A be a symmetric special Frobenius algebra such that m A A =ζA idA

(i) The left centre Cl(A) of A is the image of the idempotent ζminus1A Pl

A(ii) The full centre Z(A) is Cl(R(A))

That ζminus1A Pl

A is an idempotent follows from [FRS Lem 52] when keeping track ofthe factors ζA ([FRS] assumes normalised-special ie ζA = 1) Note that Cl(A) is againan object of C while Z(A) is an object of C2plusmn Let el

A Cl(A) rarr A be the embeddingof Cl(A) into A The left centre is in fact the maximal subobject of A such that

m A cAA (elA otimes idA) = m A (330)

see [Fr Lem 232] This observation explains the name left centre and also makes theconnection to [O Def 15]

The full centre is by definition the image of the idempotent ζminus1A Pl

R(A) R(A) rarrR(A) Since C2plusmn is abelian the idempotent splits and we obtain the embedding andrestriction morphisms

e Z(A) rarr R(A) and r R(A) Z(A) (331)

which obey r e = idZ(A) and e r = ζminus1A Pl

R(A) It follows from Proposition 225 and

[Fr Prop 237] that Z(A) is a commutative symmetric Frobenius algebra in C2plusmn withstructure morphisms5

m Z(A) = r m R(A) (e otimes e) ηZ(A) = r ηR(A)

Z(A) = ζA middot (r otimes r) R(A) e εZ(A) = ζminus1A middot εR(A) e

(332)

Moreover if A is simple then Z(A) is simple and if A is simple and dim A = 0 thenZ(A) is simple and special The normalisation of the counit is such that

εZ(A) ηZ(A) = ζminus2A dim A DimC (333)

5 The normalisation of product and unit is the standard one The factors in the coproduct and counit haveto be included in order for (A|Z(A) e) to be a Cardy algebra see Theorem 318 below The normalisation ofthe counit enters the Cardy condition (314) through the definition of ( middot )lowast

Cardy Algebras and Sewing Constraints I 901

Theorem 318 Let A be a special symmetric Frobenius C-algebra Then (A|Z(A) e)is a Cardy C|C2plusmn-algebra

The proof of this theorem makes use the following two lemmas

Lemma 319 e Z(A) rarr R(A) is an algebra map and elowast = ζA middot r

Proof It follows from [Fr Lem 229] (or by direct calculation using in particularm R(A) R(A) = ζAidR(A)) that

m R(A) (e otimes e) = ζminus1A middot Pl

R(A) m R(A) (e otimes e) (334)

Substituting e r = ζminus1A Pl

R(A) shows that e is compatible with multiplication For theunit one finds

e ηZ(A) = e r ηR(A) = ζminus1A Pl

R(A) ηR(A) = ηR(A) (335)

Thus e is an algebra map For the second statement one computes

elowast (1)= ζA

R(A)

r

R(A)

R(A)

Z(A)

R(A)

Z(A)

r

e

e

(2)=

R(A)

r

R(A)

R(A)

R(A)

Z(A)

(3)=

R(A)

r

R(A)

Z(A)

(4)= ζA middot r (336)

where in (1) the definitions (230) and (332) have been substituted step (2) is e r = ζminus1

A PlR(A) step (3) uses that R(A) is symmetric Frobenius and step (4) is again

e r = ζminus1A Pl

R(A) Lemma 320 Let A be a symmetric Frobenius algebra in C The morphism

PlR(A) m R(A)

(Pl

R(A) otimes PlR(A)

) R(A)otimes R(A) minusrarr R(A) (337)

is S-invariant

The proof of this lemma is a slightly lengthy explicit calculation and has been deferredto Appendix A2

902 L Kong I Runkel

Proof of Theorem 318 That e is an algebra map was proved in Lemma 319 The centrecondition (313) holds by property (330) of the left centre The Cardy condition (314)also is an immediate consequence of Lemma 319

ιcl-op ιlowastcl-op = e (ζA r) = PlR(A) (338)

The full centre Z(A) is a commutative symmetric Frobenius algebra It remains to provemodular invariance That θZ(A) = idZ(A) is implied by commutativity and symmetryof Z(A) [Fr Prop 225] The S-invariance condition (33) follows from Lemma 320In (337) substitute Pl

R(A) = ζA e r and then put the resulting morphism into (33)Compose the resulting equation with e otimes idW from the right (ie from the bottom) andsubstitute the definition (332) of m Z(A) This results in the statement that m Z(A) isS-invariant

The following theorem is analogous to [LR Prop 29] and [Fj Thm 426] whichroughly speaking answer the question under which circumstances the restriction of atwo-dimensional conformal field theory to the boundary already determines the entireconformal field theory The first work is set in Minkowski space and uses operator alge-bras and subfactors while the second work is set in Euclidean space and uses modulartensor categories

Theorem 321 Let (A|Acl ιcl-op) be a Cardy C|C2plusmn-algebra such that dim A = 0 andAcl is simple Then A is special and (A|Acl ιcl-op) sim= (A|Z(A) e) as Cardy algebras

Proof By Proposition 316 A is simple and special Since Acl is simple the algebramap ιcl-op Acl rarr R(A) is either zero or a monomorphism But ιcl-op ηcl = ηR(A)and so ιcl-op = 0 Thus ιcl-op is monic By Lemma 218 (ii) ζminus1

A ιlowastcl-op is epi The Cardycondition (314) implies

ιcl-op ζminus1A ιlowastcl-op = ζminus1

A PlR(A) = e r (339)

Composing this with e r from the left yields e r ιcl-op ζminus1A ιlowastcl-op = e r =

ιcl-op ζminus1A ιlowastcl-op Since ζminus1

A ιlowastcl-op is epi we have

e r ιcl-op = ιcl-op (340)

Actually (340) also follows from (313) and specialness of R(A) We will prove that

( fop fcl) (A|Acl ιcl-op) minusrarr (A|Z(A) e) where fop = idA fcl = r ιcl-op

(341)

is an isomorphism of Cardy algebrasfcl is an algebra map Compatibility with the units follows since ιcl-op is an algebramap

fcl ηcl = r ιcl-op ηcl = r ηR(A) = ηZ(A) (342)

Compatibility with the multiplication also follows since ιcl-op is an algebra map

m Z(A) ( fcl otimes fcl) = r m R(A) (e otimes e) (r otimes r) (ιcl-op otimes ιcl-op)

= r m R(A) (ιcl-op otimes ιcl-op) = r ιcl-op mcl = fcl mcl (343)

where in the second step we used (340)

Cardy Algebras and Sewing Constraints I 903

fcl is an isomorphism As above since fcl is an algebra map and since Acl is simple fcl

has to be monic By Lemma 319 rlowast = ζminus1A e Thus f lowast

cl = ιlowastcl-op rlowast = ζminus1A ιlowastcl-op e and

fcl f lowastcl = r ιcl-op ζminus1

A ιlowastcl-op e = r e r e = idZ(A) (344)

and so fcl is also epi and hence isofcl is a coalgebra map Since fcl is an algebra map so is f minus1

cl By (344) f minus1cl = f lowast

cland by Lemma 218 (iii) this implies that fcl is a also coalgebra mapThe diagram (315) commutes Commutativity of (315) is equivalent to e fcl = ιcl-opwhich holds by (340)

Let A be a special symmetric Frobenius algebra So far we have seen that (A|Z(A) e)is a Cardy algebra and that all Cardy algebras with Aop = A and simple Acl are of thisform It is now natural to ask if every simple Acl does occur as part of a Cardy algebraThe following theorem provides an affirmative answer Recall that for an A-left moduleM the object Mor otimesA M is an algebra (see eg [KR Lem 42])

Theorem 322 If Acl is a simple modular invariant commutative symmetric FrobeniusC2plusmn-algebra then there exist a simple special symmetric Frobenius C-algebra A and amorphism ιcl-op Acl rarr R(A) such that

(i) Acl sim= Z(A) as Frobenius algebras(ii) (A|Acl ιcl-op) is a Cardy C|C2plusmn-algebra

(iii) T (Acl) sim= oplusκisinJ Morκ otimesA Mκ as algebras where MκκisinJ is a set of representa-

tives of the isomorphism classes of simple A-left modules

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis haploid It then follows from Theorem 34 that Acl is special By Proposition 224T (Acl) is a special symmetric Frobenius algebra in C Thus T (Acl) = oplusi Ai where theAi are simple symmetric Frobenius algebras We will show that at least one of the Aiis special Since T (Acl) is special we have mT (Acl) T (Acl) = ζ idT (Acl) for someζ isin C

times Restricting this to the summand Ai shows mi i = ζ idAi FurthermoreεT (Acl) ηT (Acl) = ξ id1 for some ξ isin C

times But εT (Acl) ηT (Acl) = sumi εi ηi and so

at least one of the εi ηi has to be nonzero Therefore at least one of the Ai is speciallet A equiv Ai be this summand We denote the embedding A rarr T (Acl) by e0 and therestriction T (Acl) A by r0 Notice that r0 is an algebra homomorphism Define

ιcl-op = χ (r0) Acl minusrarr R(A) (345)

By Proposition 216 ιcl-op is an algebra homomorphism Next we verify the centre con-dition (313) or rather its equivalent form (320) By substituting the definitions onecan convince oneself that the commutativity mcl cAclAcl = mcl of Acl in C2plusmn implies thecondition mT (Acl) = mT (Acl) in C see [K2 Prop 36] Here T (Acl)otimes T (Acl) rarrT (Acl)otimes T (Acl) is given by

=oplusmn

(idCl

notimes cCl

m Crnotimes idCr

m

)

(cCl

m Clnotimes cminus1

Crn C

rm

)

(idCl

motimes cminus1

Cln C

rm

otimes idCrn

)

(346)

and we decomposed Acl as Acl = oplusnCln times Cr

n As a consequence we obtain the identity

r0 mT (Acl) (idAcl otimes e0) = r0 mT (Acl) (idAcl otimes e0) (347)

904 L Kong I Runkel

Using that r0 is an algebra map and that by definition ιcl-op = r0 we obtain (320)In order to show that (A|Acl ιcl-op) is a Cardy algebra it remains to show that the

Cardy condition (314) is satisfied We will demonstrate this via a detour by first provingthat Acl sim= Z(A) as Frobenius algebras

Recall the notations e and r given in (331) Using the centre condition (320) onecan check that Pl

R(A) ιcl-op = m R(A) R(A) ιcl-op By specialness of A we have

m A A = ζA idA and so together with e r = ζminus1A Pl

R(A) we get

e r ιcl-op = ιcl-op (348)

Next consider the morphism

fcl = r ιcl-op Acl minusrarr Z(A) (349)

By the same derivation as in (342) and (343) one sees that fcl is an algebra map Inparticular fcl ηcl = ηZ(A) = 0 and so fcl = 0 Since Acl is simple fcl has to be amonomorphism

By the same argument as used in the proof of Theorem 34 (ii) up to isomorphismAcl is the unique simple local Acl-(left-)module The algebra monomorphism fcl turnsZ(A) into an Acl-module Since Z(A) is commutative it is local as an Acl-module andso Z(A) sim= AoplusN

cl for some N ge 1 By construction A is a simple special symmetricFrobenius algebra Proposition 225 and Corollary 227 show that R(A) inherits all theseproperties and thus Z(A) is simple (see the comment below Eq (332)) By Theorem318 Z(A) is modular invariant and then by Theorem 34 (i) dim Z(A) = Dim C Thisimplies that N = 1 in Z(A) sim= AoplusN

cl and so fcl is in fact an isomorphismSince Acl and Z(A) are both haploid we have εZ(A) fcl = ξ εcl for some ξ isin C

timesThe counit uniquely determines the Frobenius structure on Acl and Z(A) (see eg [FRSLemma 37]) so that fcl is a coalgebra isomorphism iff ξ = 1 To compute ξ we composethe above identity with ηcl from the right Defining ζcl via εcl ηcl = ζminus1

cl Dim C middot id1 andusing (333) gives ξ = dim A ζclζ

2A By rescaling the comultiplication and the counit

of A and consequently changing ζA we can always achieve ξ = 1 This proves part (i)of the theorem

Equation (348) implies that ιcl-op = e fcl Since fcl is an isomorphism of Frobeniusalgebras by Lemmas 218 and 319 we have

ιcl-op ιlowastcl-op = e fcl f lowastcl elowast = ζA e r = Pl

R(A) (350)

Thus (A|Acl ιcl-op) is a Cardy algebra This proves part (ii) of the theoremPart (iii) can be seen as follows By [KR Prop 43] T Z(A) sim= oplusκisinJ Mor

κ otimesA Mκ asalgebras Together with the observation that T ( fcl) T (Acl) rarr T Z(A) is an isomor-phism of algebras this proves part (iii) Remark 323 Part (i) of Theorem 322 was announced by Muumlger [Muuml2] We provide anindependent proof in the setting of Cardy algebras

The above theorem together with Lemma 211 and Theorem 34 shows that a sim-ple commutative symmetric Frobenius C2plusmn-algebra Acl with dim Acl = Dim C is alwayspart of a Cardy algebra (Aop|Acl ιcl-op) for some simple special symmetric Frobeniusalgebra Aop in C However the above proof also illustrates that Aop is not unique Thisraises the question how two Cardy algebras with a given Acl can differ This question isanswered by [KR Thm 11] which in the present framework can be restated as follows

Cardy Algebras and Sewing Constraints I 905

Theorem 324 If (A(i)op |A(i)cl ι(i)cl-op) i = 1 2 are two Cardy C|C2plusmn-algebras such that

A(i)cl is simple and dim A(i)op = 0 for i = 1 2 then A(1)clsim= A(2)cl as algebras if and only

if A(1)op and A(2)op are Morita equivalent

Proof Theorem 11 in [KR] is stated for A(i)op being non-degenerate algebras and A(i)cl =Z(A(i)op ) for i = 1 2 By Proposition 316 A(i)op are simple and special for i = 1 2

Then by [KR Lem 21] A(i)op are non-degenerate algebras By Theorem 321 we have

A(i)clsim= Z(A(i)op ) as Frobenius algebras Finally by [KR Thm 11] Z(A(1)op ) sim= Z(A(2)op )

as algebras iff A(1)op and A(2)op are Morita equivalent Let Cmax(C2plusmn) be the set of equivalence classes [B] of simple modular invariant

commutative symmetric Frobenius algebras B in C2plusmn Two such algebras B and B prime areequivalent if B and B prime are isomorphic as algebras (but not necessarily as Frobenius alge-bras) Let Msimp(C) be the set of Morita classes of simple special symmetric Frobeniusalgebras in C Define the map z Msimp(C) rarr Cmax(C2plusmn) by z A rarr [Z(A)] whereA denotes the Morita class of A From Theorem 322 (i) and [KR Thm 11] we learn

Corollary 325 The map z Msimp(C) rarr Cmax(C2plusmn) is a bijection

A Appendix

A1 Proof of Lemma 27 We will show that if (F ψ F2 ψ

F0 ) is a colax tensor functor

from C1 to C2 then (G φG2 φ

G0 ) is a lax tensor functor from C2 to C1 Applying this

result to the opposed categories then gives the converse statementWe need to show that φG

0 and φG2 make the diagrams (21) and (22) commute We

first prove the commutativity of (21) Consider the following diagram

G(A)otimes (G(B)otimes G(C))

GψF2 δ

idG(A)otimesφG2 G(A)otimes G(B otimes C)

GψF2 δ

G(FG(A)otimes F(G(B)otimes G(C)))

G(F(idG(A))otimesF(φG2 ))

G(idFG(A)otimesψ F2 )

G(FG(A)otimes FG(B otimes C))

G(ρAotimesρBotimesC )

G(FG(A)otimes (FG(B)otimes FG(C)))

G(ρAotimes(ρBotimesρC )) G(A otimes (B otimes C))

(A1)

The top subdiagram is commutative because of the naturality of Gψ F2 δ The commu-

tativity of the bottom subdiagram follows from the following identities

(ρB otimes ρC ) ψ F2 = (ρB otimes ρC ) ψ F

2 ρF Fδ

= ρBotimesC FG(ρB otimes ρC ) FG(ψ F2 ) Fδ

= ρBotimesC F(φG2 ) (A2)

as a map F(G(B) otimes G(C)) rarr B otimes C The commutativity of (A1) implies that thecomposition of maps in the left column in (21) can be replaced by

G(ρA otimes (ρB otimes ρC )) G((idFG(A) otimes ψ F

2 ) ψ F2

) δ (A3)

906 L Kong I Runkel

Similarly we can show that the composition of maps in the right column in (21) can bereplaced by

G((ρA otimes ρB)otimes ρC ) G((ψ F

2 otimes idFG(C)) ψ F2

) δ (A4)

Using the commutativity of (23) it is easy to see that (21) with the left and right col-umns of (21) replaced by (A3) and (A4) respectively is commutative Hence (21) iscommutative

Now we prove the commutativity of the first diagram in (22)

φG2 (φG

0 otimes idG(A))

(1)= G(ρ12 otimes ρA) Gψ F2 δ

[(Gψ F

0 δ11)otimes idG(A)

]

(2)= G(ρ12 otimes ρA) Gψ F2 G F(Gψ0 otimes idG(A)) G F(δ11 otimes idG(A)) δ

(3)= G(ρ12 otimes ρA) G(FG(ψ F0 )otimes idFG(A)) Gψ F

2 G F(δ11 otimes idG(A)) δ(4)= G(id12 otimes ρA) G

([ψ F

0 ρF(11) (Fδ)11] otimes idFG(A)

) Gψ F

2 δ(5)= G(id12 otimes ρA) G(ψ F

0 otimes idFG(A)) Gψ F2 δ

(6)= G(id12 otimes ρA) G(lminus1FG(A)) G F(lG(A)) δ

(7)= G(lminus1A ) GρA δG lG(A)

(8)= G(lminus1A ) lG(A) (A5)

where in step (1) we substituted the definition of φG0 φ

G2 given in (210) in step (2)

we used the naturality of δ in step (3) we used the naturality of Gψ F2 in step (4) we

switched the position between Gρ12 and G FG(ψ F0 ) and the position between Gψ2 and

G F(δ12 otimes idG(A)) using the naturality of ρ and FψG2 respectively in step (5) we applied

the second identity in (28) in step (6) we used (24) in step (7) we used the naturalityof lminus1 and δ in step (8) we used the first identity in (28)

The proof of the commutativity of the second diagram in (22) is similar Thus wehave shown that G is a lax tensor functor

A2 Proof of Lemma 320 To prepare the proof recall that for a given object B isin Cthe modular group P SL(2Z) acts on the space oplusi HomC(B otimes Ui Ui ) see eg [BKSect 31] and [K3 Eq (455)] We will only need the action of S and Sminus1 Let f isinoplusi HomC(B otimes Ui Ui ) Then

S oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A6)

Cardy Algebras and Sewing Constraints I 907

Sminus1 oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A7)

By Lemma 32 to establish that (337) is S-invariant it is enough to prove the identity(34) when f is given by (337) Using (A6) and (A7) we can see that Eq (34) simplysays that oplusi j [RHS of (34)] is invariant under the action of S times S Consider the elementg of oplus jkisinIHomC2plusmn(R(A)otimes (Uor

j times Uk)Uorj times Uk) given by

g =oplusjkisinI

sumα

R(A)

R(A)

R(A)

R(A)

Uorj times Uk

Uorj times Uk

α

α

(A8)

By the above arguments proving S-invariance of (337) is equivalent to proving invari-ance of g under the action of S times S

For i isin I we denote by gi the component of g in(oplus jisinIHomC(A otimes Uor

i otimes Uorj U

orj )

)otimes (opluskisinIHomC(Ui otimes UkUk))

We view the second Hom-space in the above tensor product as a Hom-space in C+ insteadof Cminus It is enough to show that gi is invariant under the action of S times Sminus1 Note thatthe action of Sminus1 in C+ is equivalent to that of S in Cminus

The morphism gi can be canonically identified with a bilinear pairing

( middot middot )i ⎛⎝oplus

jisinIHomC(Uor

j A otimes Uori otimes Uor

j )

⎞⎠

times(oplus

kisinIHomC(UkUi otimes Uk)

)minusrarr C (A9)

as follows For h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) we set

(h1 h2)i = (dim U j dim Uk)minus1 trUor

j timesUk[gi (h1 times h2)] (A10)

908 L Kong I Runkel

When substituting the explicit form of the product m R(A) of R(A) = (A times 1) otimes R(1)after a short calculation one finds

(h1 h2)i =sumα

1

dim U j

A

A

A

A

A A

A α

α

h1

h2

U jUk

Uorj

Ui

Uk (A11)

Here the top morphism PlR(A) has been simplified with the help of the identity

PR(A) m R(A) (PR(A) otimes PR(A))

=(oplus

iisinI((m A A)otimes idUor

i)times idUi

) m R(A) (PR(A) otimes PR(A)) (A12)

which can be checked by direct calculation along the same lines as in the proof of [FrLem 310]

The action of the modular transformation S on oplusiisinI(B otimes Ui Ui ) for B isin C nat-urally induces an action on oplusiisinI(Ui B otimes Ui ) [K3 Prop 514] which we denote bySlowast In the present case we get an action of Slowast on oplus jisinIHomC(Uor

j A otimes Uori otimes Uor

j ) and

opluskisinIHomC(UkUi otimes Uk) Then to show gi is invariant under the action of S times Sminus1

amounts to showing that

(h1 h2)i = ((Sminus1)lowast h1 Slowast h2)i (A13)

Cardy Algebras and Sewing Constraints I 909

for all h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) We have

((Sminus1)lowast h1 Slowast h2)i =sum

mnα

dim Un

DimC

A

A

A

A

A A

A

Uk

α

α

h1

h2

Ui

U j

UmUn

(A14)

Now drag the upper vertex indexed by α in the above graph along its Uorm -leg until it

meets the lower vertex also indexed by α then sum over α and m This gives

((Sminus1)lowast h1 Slowast h2)i =sum

n

dim Un

DimC

A

A

A

A

A A

A

Uk

Un

h1

h2

Ui

U j

(A15)

910 L Kong I Runkel

If we just look at the neighbourhood of the Un-loop in the above graph we see thefollowing subgraphs

sumn

dim Un

DimC

A Aor

Uorj U j

Uk Uork

Un=

sumα

1

dim U j

A Aor Uk Uork

Uorj U j

αα

(A16)

where we have applied [BK Cor 3111] Substituting this subgraph back to the originalgraph in (A15) we obtain

((Sminus1)lowast h1 Slowast h2)i =sumα

1

dim U j

A

A

A

A

A A

A

A

A

Uorj

α

α

h1

h2m A

Ui

U jUk

Uk

(A17)

The graph in (A17) is equal to that in (A11) In order to see this we first drag theldquobubblerdquo (m A A) along A lines and through the m A vertex (because m A A is abimodule map) until it reaches the lower-left leg of the upper vertex indexed by α Thendrag the m A vertex along the (red) dotted line in the above graph Finally we applythe associativity of A (A12) and [Fr Lem 311] Then we see that the graph in (A17)exactly matches with the one in (A11)

Acknowledgements We would like to thank the organisers of the Oberwolfach Arbeitsgemeinschaft ldquoAlge-braic structures in conformal field theoriesrdquo (April 2007) where this work was started for an inspiring meetingWe would further like to thank the Hausdorff Institute for Mathematics in Bonn and the organisers of the stim-ulating meeting ldquoGeometry and Physicsrdquo (May 2008) We are indebted to Alexei Davydov Jens FjelstadJuumlrgen Fuchs Yi-Zhi Huang Alexei Kitaev Urs Schreiber Christoph Schweigert Stephan Stolz and PeterTeichner for helpful discussions andor comments on a draft of this paper The research of IR was partiallysupported by the EPSRC First Grant EPE0050471 the PPARC rolling grant PPC5071451 and the MarieCurie network lsquoSuperstring Theoryrsquo (MRTN-CT-2004-512194)

Cardy Algebras and Sewing Constraints I 911

References

[AN] Alexeevski A Natanzon SM Noncommutative two-dimensional topological field theories andhurwitz numbers for real algebraic curves Sel Math New Ser 12 307ndash377 (2006)

[Bi] Bichon J Cosovereign hopf algebras J Pure Appl Alg 157 121ndash133 (2001)[BK] Bakalov B Kirillov AA Lectures on Tensor Categories and Modular Functors Providence

RI Amer Math Soc 2001[C1] Cardy JL Operator content of two-dimensional conformal invariant theories Nucl Phys

B 270 186ndash204 (1986)[C2] Cardy JL Boundary conditions fusion rules and the verlinde formula Nucl Phys B 324

581ndash596 (1989)[CL] Cardy JL Lewellen DC Bulk and boundary operators in conformal field theory Phys Lett

B 259 274ndash278 (1991)[DMZ] Dong C-Y Mason G Zhu Y-C Discrete series of the Virasoro algebra and the moonshine

module In Algebraic Groups and Their Generalizations Quantum and infinite-dimensional Meth-ods Proc Symp Pure Math 56 Part 2 Providence RI Amer MathSoc 1994 pp 295ndash316

[DP] Day B Pastro C Note on frobenius monoidal functors New York J Math 14 733ndash742 (2008)[ENO1] Etingof PI Nikshych D Ostrik V On fusion categories Ann Math 162 581ndash642 (2005)[ENO2] Etingof PI Nikshych D Ostrik V An analogue of radfordrsquos s4 formula for finite tensor

categories Int Math Research Notices 54 2915ndash2933 (2004)[Fe] Felder G Froumlhlich J Fuchs J Schweigert C Correlation functions and boundary conditions in

rational conformal field theory and three-dimensional topology Comp Math 131 189ndash237 (2002)[FHL] Frenkel IB Huang Y-Z Lepowsky J On axiomatic approaches to vertex operator algebras

and modules Mem Amer Math Soc 104 (1993)[Fj] Fjelstad J Fuchs J Runkel I Schweigert C Uniqueness of openclosed rational cft with

given algebra of open states Adv Theor Math Phys 12 1283ndash1375 (2008)[FK] Froumlhlich J King C The chern-simons theory and knot polynomials Commun Math

Phys 126 167ndash199 (1989)[Fr] Froumlhlich J Fuchs J Runkel I Schweigert C Correspondences of ribbon categories Adv

Math 199 192ndash329 (2006)[FRS] Fuchs J Runkel I Schweigert C Tft construction of rcft correlators I partition functions Nucl

Phys B 646 353ndash497 (2002)[FS] Fuchs J Schweigert C Category theory for conformal boundary conditions Fields Inst Com-

mun 39 25ndash71 (2003)[H1] Huang Y-Z Two-dimensional conformal geometry and vertex operator algebras Progress in

Mathematics Vol 148 Boston Birkhaumluser 1997[H2] Huang Y-Z Rigidity and modularity of vertex tensor categories Comm Contemp Math 10

871ndash911 (2008)[HK1] Huang Y-Z Kong L Open-string vertex algebra category and operads Commun Math

Phys 250 433ndash471 (2004)[HK2] Huang Y-Z Kong L Full field algebras Commun Math Phys 272 345ndash396 (2007)[HK3] Huang Y-Z Kong L Modular invariance for conformal full field algebras httparxivorgabs

math0609570v2[mathQA] 2006[HL] Huang Y-Z Lepowsky J Tensor products of modules for a vertex operator algebra and vertex

tensor categories In Lie Theory and Geometry in honor of Bertram Kostant ed R BrylinskiJ-L Brylinski V Guillemin V Kac Boston Birkhaumluser 1994 pp 349ndash383

[JS] Joyal A Street R Braided tensor categories Adv Math 102 20ndash78 (1993)[K1] Kong L Full field algebras operads and tensor categories Adv Math 213 271ndash340 (2007)[K2] Kong L Open-closed field algebras Commun Math Phys 280 207ndash261 (2008)[K3] Kong L Cardy condition for open-closed field algebras Commun Math Phys 283 25ndash92 (2008)[Ki] Kitaev A Private communication[KO] Kirillov AA Ostrik V On q-analog of mckay correspondence and ade classification of sl(2)

conformal field theories Adv Math 171 183ndash227 (2002)[KR] Kong L Runkel I Morita classes of algebras in modular tensor categories Adv Math 219

1548ndash1576 (2008)[Ld] Lauda AD Frobenius algebras and ambidextrous adjunctions Theo Appl Cat 16 84ndash

122 (2006)[Li1] Li H-S Regular representations of vertex operator algebras Commun Contemp Math 4

639ndash683 (2002)[Li2] Li H-S Regular representations and huang-lepowsky tensor functors for vertex operator

algebras J Alge 255 423ndash462 (2002)

Cardy Algebras and Sewing Constraints I 881

Proposition 2133 If F C1 rarr C2 is a Frobenius functor and (Am A ηAA εA) aFrobenius algebra in C1 then (F(A)m F(A) ηF(A)F(A) εF(A)) is a Frobenius alge-bra in C2

Proof One Frobenius property (m F(A)otimes idF(A)) (idF(A)otimesF(A)) = F(A) m F(A)follows from the commutativity of the following diagram (we spell out the associativityisomorphisms)

F(A)otimes F(A)idF(A)otimesF(A)

φF2

F(A)otimes F(A otimes A)idF(A)otimesψF

2

φF2

F(A)otimes (F(A)otimes F(A))

αF(A)F(A)F(A)

F(A otimes A)

F(idAotimesA)

F(m A)

F(A otimes (A otimes A))

F(αAAA)

(F(A)otimes F(A))otimes F(A)

φF2 otimesidF(A)

F((A otimes A)otimes A)

F(m AotimesidA)

ψF2 F(A otimes A)otimes F(A)

F(m A)otimesidF(A)

F(A)

F(A) F(A otimes A)ψF

2 F(A)otimes F(A)

(219)

The commutativity of the upper-left subdiagram follows from the naturalness of φF2 that

of the upper-right subdiagram follows from (25) that of the lower-left subdiagram fol-lows from the Frobenius properties of A and that of the lower-right subdiagram followsfrom the naturalness of ψ F

2 The proof of the other Frobenius property is similar Proposition 214 If F C1 rarr C2 is a tensor functor and A a Frobenius algebra in C1then

(i) F(A) has a natural structure of Frobenius algebra as given in Proposition 213(ii) If A is (normalised-)special so is F(A)

Proof Part (i) follows from Propositions 25 and 213 Part (ii) is a straightforwardverification of the definition using ψ F

2 = (φF2 )

minus1 and ψ F0 = (φF

0 )minus1

Let C1 C2 be sovereign tensor categories and F C1 rarr C2 a Frobenius functor Wedefine two morphisms IF(Aor) I prime

F(Aor) F(Aor) rarr F(A)or for a Frobenius algebra A inC1 as follows

IF(Aor) = ((ψ F0 F(dA) φF

2 )otimes idF(A)or) (idF(Aor) otimes bF(A))

I primeF(Aor) = (idF(A)or otimes (ψ F

0 F(dA) φF2 )) (bF(A) otimes idF(Aor))

(220)

It is easy to see that these are isomorphisms

Lemma 215 If F C1 rarr C2 is a Frobenius functor and A a Frobenius algebra in C1then

F(A) = IF(Aor) F(A) primeF(A) = I prime

F(Aor) F(primeA) (221)

3 After the preprint of the present paper appeared we noticed that this proposition is also proved in[DP Cor 5]

882 L Kong I Runkel

Proof We only prove the first equality the second one can be seen in the same way Bydefinition we have

IF(Aor) F(A)

= ((ψ F0 F(dA) φF

2 )otimes idF(A)or) (idF(Aor) otimes bF(A)) F(A)

=[(ψ F

0 F(dA) φF2 ) (F(A)otimes idF(A))

]otimes idF(A)or

(idF(A) otimes bF(A))

For the term inside the square brackets we find

ψ F0 F(dA) φF

2 (F(A)otimes idF(A)) = ψ F0 F(dA) F(A otimes idA) φF

2

= ψ F0 F(dA (A otimes idA)) φF

2 = ψ F0 F(εA m A) φF

2 (222)

On the other hand by definitionF(A) = [((ψ F0 F(εA) (F(m A)φF

2 ))otimes idF(A)or] (idF(A) otimes bF(A)) This demonstrates the first equality in (221) Proposition 216 Let F C1 rarr C2 be a tensor functor G C2 rarr C1 a functor〈FG χ〉 an adjunction A a C1-algebra and B a C2-algebra Then f A rarr G(B)is an algebra homomorphism if and only if f = χminus1( f ) F(A) rarr B is an algebrahomomorphism

Proof We need to show that

mG(B) ( f otimes f ) = f m A and f ηA = ηG(B) (223)

is equivalent to

m B ( f otimes f ) = f m F(A) and f ηF(A) = ηBs (224)

We first prove that the first identity in (223) is equivalent to the first identity in (224)For the left-hand side of the first identity in (223) we have the following equalities

mG(B) ( f otimes f )(1)= G(m B) φG

2 ( f otimes f )(2)= G(m B) G(ρ otimes ρ) G(ψ F

2 ) δ ( f otimes f )(3)= G(m B) G(ρ otimes ρ) G(ψ F

2 ) G F( f otimes f ) δ(4)= G(m B) G(ρ otimes ρ) G(F( f )otimes F( f )) G(ψ F

2 ) δ(5)= G

(m B (ρ otimes ρ) (F( f )otimes F( f )) ψ F

2

) δ

(6)= χ(

m B (ρ otimes ρ) (F( f )otimes F( f )) ψ F2

) (225)

where (1) is the definition of mG(B) in (217) (2) is the second identity in (210) (3)and (4) are naturality of δ and ψ F

2 respectively step (5) is functoriality of G and finallystep (6) is (29) For the right hand side of the first identity in (223) we get

f m A(1)= Gρ δG ( f m A)

(2)= Gρ G F( f m A) δ(3)= G(ρ F( f m A)) δ(4)= χ(ρ F( f ) F(m A)) (226)

Cardy Algebras and Sewing Constraints I 883

where (1) is the adjunction property (28) (2) is naturality of δ (3) functoriality of Gand (4) amounts to (29) and functoriality of F

On the other hand we see that the first equality in (224) is equivalent to

m B (ρ otimes ρ) (F( f )otimes F( f )) = ρ F( f ) F(m A) φF2 (227)

Using that φF2 is invertible with inverse (φF

2 )minus1 = ψ F

2 and that χ is an isomorphismit follows that the statement that (225) is equal to (226) is equivalent to the identity(227)

Now we prove that the second identity in (223) is equivalent to the second identity in(224) Using (217) and (210) we can write ηG(B) = G(ηB)φF

0 = G(ηB)G(ψ F0 )δ1

Together with (29) this shows that the second identity in (223) is equivalent to

f ηA = χ(ηB ψ F0 ) (228)

On the other hand the second identity in (224) is equivalent to

ρ F( f ) F(ηA) φF0 = ηB (229)

which by φF0 = (ψ F

0 )minus1 and (29) is further equivalent to (228)

Definition 217 Let (Am A ηAA εA) and (Bm B ηBB εB) be two Frobeniusalgebras in a tensor category C For f A rarr B we define f lowast B rarr A by

f lowast = ((εB m B)otimes idA) (idB otimes f otimes idA) (idB otimes (A ηA)) (230)

The following lemma is immediate from the definition of (middot)lowast and the properties ofFrobenius algebras We omit the proof

Lemma 218 Let C be a tensor category let A BC be Frobenius algebras in C andlet f A rarr B and g B rarr C be morphisms

(i) (g f )lowast = f lowast glowast(ii) f is a monomorphism iff f lowast is an epimorphism

(iii) f is an algebra map iff f lowast is a coalgebra map(iv) If f is a homomorphism of Frobenius algebras then f lowast f = idA and f f lowast =

idB(v) If C is sovereign and if A and B are symmetric then f lowastlowast = f

Let C and D be tensor categories and let F C rarr D be a Frobenius functor GivenFrobenius algebras A B in C and a morphism f A rarr B the next lemma shows how(middot)lowast behaves under F

Lemma 219 F( f lowast) = F( f )lowast

Proof The definition of the structure morphisms of the Frobenius algebra F(A) is givenin (217) and (218) Substituting these definitions gives

F( f )lowast =[(ψ F

0 F(εB) F(m B) φF2

)otimes idF(A)

] [

idF(B) otimes F( f )otimes idF(A)]

[idF(B) otimes

(ψ F

2 F(A) F(ηA) φF0

)]

= (ψ F0 otimes idF(A)) [

F (εB m B (idB otimes f ))otimes idF(A)]

(φF2 otimes idF(A)) (idF(B) otimes ψ F

2 )

[idF(B) otimes F(A ηA)

] (idF(B) otimes φF0 ) (231)

884 L Kong I Runkel

In the middle line of the last expression we can use the defining property (25) of F namely we substitute (φF

2 otimes idF(A)) (idF(B) otimes ψ F2 ) = ψ F

2 φF2 Then ψ F

2 can bemoved to the left and φF

2 to the right until they can be omitted against ψ F0 and φF

0 respectively using (22) and (24) This results in

F( f )lowast = F ((εB m B (idB otimes f ))otimes idA) F (idB otimes (A ηA)) (232)

which is nothing but F( f lowast)

23 Modular tensor categories Let C be a modular tensor category [TBK] ie anabelian semi-simple finite C-linear ribbon category with simple tensor unit 1 and a non-degeneracy condition on the braiding (to be stated in a moment) We denote the set ofequivalence classes of simple objects in C by I elements in I by i j k isin I and theirrepresentatives by Ui U j Uk We also set U0 = 1 and for an index k isin I we define kby Uk

sim= Uork

Since the tensor unit is simple we shall for modular tensor categories identifyHom(1 1) sim= C (cf footnote 2) Define numbers si j isin C by4

si j = UiU j (233)

They obey si j = s ji and s0i = dim Ui see eg [BK Sect 31] (In a ribbon cate-gory the left and right dimension (214) of Ui coincide and are denoted by dim Ui )The non-degeneracy condition on the braiding of a modular tensor category is that the|I|times|I|-matrix s should be invertible In fact [BK Thm 317]

sumkisinI

sik sk j = Dim C δij (234)

where Dim C = sumiisinI(dim Ui )

2 One can show (even in the weaker context of fusioncategories over C) that Dim C ge 1 [ENO1 Thm 23] In particular Dim C = 0 We fixonce and for all a square root

radicDim C of Dim C

Let us fix a basis λα(i j)kN k

i jα=1 in HomC(Ui otimesU j Uk) and the dual basis ϒ(i j)k

α N ki j

α=1

in HomC(UkUi otimesU j ) The duality of the bases means that λα(i j)k ϒ(i j)kβ = δαβ idUk

We also fix λ(0i)i = λ(i0)i = idUi We denote the basis vectors graphically as follows

λα(i j)k = α

Uk

Ui U j

ϒ(i j)kα = α

Uk

Ui U j

(235)

4 In the graphical notation used below we have given an orientation to the ribbons indicated by the arrowsFor example it is understood that this orientation determines which of the duality morphisms in (213) to use

Cardy Algebras and Sewing Constraints I 885

For V isin C we also choose a basis b(iα)V of HomC(VUi ) and the dual basis bV(iβ)

of HomC(Ui V ) for i isin I such that b(iα)V bV(iβ) = δαβ idUi We use the graphical

notation

b(iα)V = α

Ui

V

bV(iα) = α

Ui

V

(236)

Given two modular tensor categories C and D by C D we mean the tensor productof additive categories over C [BK Def 1115] ie the category whose objects are directsums of pairs V times W of objects V isin C and W isin D and whose morphism spaces are

HomCD(V times W V prime times W prime) = HomC(V V prime)otimesC HomC(WW prime) (237)

for pairs and direct sums of these if the objects are direct sums of pairsIf we replace the braiding and the twist in C by the antibraiding cminus1 and the antitwist

θminus1 respectively we obtain another ribbon category structure on C In order to distin-guish these two distinct structures we denote (C c θ) and (C cminus1 θminus1) by C+ and Cminusrespectively As in the Introduction we will abbreviate

C2plusmn = C+ Cminus (238)

Note that a set of representatives of the simple objects in C2plusmn is given by Ui times U j fori j isin I

For the remainder of Sect 2 we fix a modular tensor category C

24 The functors T and R The tensor product bifunctor otimes can be naturally extendedto a functor T C2plusmn rarr C Namely T (oplusN

i=1Vi times Wi ) = oplusNi=1Vi otimes Wi for all Vi Wi isin C

and N isin N The functor T becomes a tensor functor as follows For φT0 1 rarr T (1 times 1)

take φT0 = id1 (or lminus1

1 in the non-strict case) Next notice that for U VW X isin C

T (U times V )otimes T (W times X)= (U otimes V )otimes (W otimes X)

T ((U times V )otimes (W times X))= (U otimes W )otimes (V otimes X)(239)

We define φT2 T (U times V )otimes T (W times X) rarr T ((U times V )otimes (W times X)) by

φT2 = idU otimes cminus1

W V otimes idX (240)

(In the non-strict case the appropriate associators have to be added) The above definitionof φT

2 can be naturally extended to a morphism φT2 T (M1 otimes M2) rarr T (M1)otimes T (M2)

for any pair of objects M1M2 in C2plusmn The following result can be checked by directcalculation [JS Prop 52]

Lemma 220 The triple (T φT2 φ

T0 ) gives a tensor functor

886 L Kong I Runkel

In particular (T φT2 φ

T0 ψ

T2 ψ

T0 ) where ψT

2 = (φT2 )

minus1 and ψT0 = (φT

0 )minus1 gives

a Frobenius functorDefine the functor R C rarr C2plusmn as follows for A isin C and f isin HomC(A B)

R(A) =oplusiisinI

(A otimes Uori )times Ui R( f ) =

oplusiisinI

( f otimes idUori)times idUi (241)

This functor was also considered in a slightly different context in [ENO2 Prop 23]The family of isomorphisms γ R

A = oplusiisinI DimCdim Ui

id(AotimesUori )timesUi

isin Aut(R(A)) defines a

natural isomorphism γ R R rarr ROur next aim is to show that R is left and right adjoint to T in other words R

and T form an ambidextrous adjunction (see eg [Ld] for a discussion of ambidextrousadjunctions) To this end we introduce two linear isomorphisms for A isin C and M isin C2plusmn

χ HomC(T (M) A) minusrarr HomC2plusmn(M R(A))

χ HomC(A T (M)) minusrarr HomC2plusmn(R(A)M)(242)

If we decompose M as M = oplusNn=1 Ml

n times Mrn then χ and χ are given by

χ Noplus

n=1

fn

Mln Mr

n

A

rarrNoplus

n=1

oplusiisinI

sumα

fn

Mln

Mrn

A

α

Uori

times α

Mrn

Ui

(243)

and

χ Noplus

n=1 gn

Mln Mr

n

A

rarrNoplus

n=1

oplusiisinI

sumα gn

Mln

Mrn

A

α

Uori

times α

Ui

Mrn

DimCdim Ui

(244)

Notice that χ and χ are independent of the choice of basis

Theorem 221 〈T R χ〉 and 〈R T χminus1〉 are adjunctions ie R is both left and rightadjoint of T

Proof Write M as M = oplusNn=1 Ml

n times Mrn The isomorphism χ amounts to the following

composition of natural isomorphisms

HomC(T (M) A) = oplusnHomC(Mln otimes Mr

n A)sim= oplusni HomC(Ml

n otimes Ui A)otimes HomC(Mrn Ui )

sim= oplusni HomC(Mln A otimes Uor

i )otimes HomC(Mrn Ui ) = HomC2plusmn(M R(A)) (245)

Cardy Algebras and Sewing Constraints I 887

Thus χ is natural Let (γ RA )

lowast HomC2plusmn(R(A)M) rarr HomC2plusmn(R(A)M) denote the

pull-back of γ RA The isomorphism χ is equal to the composition of (γ R

A )lowast and the

following sequence of natural isomorphisms

HomC(A T (M)) = oplusn HomC(AMln otimes Mr

n)

sim= oplusni HomC(AMln otimes Ui )otimes HomC(Ui Mr

n)

sim= oplusni HomC(A otimes Uori Ml

n)otimes HomC(Ui Mrn) = HomC2plusmn(R(A)M) (246)

We have proved that both χ and χ are natural isomorphisms There are four natural transformations associated to χ and χ namely

idC2plusmnδminusrarr RT

ρminusrarr idC2plusmn and idCδminusrarr T R

ρminusrarr idC (247)

defined by for A isin C M isin C2plusmn

δM = χ(idT (M)) ρA = χminus1(idR(A))

ρM = χ (idT (M)) δA = χminus1(idR(A))(248)

They can be expressed graphically as follows with M = oplusNn=1 Ml

n times Mrn

δM =oplusni

sumα

Mln

Mln Mr

n Uori

αtimes

Ui

Mrn

α

ρA =oplusiisinI

A

A

Uori Ui

(249)

ρM =oplusni

sumα

Mln

Mln

Mrn Uor

i

α times

Ui

Mrn

α DimCdim Ui

δA =oplusiisinI

A

A Uori Ui

dim UiDimC

Note that

ρM δM = Dim C middot idM and ρA δA = idA (250)

Lemma 222 The functors T and R as maps on the sets of morphisms have left inversesand thus are injective

Proof Let f A rarr B be a morphism in C We define a map Q R HomC2plusmn(R(A)

R(B)) rarr HomC(A B) by f prime rarr ρB T ( f prime) δA Then we have

Q R R( f ) = ρB T R( f ) δA = ρB δB f = f (251)

888 L Kong I Runkel

where we used naturality of δ and (250) in the second and third equalities respectivelySo Q R is a left inverse of R on morphisms Thus R is injective on morphisms Similarlylet g M rarr N be a morphism in C2plusmn We define a map QT HomC(T (M) T (N )) rarrHomC2plusmn(M N ) by gprime rarr (Dim C)minus1 middot ρN R(gprime) δM Then we have

QT T (g) = (Dim C)minus1 middot ρN RT (g) δM = (Dim C)minus1 middot ρN δN g = g (252)

So QT is a left inverse of T on morphisms Thus T is injective on morphisms Using (29) and (250) one can express the two inverse maps χminus1 χminus1 as follows

for f isin HomC2plusmn(M R(A)) and g isin HomC2plusmn(R(A)M)

χminus1( f ) = ρ T ( f ) χminus1(g) = T (g) δ (253)

By Proposition 25 and Lemma 27 R is both a lax and colax tensor functor In particularφR

0 1 times 1 rarr R(1) is given by

φR0 = χ (ψT

0 ) = R(ψT0 ) δ1 times 1 = id1times1 (254)

and φR2 R(A)otimes R(B) rarr R(A otimes B) by φR

2 = R(ρA otimes ρB) R(ψT2 ) δ which can

be expressed graphically as

φR2 =

oplusi jkisinI

sumα

A

A

Uori B

B

Uorj

Uork

α

times

Ui U j

α

Uk

(255)

Similarly ψ R0 R(1) rarr 1 times 1 is given by

ψ R0 = ρ1 R(φT

0 ) = Dim C id1times1 (256)

and ψ R2 R(A otimes B) rarr R(A) otimes R(B) by ψ R

2 = ρ R(φT2 ) R(δA otimes δB) which in

graphical notation reads

ψ R2 =

oplusi jkisinI

sumα

A

A Uori

B

B Uorj

Uork

α

times

Ui U j

α

Uk

dim Ui dim U j

dim Uk Dim C

(257)

If C has more than one simple object then R does not take the tensor unit of C to thetensor unit of C2plusmn and so is clearly not a tensor functor However we will show that R isstill a Frobenius functor This will imply that if A is a Frobenius algebra in C then

R(A) = (R(A)m R(A) ηR(A)R(A) εR(A)) (258)

Cardy Algebras and Sewing Constraints I 889

is a Frobenius algebra in C2plusmn where the structure morphisms were given in (217) and(218) In the case A = 1 it was proved in [Muuml1 Prop 41] (see also [Fr Lem 619]and [K1 Thm 52]) that (258) is a commutative simple symmetric normalised-specialFrobenius algebra in C2plusmn In fact given a Frobenius algebra A in C it is straightforwardto verify that the structure morphisms in (258) are precisely those of (A times 1)otimes R(1)cf Sect 22

Proposition 223 (R φR2 φ

R0 ψ

R2 ψ

R0 ) is a Frobenius functor

Proof Using the explicit graphical expression of φR2 φ

R0 ψ

R2 ψ

R0 it is easy to see that

the commutativity of the diagrams (25) and (26) are equivalent to the statement thatR(1) with structure morphisms as in (258) is a Frobenius algebra in C2plusmn The latterstatement is true by [Muuml1 Prop 41]

From Lemma 220 and Proposition 223 we see that T and R take Frobenius alge-bras to Frobenius algebras The following two propositions show how the properties ofFrobenius algebras are transported

Proposition 224 Let A be a Frobenius algebra in C2plusmn Then T (A) is a Frobenius algebrain C and

(i) A is symmetric iff T (A) is symmetric(ii) A is (normalised-)special iff T (A) is (normalised-)special

Proof For part (i) write A as a direct sum oplusNn=1 Al

n times Arn Then the maps IT (A) I prime

T (A) T (Aor) rarr T (A)or defined in (220) are given by

IT (A) = I primeT (A) = oplusN

n=1c(Aln)

or(Arn)

or (259)

Therefore by (221) T (A) = primeT (A) is equivalent to T (A) = T (prime

A) Since byLemma 222 T is injective on morphisms this proves part (i) Part (ii) can be checkedin the same way for example the condition mT (A) T (A) = ζ idT (A) is easily checkedto be equivalent to T (m A A) = ζ T (idA) Proposition 225 Let A be a Frobenius algebra A in C Then R(A) is a Frobeniusalgebra in C2plusmn and

(i) A is symmetric iff R(A) is symmetric(ii) A is (normalised-)special iff R(A) is (normalised-)special

Proof Recall that the structure morphisms of the Frobenius algebra R(A) are equal tothose of (A times 1) otimes R(1) Using this equality part (i) and (ii) follow because R(1) issymmetric and normalised-special For example

m R(A) R(A) = [(m A A)times id1] otimes idR(1) = R(m A A) (260)

so that m R(A) R(A) = ζ idR(A) is equivalent to R(m A A) = ζ R(idA) which byLemma 222 is equivalent to m A A = ζ idA

The functor R has one additional property not shared by T namely R takes absolutelysimple algebras to absolutely simple algebras We will see explicitly in Sect 33 thatthis is not true for T

890 L Kong I Runkel

Lemma 226 For a C-algebra A the map

R HomA|A(A A) rarr HomR(A)|R(A)(R(A) R(A)) (261)

given by f rarr R( f ) is well-defined and an isomorphism

Proof Since R is a lax tensor functor R(A) is naturally a R(A)-bimodule It is easy tosee that R in (261) is a well-defined map R(A) is also naturally a R(1)-bimodule whichcan be identified with the induced R(1)-bimodule structure on (A times 1)otimes R(1) wherethe left R(1) action on (Atimes1)otimes R(1) is given by (idAtimes1 otimesm R(1))(cminus1

Atimes1R(1)otimes idR(1))We have the following natural isomorphisms

HomR(1)|R(1)(R(A) R(A))sim=minusrarr HomC2plusmn(A times 1 R(A))

χminus1

minusminusrarr HomC(A A) (262)

which by (253) are given by for f isin HomR(1)|R(1)(R(A) R(A))

f rarr f prime = f (idAtimes1 otimes ηR(1)) rarr f primeprime = ρ T ( f prime) (263)

and its inverse is given by for g isin HomC(A A)

g rarr gprime = R(g) δ rarr gprimeprime = (idAtimes1 otimes m R(1)) (gprime otimes idR(1)) (264)

where gprimeprime is indeed a R(1)-bimodule map due to the commutativity of R(1) It is easy tocheck that gprimeprime = R(g) in (264) Therefore R gives an isomorphism from HomC(A A)to HomR(1)|R(1)(R(A) R(A)) Moreover one verifies that R(g) is an R(A)-bimodulemap iff g is an A-bimodule map In other words R g rarr R(g) gives an isomorphism

HomA|A(A A)sim=minusrarr HomR(A)|R(A)(R(A) R(A))

Corollary 227 Let A be a C-algebra

(i) A is absolutely simple iff R(A) is absolutely simple(ii) Let A be in addition Frobenius Then A is simple and special iff R(A) is simple and

special

Proof Part (i) immediately follows from Lemma 226 The statement of part (ii) withoutthe qualifier lsquosimplersquo is proved in Proposition 225 But as in the proof of (iii)rArr(i) inLemma 211 a special Frobenius algebra in a semi-simple category has a semi-simplecategory of bimodules and for a semi-simple C-linear category simple and absolutelysimple are equivalent Part (ii) then follows from part (i)

The following lemma will be needed in Sect 32 below to discuss the properties ofCardy algebras

Lemma 228 Let A be a Frobenius algebra in C Then (δA)lowast = ρA

Proof Recall from (250) that δA is a morphism A rarr T R(A) Since T and R are bothFrobenius functors T R(A) is a Frobenius algebra in C Substituting the definitions aftera short calculation one finds

Cardy Algebras and Sewing Constraints I 891

εT R(A) mT R(A) = Dim(C)oplusiisinI

A Uori Ui A U or

ı Uı

(265)

Substituting this in the definition of (δA)lowast gives again after a short calculation the mor-

phism ρA At an intermediate step one uses that the part of the morphism (δA)lowast which

is made up of Ui and Uı ribbons their duals and the basis morphisms λ(iı)0 andϒ(iı)0can be replaced by 1

dim Uimiddot dUi

3 Cardy Algebras

In this section we start by investigating the properties of Frobenius algebras which sat-isfy the so-called modular invariance condition We then give two definitions of a Cardyalgebra and prove their equivalence Finally in Sect 33 we study the properties of thesealgebras and state our main results

We fix a modular tensor category C Recall that C2plusmn is an abbreviation for C+ Cminus

31 Modular invariance In C2plusmn we define the object K and the morphism ω K rarr Kas

K =oplus

i jisinIUi times U j ω =

sumi jisinI

dim Ui dim U j

Dim C idUi timesU j (31)

They have the property (see eg [BK Cor 3111])

ω

K

(Uk times Ul )or

(Uk times Ul )or

Ui times U j

Ui times U j

= δik δ jlDim C

dim Ui dim U jbUi timesU j dUi timesU j (32)

892 L Kong I Runkel

Definition 31

(i) Let A B be objects of C2plusmn A morphism f A otimes B rarr B is called S-invariant iff

A

B

W

W

f =

A

B

W

W

f

K

ω(33)

holds for all for W isin C2plusmn(ii) A C2plusmn-algebra (Aclmcl ηcl) is called modular invariant iff θAcl = idAcl and mcl

is S-invariant

Lemma 32 The morphism f A otimes B rarr B is S-invariant if and only if

A

B

Ui times U j

Ui times U j

f = Dim Cdim Ui dim U j

sumα

A

B

B

f

Ui times U j

Ui times U j

α

α

(34)

holds for all i j isin I

Proof Condition (33) holds for all W iff it holds for all W = Ui times U j i j isin I so itis enough to show that the right-hand side of (33) with W = Ui times U j is equal to theright-hand side of (34) Recall the notation for basis morphisms in (236) Starting from(33) write

idBor otimes idUi timesU j =⎛⎝sum

klα

(b(ktimeslα)

B

)or (

bB(ktimeslα)

)or⎞⎠ otimes idUi timesU j (35)

and then apply (32) The graphical representation of the resulting morphism can bedeformed to give (34) Remark 33 As shown in [K3 Sect 61] the modular invariance condition of a C2plusmn-alge-bra exactly coincides with the modular invariance condition for torus 1-point correlationfunctions of a genus-01 closed CFT In particular the condition θAcl = idAcl is equiv-alent to invariance under the modular transformation T τ rarr τ + 1 and the condition(34) with f = mcl is equivalent to invariance under S τ rarr minus 1

τ Combining the

modular invariance condition with the genus zero properties of a genus-01 closed CFTresults in a modular invariant commutative symmetric Frobenius algebra in C2plusmn

Cardy Algebras and Sewing Constraints I 893

Let Acl be a modular invariant C2plusmn-algebra Evaluating (34) for f = mcl composingit with ηcl otimes idUi timesU j and taking the trace implies the following identity

Zi j = 1

DimCAclUi timesU j where

(36)

Zi j = dimC HomC2plusmn(UitimesU j Acl)

Decomposing Acl into simple objects this gives

Zi j =sum

klisinISik Zkl Sminus1

l j where Si j = si jradic

DimC (37)

which in CFT terms is of course nothing but the invariance of the torus partition functionunder the modular S-transformation

The following theorem gives a simple criterion for modular invariance

Theorem 34 Let Acl be a haploid commutative symmetric Frobenius algebra in C2plusmn

(i) If Acl is modular invariant then dim Acl = Dim C(ii) If dim Acl = Dim C then Acl is special and modular invariant

Proof Part (i) Since Acl is haploid for i = j = 0 Equation (36) reduces to 1 =dim AclDimCPart (ii) By the same reasoning as in the proof of (iii)rArr(i) in Lemma 211 one showsthat Acl is special Thus mcl cl = ζclidAcl for some ζcl = 0

By [KO Thm 45] the category (C2plusmn)locAcl

of local Acl-modules is again a mod-

ular tensor category and Dim (C2plusmn)locAcl

= (Dim C dim Acl)2 (see [Fr Prop 321 amp

Rem 323] for the same statement in the notation used here) Thus by assumption wehave Dim(C2plusmn)loc

Acl= 1 It then follows from [ENO1 Thm 23] that up to isomorphism

(C2plusmn)locAcl

has a unique simple object (namely the tensor unit) In other words every simplelocal Acl-module is isomorphic to Acl (seen as a left-module over itself)

We have the following isomorphisms between morphism spaces [FS Prop 47 amp 411]

HomAcl(Acl otimes (Ui times U j ) Acl) sim= HomC2plusmn(Ui times U j Acl)

HomAcl(Acl Acl otimes (Ui times U j )) sim= HomC2plusmn(AclUi times U j )

Using these to transport the bases (236) from the right to the left we obtain bases bα(i j)αof HomAcl(Acl otimes (Ui times U j ) Acl) and b(i j)

β β of HomAcl(Acl Acl otimes (Ui times U j )) Thesecan be expressed graphically as

894 L Kong I Runkel

bα(i j) =

Acl

Acl

Acl

Ui times U j

α

b(i j)β = DimC

dim Ui dim U j

1

ζcl

Acl

Ui times U jAcl

Acl

β

(38)

where the nonzero factor in b(i j)β is included for convenience Notice that bα(i j) b(i j)

β isa left Acl-module map Since Acl is simple as a left module over itself we have

bα(i j) b(i j)β = λαβ idAcl (39)

for some λαβ isin C By computing tr(bα(i j) b(i j)β ) it is easy to verify that λαβ = δαβ

We will now prove the following identity

1

ζcl

Acl

Acl

Ui times U j

Ui times U j

Acl

=sumα

DimCdim Ui dim U j

1

ζcl

Acl

Ui times U j

Ui times U j

Acl

Acl

Acl

α

α

(310)

One checks that the left-hand side of this equation is an idempotent which we denote byζminus1

cl PlAcl(Ui timesU j ) cf [Fr Sect 31] By [Fr Prop 41] the image Im(ζminus1

cl PlAcl(Ui timesU j ))

is a local Acl-module and hence isomorphic to AoplusNcl for some N isin Zge0

All left-module morphisms from Acl to Aclotimes(Ui timesU j ) are linear combinations of the

b(i j)β Furthermore one verifies that ζminus1

cl PlAcl(Ui timesU j )b(i j)

β = b(i j)β Therefore the b(i j)

β

describe precisely the image of the idempotent ie ζminus1cl Pl

Acl(Ui timesU j ) = sum

α b(i j)α bα(i j)

which is nothing but (310) Composing (310) with ζcl middot εcl otimes idUi timesU j from the left (iefrom the top) produces (34)

In addition since Acl is commutative symmetric Frobenius it satisfies θAcl = idAcl

[Fr Prop 225] Altogether this shows that Acl is modular invariant Remark 35 As we were writing this paper we heard that the results in Theorem 34were obtained independently by Kitaev and Muumlger [Ki]

Remark 36 Setting i = j = 0 in (36) gives the identity dim Acl = Z00 Dim C [KRProp 23] Combining this with Theorem 34 (ii) one may wonder if a general modularinvariant commutative symmetric Frobenius algebra Acl in C2plusmn is isomorphic to a directsum of simple such algebras However this is not so For example one can take the com-mutative symmetric Frobenius algebra Acl = C[x]x2 in the category of vector spacesequipped with the non-degenerate trace ε(ax + b) = a In this case the modular invari-ance condition holds automatically but Acl is clearly not a direct sum of two algebras

Cardy Algebras and Sewing Constraints I 895

For a general modular tensor category C the algebra C[x]x2 R(1) understood as analgebra in C2plusmn via the braided monoidal isomorphism Vect f (C) C2plusmn rarr C2plusmn providesanother counter-example

32 Two definitions Define a morphism PlA A rarr A for a Frobenius algebra A in C

or C2plusmn as follows [Fr Sect 24]

PlA =

A

A

A

m

(311)

If A is also commutative and obeys m A A = ζA idA we have PlA = ζA idA In partic-

ular this holds if A is commutative and special Using the fact that the Frobenius algebraR(1) is commutative and normalised-special one can check that Pl

R(A) R(A) rarr R(A)takes the following form

PlR(A) =

oplusiisinI

A

A

Uori

Uori

A

m

times

Ui

Ui

(312)

With these ingredients we can now give the first definition of a Cardy C|C2plusmn-algebrawhich was introduced in [K3 Def 514] cf Remark 315 below

Definition 37 (CardyC|C2plusmn-algebra I) A CardyC|C2plusmn-algebra is a triple (Aop|Acl ιcl-op)where (Aclmcl ηclcl εcl) is a modular invariant commutative symmetric Frobe-nius C2plusmn-algebra (Aopmop ηopop εop) is a symmetric Frobenius C-algebra andιcl-op Acl rarr R(Aop) an algebra homomorphism such that the following conditionsare satisfied

(i) Centre condition

Acl R(Aop)

R(Aop)

R(Aop)

ιcl-op

m R(Aop)

=

Acl

ιcl-op

R(Aop)

R(Aop)

R(Aop)

m R(Aop)

(313)

896 L Kong I Runkel

(ii) Cardy condition

ιcl-op ιlowastcl-op =

R(Aop)

R(Aop)

R(Aop)

(314)

Remark 38

(i) The name ldquoCardy C|C2plusmn-algebrardquo in Definition 37 was chosen because many ofthe important ingredients were first studied by Cardy the modular invariance ofthe closed theory [C1] the consistency of the annulus amplitude [C2] and thebulk-boundary OPE [CL] On the other hand the boundary-boundary OPE andthe OPE analogue of the centre condition were first considered in [Lw]

(ii) One can easily see that in the special case that C is the category Vect f (C) of finite-dimensional C-vector spaces a Cardy C|C2plusmn-algebra gives exactly the algebraicformulation of two-dimensional open-closed topological field theory over C (cfRemark 614 in [K3]) see [Lz Sect 48] [Mo Thm 11] [AN Thm 45] [LPCor 43] [MS Sect 22] When passing to a general modular tensor category Cthere are two important differences to the two-dimensional topological field the-ory Firstly the algebras Acl and Aop now live in different categories which inparticular affects the formulation of the centre condition and the Cardy conditionSecondly the modular invariance condition has to be imposed on Acl In the caseC = Vect f (C) modular invariance holds automatically

Definition 39 A homomorphism of Cardy C|C2plusmn-algebras (A(1)op |A(1)cl ι(1)cl-op) rarr (A(2)op

|A(2)cl ι(2)cl-op) is a pair ( fop fcl) of Frobenius algebra homomorphisms fop A(1)op rarr A(2)op

and fcl A(1)cl rarr A(2)cl such that the diagram

A(1)cl

fcl

ι(1)cl-op

A(2)cl

ι(2)cl-op

R(A(1)op )

R( fop) R(A(2)op )

(315)

commutes

Remark 310 Since a homomorphism of Frobenius algebras is invertible (cf Lemma218 (iv)) a homomorphism of Cardy algebras is always an isomorphism

For a homomorphism ( fop fcl) of Cardy C|C2plusmn-algebras using the commutativity of(315) and the fact that fcl and fop are both algebra and coalgebra homomorphisms itis easy to show that (315) commutes iff

R(A(1)op )R( fop)

(ι(1)cl-op)

lowast

R(A(2)op )

(ι(2)cl-op)

lowast

A(1)cl

fcl A(2)cl

(316)

Cardy Algebras and Sewing Constraints I 897

commutesLet (Aop|Acl ιcl-op) be a Cardy C|C2plusmn-algebra Define the morphism

ιcl-op = χminus1(ιcl-op) T (Acl) minusrarr Aop (317)

Decompose Acl as Acl = oplusNn=1Cl

n times Crn such that Cl

1 times Cr1 = ηcl(1 times 1) We use ι(n)cl-op

to denote the restriction of ιcl-op to Cln times Cr

n and ι(n)cl-op to denote the restriction of ιcl-op

to Cln otimes Cr

n We introduce the following graphical notation

ι(n)cl-op =

Cln Cr

n

Aop

(318)

By (243) ιcl-op can be expressed in terms of ιcl-op as follows

ιcl-op =Noplus

n=1

oplusiisinI

sumα

Cln

Crn

Aop

α

Uori

times α

Crn

Ui

(319)

Lemma 311 The centre condition (313) is equivalent to the following condition in C

Cln Cr

n

Aop

Aop

Aop

=

Cln Cr

n

Aop

Aop

Aop

for n = 1 N (320)

Proof First insert (319) and the definition (217) of m R(Aop) into (313) Then apply thecommutativity of R(1) to the left hand side of (313) The equivalence between (313)and (320) follows immediately Remark 312 The centre condition (320) is very natural from the open-closed confor-mal field theory point of view Correlators on the upper half plane are expressed in termsof conformal blocks on the full complex plane The objects Cl

n and Crn are associated to

the field insertion at a point z in the upper half plane and at the complex conjugate pointz in the lower half plane respectively The object Aop corresponds to a field inserted ata point r on the real axis The centre condition (320) simply says that the correlationfunctions in the disjoint domains |z| gt r gt 0 and r gt |z| gt 0 are analytic continuationsof each other see [K2 Prop 118]

898 L Kong I Runkel

Recall that we define ιlowastcl-op Aop rarr T (Acl) as in (230) We introduce the graphicalnotation

ιlowastcl-op =Noplus

n=1

Cln Cr

n

Aop

=Noplus

n=1

Cln Cr

n

Aop

Aop T (minus1Acl) (321)

where the second equality follows from (215) (221) and (259)

Lemma 313 The Cardy condition (314) is equivalent to the following identity in C

Noplusn=1

DimCdim Ui

sumα

Cln

Crn

Crn

α

α

Aop

Aop Uori

Uori

=Aop

Aop

Aop

Uori

Uori

op

mop for all i isin I (322)

Proof By (244) (312) and (319) it is easy to see that (322) is equivalent to thefollowing identity

ιcl-op χ(ιlowastcl-op) = PlR(Aop)

(323)

Therefore it is enough to show that

χ(ιlowastcl-op) = ιlowastcl-op (324)

We have

χminus1(ιlowastcl-op)(1)= T (ιlowastcl-op) δ (2)= T (ιcl-op)

lowast δlowastlowast (3)= (ρ T (ιcl-op)

)lowast (4)= (ιcl-op

)lowast (325)

In step (1) we use the expression (253) for χminus1 step (2) follows from Lemma 218 (v)and Lemma 219 Step (3) is Lemma 218 (i) and Lemma 228 and finally step (4)amounts to substituting (253) and (317) Acting with χ on both sides of the aboveequality produces (324)

Combining Lemmas 311 and 313 and Proposition 216 we obtain the followingequivalent definition of Cardy C|C2plusmn-algebra (recall the graphical notation (318) forιcl-op and (321) for ιlowastcl-op)

Cardy Algebras and Sewing Constraints I 899

Definition 314 (CardyC|C2plusmn-algebra II) A CardyC-algebra is a triple (Aop|Acl ιcl-op)where Acl is a commutative symmetric Frobenius C2plusmn-algebra satisfying property (34)with f = mcl Aop is a symmetric Frobenius C-algebra and ιcl-op T (Acl) rarr Aop isan algebra homomorphism satisfying the conditions (320) and (322)

Remark 315 Up to a choice of normalisation Definition 314 is the same as the orig-inal one in [K3 Def 613] The difference between the two definitions is the factorDim C dim Ui on the left hand side of (322) which in [K3 Def 613] is given byradic

Dim C dim Ui The two definitions are related by rescaling the coproduct cl andcounit εcl of Acl by 1

radicDim C and

radicDim C respectively We chose the convention in

(322) to remove all dimension factors from the expression (314) for the Cardy condition

33 Uniqueness and existence theorems In this subsection we investigate the structureof Cardy algebras We start with the following proposition which when combined withthe results of part II provides an alternative proof of [Fj Prop 422]

Proposition 316 Let (Aop|Acl ιcl-op) be a Cardy C|C2plusmn-algebra If Acl is simple anddim Aop = 0 then Aop is simple and special

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis therefore haploid Restricting the Cardy condition (322) to the case Ui = 1 andcomposing both sides with εop from the left we see that εop kills all terms associated toU j times 1 isin Acl in the sum except for a single 1 times 1 term Thus we obtain the followingidentity

β εop = dAop (mop otimes idAorop) (idAop otimes bAop) (326)

where β isin C Composing with ηop from the right in turn implies that βεop ηop =dim Aop which is nonzero by assumption Thus also β = 0 and εop is a nonzero multipleof the morphism on the right hand side of (326) By [FRS Lem 311] Aop is special

Since Aop is a special Frobenius algebra Aop is semi-simple as an Aop-bimodule(apply [FS Prop 524] to Aop tensored with its opposite algebra) Suppose Aop is not

simple so that we can write Aop = A(1)op oplus A(2)op for nonzero Aop-bimodules A(1)op and A(2)op We denote the canonical embeddings and projections associated to this decompositionas ι12 and π12 We have the identities

mop (ι1 otimes ι2) = 0 εop ηop =2sum

i=1

εop ιi πi ηop (327)

The first identity follows since π1 mop (ι1 otimes ι2) = 0 (as mop gives the left action of

Aop on Aop and hence it preserves A(2)op ) and similarly π2 mop (ι1 otimes ι2) = 0 Thesecond identity is just the completeness of ι12 π12

Since εop ηop = 0 without losing generality we can assume εop ι1 π1 ηop = 0Using that π2 is a bimodule map we compute

π2 [LHS of (322)]Ui =1 ι1 π1 ηop = π2 [RHS of (322)]Ui =1 ι1 π1 ηop

= PlA(2)op

π2 ι1 π1 ηop = 0 (328)

900 L Kong I Runkel

On the other hand using that Acl is haploid that ιcl-op is an algebra map and thatι lowastcl-op is a coalgebra map one can check that the left-hand side of (328) is equal toλ(εop ι1 π1 ηop) π2 ηop for some λ = 0 This implies that π2 ηop = 0 Thusηop = sum

i ιi πi ηop = ι1 π1 ηop Hence we have

0 = π2 ι2 = π2 mop (ηop otimes ι2) = π2 mop ((ι1 π1 ηop)otimes ι2) (329)

However the right-hand side is zero by (327) This is a contradiction and hence Aopmust be simple

To formulate the next theorem we need the notion of the full centre of an algebra [FjDef 49] Recall that an algebra A in a braided tensor category has a left centre and aright centre [VZO] both of which are sub-algebras of A Of these two we will onlyneed the left centre The following definition is [Fr Def 231] which in our setting isequivalent to that of [VZO]

Definition 317 Let A be a symmetric special Frobenius algebra such that m A A =ζA idA

(i) The left centre Cl(A) of A is the image of the idempotent ζminus1A Pl

A(ii) The full centre Z(A) is Cl(R(A))

That ζminus1A Pl

A is an idempotent follows from [FRS Lem 52] when keeping track ofthe factors ζA ([FRS] assumes normalised-special ie ζA = 1) Note that Cl(A) is againan object of C while Z(A) is an object of C2plusmn Let el

A Cl(A) rarr A be the embeddingof Cl(A) into A The left centre is in fact the maximal subobject of A such that

m A cAA (elA otimes idA) = m A (330)

see [Fr Lem 232] This observation explains the name left centre and also makes theconnection to [O Def 15]

The full centre is by definition the image of the idempotent ζminus1A Pl

R(A) R(A) rarrR(A) Since C2plusmn is abelian the idempotent splits and we obtain the embedding andrestriction morphisms

e Z(A) rarr R(A) and r R(A) Z(A) (331)

which obey r e = idZ(A) and e r = ζminus1A Pl

R(A) It follows from Proposition 225 and

[Fr Prop 237] that Z(A) is a commutative symmetric Frobenius algebra in C2plusmn withstructure morphisms5

m Z(A) = r m R(A) (e otimes e) ηZ(A) = r ηR(A)

Z(A) = ζA middot (r otimes r) R(A) e εZ(A) = ζminus1A middot εR(A) e

(332)

Moreover if A is simple then Z(A) is simple and if A is simple and dim A = 0 thenZ(A) is simple and special The normalisation of the counit is such that

εZ(A) ηZ(A) = ζminus2A dim A DimC (333)

5 The normalisation of product and unit is the standard one The factors in the coproduct and counit haveto be included in order for (A|Z(A) e) to be a Cardy algebra see Theorem 318 below The normalisation ofthe counit enters the Cardy condition (314) through the definition of ( middot )lowast

Cardy Algebras and Sewing Constraints I 901

Theorem 318 Let A be a special symmetric Frobenius C-algebra Then (A|Z(A) e)is a Cardy C|C2plusmn-algebra

The proof of this theorem makes use the following two lemmas

Lemma 319 e Z(A) rarr R(A) is an algebra map and elowast = ζA middot r

Proof It follows from [Fr Lem 229] (or by direct calculation using in particularm R(A) R(A) = ζAidR(A)) that

m R(A) (e otimes e) = ζminus1A middot Pl

R(A) m R(A) (e otimes e) (334)

Substituting e r = ζminus1A Pl

R(A) shows that e is compatible with multiplication For theunit one finds

e ηZ(A) = e r ηR(A) = ζminus1A Pl

R(A) ηR(A) = ηR(A) (335)

Thus e is an algebra map For the second statement one computes

elowast (1)= ζA

R(A)

r

R(A)

R(A)

Z(A)

R(A)

Z(A)

r

e

e

(2)=

R(A)

r

R(A)

R(A)

R(A)

Z(A)

(3)=

R(A)

r

R(A)

Z(A)

(4)= ζA middot r (336)

where in (1) the definitions (230) and (332) have been substituted step (2) is e r = ζminus1

A PlR(A) step (3) uses that R(A) is symmetric Frobenius and step (4) is again

e r = ζminus1A Pl

R(A) Lemma 320 Let A be a symmetric Frobenius algebra in C The morphism

PlR(A) m R(A)

(Pl

R(A) otimes PlR(A)

) R(A)otimes R(A) minusrarr R(A) (337)

is S-invariant

The proof of this lemma is a slightly lengthy explicit calculation and has been deferredto Appendix A2

902 L Kong I Runkel

Proof of Theorem 318 That e is an algebra map was proved in Lemma 319 The centrecondition (313) holds by property (330) of the left centre The Cardy condition (314)also is an immediate consequence of Lemma 319

ιcl-op ιlowastcl-op = e (ζA r) = PlR(A) (338)

The full centre Z(A) is a commutative symmetric Frobenius algebra It remains to provemodular invariance That θZ(A) = idZ(A) is implied by commutativity and symmetryof Z(A) [Fr Prop 225] The S-invariance condition (33) follows from Lemma 320In (337) substitute Pl

R(A) = ζA e r and then put the resulting morphism into (33)Compose the resulting equation with e otimes idW from the right (ie from the bottom) andsubstitute the definition (332) of m Z(A) This results in the statement that m Z(A) isS-invariant

The following theorem is analogous to [LR Prop 29] and [Fj Thm 426] whichroughly speaking answer the question under which circumstances the restriction of atwo-dimensional conformal field theory to the boundary already determines the entireconformal field theory The first work is set in Minkowski space and uses operator alge-bras and subfactors while the second work is set in Euclidean space and uses modulartensor categories

Theorem 321 Let (A|Acl ιcl-op) be a Cardy C|C2plusmn-algebra such that dim A = 0 andAcl is simple Then A is special and (A|Acl ιcl-op) sim= (A|Z(A) e) as Cardy algebras

Proof By Proposition 316 A is simple and special Since Acl is simple the algebramap ιcl-op Acl rarr R(A) is either zero or a monomorphism But ιcl-op ηcl = ηR(A)and so ιcl-op = 0 Thus ιcl-op is monic By Lemma 218 (ii) ζminus1

A ιlowastcl-op is epi The Cardycondition (314) implies

ιcl-op ζminus1A ιlowastcl-op = ζminus1

A PlR(A) = e r (339)

Composing this with e r from the left yields e r ιcl-op ζminus1A ιlowastcl-op = e r =

ιcl-op ζminus1A ιlowastcl-op Since ζminus1

A ιlowastcl-op is epi we have

e r ιcl-op = ιcl-op (340)

Actually (340) also follows from (313) and specialness of R(A) We will prove that

( fop fcl) (A|Acl ιcl-op) minusrarr (A|Z(A) e) where fop = idA fcl = r ιcl-op

(341)

is an isomorphism of Cardy algebrasfcl is an algebra map Compatibility with the units follows since ιcl-op is an algebramap

fcl ηcl = r ιcl-op ηcl = r ηR(A) = ηZ(A) (342)

Compatibility with the multiplication also follows since ιcl-op is an algebra map

m Z(A) ( fcl otimes fcl) = r m R(A) (e otimes e) (r otimes r) (ιcl-op otimes ιcl-op)

= r m R(A) (ιcl-op otimes ιcl-op) = r ιcl-op mcl = fcl mcl (343)

where in the second step we used (340)

Cardy Algebras and Sewing Constraints I 903

fcl is an isomorphism As above since fcl is an algebra map and since Acl is simple fcl

has to be monic By Lemma 319 rlowast = ζminus1A e Thus f lowast

cl = ιlowastcl-op rlowast = ζminus1A ιlowastcl-op e and

fcl f lowastcl = r ιcl-op ζminus1

A ιlowastcl-op e = r e r e = idZ(A) (344)

and so fcl is also epi and hence isofcl is a coalgebra map Since fcl is an algebra map so is f minus1

cl By (344) f minus1cl = f lowast

cland by Lemma 218 (iii) this implies that fcl is a also coalgebra mapThe diagram (315) commutes Commutativity of (315) is equivalent to e fcl = ιcl-opwhich holds by (340)

Let A be a special symmetric Frobenius algebra So far we have seen that (A|Z(A) e)is a Cardy algebra and that all Cardy algebras with Aop = A and simple Acl are of thisform It is now natural to ask if every simple Acl does occur as part of a Cardy algebraThe following theorem provides an affirmative answer Recall that for an A-left moduleM the object Mor otimesA M is an algebra (see eg [KR Lem 42])

Theorem 322 If Acl is a simple modular invariant commutative symmetric FrobeniusC2plusmn-algebra then there exist a simple special symmetric Frobenius C-algebra A and amorphism ιcl-op Acl rarr R(A) such that

(i) Acl sim= Z(A) as Frobenius algebras(ii) (A|Acl ιcl-op) is a Cardy C|C2plusmn-algebra

(iii) T (Acl) sim= oplusκisinJ Morκ otimesA Mκ as algebras where MκκisinJ is a set of representa-

tives of the isomorphism classes of simple A-left modules

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis haploid It then follows from Theorem 34 that Acl is special By Proposition 224T (Acl) is a special symmetric Frobenius algebra in C Thus T (Acl) = oplusi Ai where theAi are simple symmetric Frobenius algebras We will show that at least one of the Aiis special Since T (Acl) is special we have mT (Acl) T (Acl) = ζ idT (Acl) for someζ isin C

times Restricting this to the summand Ai shows mi i = ζ idAi FurthermoreεT (Acl) ηT (Acl) = ξ id1 for some ξ isin C

times But εT (Acl) ηT (Acl) = sumi εi ηi and so

at least one of the εi ηi has to be nonzero Therefore at least one of the Ai is speciallet A equiv Ai be this summand We denote the embedding A rarr T (Acl) by e0 and therestriction T (Acl) A by r0 Notice that r0 is an algebra homomorphism Define

ιcl-op = χ (r0) Acl minusrarr R(A) (345)

By Proposition 216 ιcl-op is an algebra homomorphism Next we verify the centre con-dition (313) or rather its equivalent form (320) By substituting the definitions onecan convince oneself that the commutativity mcl cAclAcl = mcl of Acl in C2plusmn implies thecondition mT (Acl) = mT (Acl) in C see [K2 Prop 36] Here T (Acl)otimes T (Acl) rarrT (Acl)otimes T (Acl) is given by

=oplusmn

(idCl

notimes cCl

m Crnotimes idCr

m

)

(cCl

m Clnotimes cminus1

Crn C

rm

)

(idCl

motimes cminus1

Cln C

rm

otimes idCrn

)

(346)

and we decomposed Acl as Acl = oplusnCln times Cr

n As a consequence we obtain the identity

r0 mT (Acl) (idAcl otimes e0) = r0 mT (Acl) (idAcl otimes e0) (347)

904 L Kong I Runkel

Using that r0 is an algebra map and that by definition ιcl-op = r0 we obtain (320)In order to show that (A|Acl ιcl-op) is a Cardy algebra it remains to show that the

Cardy condition (314) is satisfied We will demonstrate this via a detour by first provingthat Acl sim= Z(A) as Frobenius algebras

Recall the notations e and r given in (331) Using the centre condition (320) onecan check that Pl

R(A) ιcl-op = m R(A) R(A) ιcl-op By specialness of A we have

m A A = ζA idA and so together with e r = ζminus1A Pl

R(A) we get

e r ιcl-op = ιcl-op (348)

Next consider the morphism

fcl = r ιcl-op Acl minusrarr Z(A) (349)

By the same derivation as in (342) and (343) one sees that fcl is an algebra map Inparticular fcl ηcl = ηZ(A) = 0 and so fcl = 0 Since Acl is simple fcl has to be amonomorphism

By the same argument as used in the proof of Theorem 34 (ii) up to isomorphismAcl is the unique simple local Acl-(left-)module The algebra monomorphism fcl turnsZ(A) into an Acl-module Since Z(A) is commutative it is local as an Acl-module andso Z(A) sim= AoplusN

cl for some N ge 1 By construction A is a simple special symmetricFrobenius algebra Proposition 225 and Corollary 227 show that R(A) inherits all theseproperties and thus Z(A) is simple (see the comment below Eq (332)) By Theorem318 Z(A) is modular invariant and then by Theorem 34 (i) dim Z(A) = Dim C Thisimplies that N = 1 in Z(A) sim= AoplusN

cl and so fcl is in fact an isomorphismSince Acl and Z(A) are both haploid we have εZ(A) fcl = ξ εcl for some ξ isin C

timesThe counit uniquely determines the Frobenius structure on Acl and Z(A) (see eg [FRSLemma 37]) so that fcl is a coalgebra isomorphism iff ξ = 1 To compute ξ we composethe above identity with ηcl from the right Defining ζcl via εcl ηcl = ζminus1

cl Dim C middot id1 andusing (333) gives ξ = dim A ζclζ

2A By rescaling the comultiplication and the counit

of A and consequently changing ζA we can always achieve ξ = 1 This proves part (i)of the theorem

Equation (348) implies that ιcl-op = e fcl Since fcl is an isomorphism of Frobeniusalgebras by Lemmas 218 and 319 we have

ιcl-op ιlowastcl-op = e fcl f lowastcl elowast = ζA e r = Pl

R(A) (350)

Thus (A|Acl ιcl-op) is a Cardy algebra This proves part (ii) of the theoremPart (iii) can be seen as follows By [KR Prop 43] T Z(A) sim= oplusκisinJ Mor

κ otimesA Mκ asalgebras Together with the observation that T ( fcl) T (Acl) rarr T Z(A) is an isomor-phism of algebras this proves part (iii) Remark 323 Part (i) of Theorem 322 was announced by Muumlger [Muuml2] We provide anindependent proof in the setting of Cardy algebras

The above theorem together with Lemma 211 and Theorem 34 shows that a sim-ple commutative symmetric Frobenius C2plusmn-algebra Acl with dim Acl = Dim C is alwayspart of a Cardy algebra (Aop|Acl ιcl-op) for some simple special symmetric Frobeniusalgebra Aop in C However the above proof also illustrates that Aop is not unique Thisraises the question how two Cardy algebras with a given Acl can differ This question isanswered by [KR Thm 11] which in the present framework can be restated as follows

Cardy Algebras and Sewing Constraints I 905

Theorem 324 If (A(i)op |A(i)cl ι(i)cl-op) i = 1 2 are two Cardy C|C2plusmn-algebras such that

A(i)cl is simple and dim A(i)op = 0 for i = 1 2 then A(1)clsim= A(2)cl as algebras if and only

if A(1)op and A(2)op are Morita equivalent

Proof Theorem 11 in [KR] is stated for A(i)op being non-degenerate algebras and A(i)cl =Z(A(i)op ) for i = 1 2 By Proposition 316 A(i)op are simple and special for i = 1 2

Then by [KR Lem 21] A(i)op are non-degenerate algebras By Theorem 321 we have

A(i)clsim= Z(A(i)op ) as Frobenius algebras Finally by [KR Thm 11] Z(A(1)op ) sim= Z(A(2)op )

as algebras iff A(1)op and A(2)op are Morita equivalent Let Cmax(C2plusmn) be the set of equivalence classes [B] of simple modular invariant

commutative symmetric Frobenius algebras B in C2plusmn Two such algebras B and B prime areequivalent if B and B prime are isomorphic as algebras (but not necessarily as Frobenius alge-bras) Let Msimp(C) be the set of Morita classes of simple special symmetric Frobeniusalgebras in C Define the map z Msimp(C) rarr Cmax(C2plusmn) by z A rarr [Z(A)] whereA denotes the Morita class of A From Theorem 322 (i) and [KR Thm 11] we learn

Corollary 325 The map z Msimp(C) rarr Cmax(C2plusmn) is a bijection

A Appendix

A1 Proof of Lemma 27 We will show that if (F ψ F2 ψ

F0 ) is a colax tensor functor

from C1 to C2 then (G φG2 φ

G0 ) is a lax tensor functor from C2 to C1 Applying this

result to the opposed categories then gives the converse statementWe need to show that φG

0 and φG2 make the diagrams (21) and (22) commute We

first prove the commutativity of (21) Consider the following diagram

G(A)otimes (G(B)otimes G(C))

GψF2 δ

idG(A)otimesφG2 G(A)otimes G(B otimes C)

GψF2 δ

G(FG(A)otimes F(G(B)otimes G(C)))

G(F(idG(A))otimesF(φG2 ))

G(idFG(A)otimesψ F2 )

G(FG(A)otimes FG(B otimes C))

G(ρAotimesρBotimesC )

G(FG(A)otimes (FG(B)otimes FG(C)))

G(ρAotimes(ρBotimesρC )) G(A otimes (B otimes C))

(A1)

The top subdiagram is commutative because of the naturality of Gψ F2 δ The commu-

tativity of the bottom subdiagram follows from the following identities

(ρB otimes ρC ) ψ F2 = (ρB otimes ρC ) ψ F

2 ρF Fδ

= ρBotimesC FG(ρB otimes ρC ) FG(ψ F2 ) Fδ

= ρBotimesC F(φG2 ) (A2)

as a map F(G(B) otimes G(C)) rarr B otimes C The commutativity of (A1) implies that thecomposition of maps in the left column in (21) can be replaced by

G(ρA otimes (ρB otimes ρC )) G((idFG(A) otimes ψ F

2 ) ψ F2

) δ (A3)

906 L Kong I Runkel

Similarly we can show that the composition of maps in the right column in (21) can bereplaced by

G((ρA otimes ρB)otimes ρC ) G((ψ F

2 otimes idFG(C)) ψ F2

) δ (A4)

Using the commutativity of (23) it is easy to see that (21) with the left and right col-umns of (21) replaced by (A3) and (A4) respectively is commutative Hence (21) iscommutative

Now we prove the commutativity of the first diagram in (22)

φG2 (φG

0 otimes idG(A))

(1)= G(ρ12 otimes ρA) Gψ F2 δ

[(Gψ F

0 δ11)otimes idG(A)

]

(2)= G(ρ12 otimes ρA) Gψ F2 G F(Gψ0 otimes idG(A)) G F(δ11 otimes idG(A)) δ

(3)= G(ρ12 otimes ρA) G(FG(ψ F0 )otimes idFG(A)) Gψ F

2 G F(δ11 otimes idG(A)) δ(4)= G(id12 otimes ρA) G

([ψ F

0 ρF(11) (Fδ)11] otimes idFG(A)

) Gψ F

2 δ(5)= G(id12 otimes ρA) G(ψ F

0 otimes idFG(A)) Gψ F2 δ

(6)= G(id12 otimes ρA) G(lminus1FG(A)) G F(lG(A)) δ

(7)= G(lminus1A ) GρA δG lG(A)

(8)= G(lminus1A ) lG(A) (A5)

where in step (1) we substituted the definition of φG0 φ

G2 given in (210) in step (2)

we used the naturality of δ in step (3) we used the naturality of Gψ F2 in step (4) we

switched the position between Gρ12 and G FG(ψ F0 ) and the position between Gψ2 and

G F(δ12 otimes idG(A)) using the naturality of ρ and FψG2 respectively in step (5) we applied

the second identity in (28) in step (6) we used (24) in step (7) we used the naturalityof lminus1 and δ in step (8) we used the first identity in (28)

The proof of the commutativity of the second diagram in (22) is similar Thus wehave shown that G is a lax tensor functor

A2 Proof of Lemma 320 To prepare the proof recall that for a given object B isin Cthe modular group P SL(2Z) acts on the space oplusi HomC(B otimes Ui Ui ) see eg [BKSect 31] and [K3 Eq (455)] We will only need the action of S and Sminus1 Let f isinoplusi HomC(B otimes Ui Ui ) Then

S oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A6)

Cardy Algebras and Sewing Constraints I 907

Sminus1 oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A7)

By Lemma 32 to establish that (337) is S-invariant it is enough to prove the identity(34) when f is given by (337) Using (A6) and (A7) we can see that Eq (34) simplysays that oplusi j [RHS of (34)] is invariant under the action of S times S Consider the elementg of oplus jkisinIHomC2plusmn(R(A)otimes (Uor

j times Uk)Uorj times Uk) given by

g =oplusjkisinI

sumα

R(A)

R(A)

R(A)

R(A)

Uorj times Uk

Uorj times Uk

α

α

(A8)

By the above arguments proving S-invariance of (337) is equivalent to proving invari-ance of g under the action of S times S

For i isin I we denote by gi the component of g in(oplus jisinIHomC(A otimes Uor

i otimes Uorj U

orj )

)otimes (opluskisinIHomC(Ui otimes UkUk))

We view the second Hom-space in the above tensor product as a Hom-space in C+ insteadof Cminus It is enough to show that gi is invariant under the action of S times Sminus1 Note thatthe action of Sminus1 in C+ is equivalent to that of S in Cminus

The morphism gi can be canonically identified with a bilinear pairing

( middot middot )i ⎛⎝oplus

jisinIHomC(Uor

j A otimes Uori otimes Uor

j )

⎞⎠

times(oplus

kisinIHomC(UkUi otimes Uk)

)minusrarr C (A9)

as follows For h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) we set

(h1 h2)i = (dim U j dim Uk)minus1 trUor

j timesUk[gi (h1 times h2)] (A10)

908 L Kong I Runkel

When substituting the explicit form of the product m R(A) of R(A) = (A times 1) otimes R(1)after a short calculation one finds

(h1 h2)i =sumα

1

dim U j

A

A

A

A

A A

A α

α

h1

h2

U jUk

Uorj

Ui

Uk (A11)

Here the top morphism PlR(A) has been simplified with the help of the identity

PR(A) m R(A) (PR(A) otimes PR(A))

=(oplus

iisinI((m A A)otimes idUor

i)times idUi

) m R(A) (PR(A) otimes PR(A)) (A12)

which can be checked by direct calculation along the same lines as in the proof of [FrLem 310]

The action of the modular transformation S on oplusiisinI(B otimes Ui Ui ) for B isin C nat-urally induces an action on oplusiisinI(Ui B otimes Ui ) [K3 Prop 514] which we denote bySlowast In the present case we get an action of Slowast on oplus jisinIHomC(Uor

j A otimes Uori otimes Uor

j ) and

opluskisinIHomC(UkUi otimes Uk) Then to show gi is invariant under the action of S times Sminus1

amounts to showing that

(h1 h2)i = ((Sminus1)lowast h1 Slowast h2)i (A13)

Cardy Algebras and Sewing Constraints I 909

for all h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) We have

((Sminus1)lowast h1 Slowast h2)i =sum

mnα

dim Un

DimC

A

A

A

A

A A

A

Uk

α

α

h1

h2

Ui

U j

UmUn

(A14)

Now drag the upper vertex indexed by α in the above graph along its Uorm -leg until it

meets the lower vertex also indexed by α then sum over α and m This gives

((Sminus1)lowast h1 Slowast h2)i =sum

n

dim Un

DimC

A

A

A

A

A A

A

Uk

Un

h1

h2

Ui

U j

(A15)

910 L Kong I Runkel

If we just look at the neighbourhood of the Un-loop in the above graph we see thefollowing subgraphs

sumn

dim Un

DimC

A Aor

Uorj U j

Uk Uork

Un=

sumα

1

dim U j

A Aor Uk Uork

Uorj U j

αα

(A16)

where we have applied [BK Cor 3111] Substituting this subgraph back to the originalgraph in (A15) we obtain

((Sminus1)lowast h1 Slowast h2)i =sumα

1

dim U j

A

A

A

A

A A

A

A

A

Uorj

α

α

h1

h2m A

Ui

U jUk

Uk

(A17)

The graph in (A17) is equal to that in (A11) In order to see this we first drag theldquobubblerdquo (m A A) along A lines and through the m A vertex (because m A A is abimodule map) until it reaches the lower-left leg of the upper vertex indexed by α Thendrag the m A vertex along the (red) dotted line in the above graph Finally we applythe associativity of A (A12) and [Fr Lem 311] Then we see that the graph in (A17)exactly matches with the one in (A11)

Acknowledgements We would like to thank the organisers of the Oberwolfach Arbeitsgemeinschaft ldquoAlge-braic structures in conformal field theoriesrdquo (April 2007) where this work was started for an inspiring meetingWe would further like to thank the Hausdorff Institute for Mathematics in Bonn and the organisers of the stim-ulating meeting ldquoGeometry and Physicsrdquo (May 2008) We are indebted to Alexei Davydov Jens FjelstadJuumlrgen Fuchs Yi-Zhi Huang Alexei Kitaev Urs Schreiber Christoph Schweigert Stephan Stolz and PeterTeichner for helpful discussions andor comments on a draft of this paper The research of IR was partiallysupported by the EPSRC First Grant EPE0050471 the PPARC rolling grant PPC5071451 and the MarieCurie network lsquoSuperstring Theoryrsquo (MRTN-CT-2004-512194)

Cardy Algebras and Sewing Constraints I 911

References

[AN] Alexeevski A Natanzon SM Noncommutative two-dimensional topological field theories andhurwitz numbers for real algebraic curves Sel Math New Ser 12 307ndash377 (2006)

[Bi] Bichon J Cosovereign hopf algebras J Pure Appl Alg 157 121ndash133 (2001)[BK] Bakalov B Kirillov AA Lectures on Tensor Categories and Modular Functors Providence

RI Amer Math Soc 2001[C1] Cardy JL Operator content of two-dimensional conformal invariant theories Nucl Phys

B 270 186ndash204 (1986)[C2] Cardy JL Boundary conditions fusion rules and the verlinde formula Nucl Phys B 324

581ndash596 (1989)[CL] Cardy JL Lewellen DC Bulk and boundary operators in conformal field theory Phys Lett

B 259 274ndash278 (1991)[DMZ] Dong C-Y Mason G Zhu Y-C Discrete series of the Virasoro algebra and the moonshine

module In Algebraic Groups and Their Generalizations Quantum and infinite-dimensional Meth-ods Proc Symp Pure Math 56 Part 2 Providence RI Amer MathSoc 1994 pp 295ndash316

[DP] Day B Pastro C Note on frobenius monoidal functors New York J Math 14 733ndash742 (2008)[ENO1] Etingof PI Nikshych D Ostrik V On fusion categories Ann Math 162 581ndash642 (2005)[ENO2] Etingof PI Nikshych D Ostrik V An analogue of radfordrsquos s4 formula for finite tensor

categories Int Math Research Notices 54 2915ndash2933 (2004)[Fe] Felder G Froumlhlich J Fuchs J Schweigert C Correlation functions and boundary conditions in

rational conformal field theory and three-dimensional topology Comp Math 131 189ndash237 (2002)[FHL] Frenkel IB Huang Y-Z Lepowsky J On axiomatic approaches to vertex operator algebras

and modules Mem Amer Math Soc 104 (1993)[Fj] Fjelstad J Fuchs J Runkel I Schweigert C Uniqueness of openclosed rational cft with

given algebra of open states Adv Theor Math Phys 12 1283ndash1375 (2008)[FK] Froumlhlich J King C The chern-simons theory and knot polynomials Commun Math

Phys 126 167ndash199 (1989)[Fr] Froumlhlich J Fuchs J Runkel I Schweigert C Correspondences of ribbon categories Adv

Math 199 192ndash329 (2006)[FRS] Fuchs J Runkel I Schweigert C Tft construction of rcft correlators I partition functions Nucl

Phys B 646 353ndash497 (2002)[FS] Fuchs J Schweigert C Category theory for conformal boundary conditions Fields Inst Com-

mun 39 25ndash71 (2003)[H1] Huang Y-Z Two-dimensional conformal geometry and vertex operator algebras Progress in

Mathematics Vol 148 Boston Birkhaumluser 1997[H2] Huang Y-Z Rigidity and modularity of vertex tensor categories Comm Contemp Math 10

871ndash911 (2008)[HK1] Huang Y-Z Kong L Open-string vertex algebra category and operads Commun Math

Phys 250 433ndash471 (2004)[HK2] Huang Y-Z Kong L Full field algebras Commun Math Phys 272 345ndash396 (2007)[HK3] Huang Y-Z Kong L Modular invariance for conformal full field algebras httparxivorgabs

math0609570v2[mathQA] 2006[HL] Huang Y-Z Lepowsky J Tensor products of modules for a vertex operator algebra and vertex

tensor categories In Lie Theory and Geometry in honor of Bertram Kostant ed R BrylinskiJ-L Brylinski V Guillemin V Kac Boston Birkhaumluser 1994 pp 349ndash383

[JS] Joyal A Street R Braided tensor categories Adv Math 102 20ndash78 (1993)[K1] Kong L Full field algebras operads and tensor categories Adv Math 213 271ndash340 (2007)[K2] Kong L Open-closed field algebras Commun Math Phys 280 207ndash261 (2008)[K3] Kong L Cardy condition for open-closed field algebras Commun Math Phys 283 25ndash92 (2008)[Ki] Kitaev A Private communication[KO] Kirillov AA Ostrik V On q-analog of mckay correspondence and ade classification of sl(2)

conformal field theories Adv Math 171 183ndash227 (2002)[KR] Kong L Runkel I Morita classes of algebras in modular tensor categories Adv Math 219

1548ndash1576 (2008)[Ld] Lauda AD Frobenius algebras and ambidextrous adjunctions Theo Appl Cat 16 84ndash

122 (2006)[Li1] Li H-S Regular representations of vertex operator algebras Commun Contemp Math 4

639ndash683 (2002)[Li2] Li H-S Regular representations and huang-lepowsky tensor functors for vertex operator

algebras J Alge 255 423ndash462 (2002)

882 L Kong I Runkel

Proof We only prove the first equality the second one can be seen in the same way Bydefinition we have

IF(Aor) F(A)

= ((ψ F0 F(dA) φF

2 )otimes idF(A)or) (idF(Aor) otimes bF(A)) F(A)

=[(ψ F

0 F(dA) φF2 ) (F(A)otimes idF(A))

]otimes idF(A)or

(idF(A) otimes bF(A))

For the term inside the square brackets we find

ψ F0 F(dA) φF

2 (F(A)otimes idF(A)) = ψ F0 F(dA) F(A otimes idA) φF

2

= ψ F0 F(dA (A otimes idA)) φF

2 = ψ F0 F(εA m A) φF

2 (222)

On the other hand by definitionF(A) = [((ψ F0 F(εA) (F(m A)φF

2 ))otimes idF(A)or] (idF(A) otimes bF(A)) This demonstrates the first equality in (221) Proposition 216 Let F C1 rarr C2 be a tensor functor G C2 rarr C1 a functor〈FG χ〉 an adjunction A a C1-algebra and B a C2-algebra Then f A rarr G(B)is an algebra homomorphism if and only if f = χminus1( f ) F(A) rarr B is an algebrahomomorphism

Proof We need to show that

mG(B) ( f otimes f ) = f m A and f ηA = ηG(B) (223)

is equivalent to

m B ( f otimes f ) = f m F(A) and f ηF(A) = ηBs (224)

We first prove that the first identity in (223) is equivalent to the first identity in (224)For the left-hand side of the first identity in (223) we have the following equalities

mG(B) ( f otimes f )(1)= G(m B) φG

2 ( f otimes f )(2)= G(m B) G(ρ otimes ρ) G(ψ F

2 ) δ ( f otimes f )(3)= G(m B) G(ρ otimes ρ) G(ψ F

2 ) G F( f otimes f ) δ(4)= G(m B) G(ρ otimes ρ) G(F( f )otimes F( f )) G(ψ F

2 ) δ(5)= G

(m B (ρ otimes ρ) (F( f )otimes F( f )) ψ F

2

) δ

(6)= χ(

m B (ρ otimes ρ) (F( f )otimes F( f )) ψ F2

) (225)

where (1) is the definition of mG(B) in (217) (2) is the second identity in (210) (3)and (4) are naturality of δ and ψ F

2 respectively step (5) is functoriality of G and finallystep (6) is (29) For the right hand side of the first identity in (223) we get

f m A(1)= Gρ δG ( f m A)

(2)= Gρ G F( f m A) δ(3)= G(ρ F( f m A)) δ(4)= χ(ρ F( f ) F(m A)) (226)

Cardy Algebras and Sewing Constraints I 883

where (1) is the adjunction property (28) (2) is naturality of δ (3) functoriality of Gand (4) amounts to (29) and functoriality of F

On the other hand we see that the first equality in (224) is equivalent to

m B (ρ otimes ρ) (F( f )otimes F( f )) = ρ F( f ) F(m A) φF2 (227)

Using that φF2 is invertible with inverse (φF

2 )minus1 = ψ F

2 and that χ is an isomorphismit follows that the statement that (225) is equal to (226) is equivalent to the identity(227)

Now we prove that the second identity in (223) is equivalent to the second identity in(224) Using (217) and (210) we can write ηG(B) = G(ηB)φF

0 = G(ηB)G(ψ F0 )δ1

Together with (29) this shows that the second identity in (223) is equivalent to

f ηA = χ(ηB ψ F0 ) (228)

On the other hand the second identity in (224) is equivalent to

ρ F( f ) F(ηA) φF0 = ηB (229)

which by φF0 = (ψ F

0 )minus1 and (29) is further equivalent to (228)

Definition 217 Let (Am A ηAA εA) and (Bm B ηBB εB) be two Frobeniusalgebras in a tensor category C For f A rarr B we define f lowast B rarr A by

f lowast = ((εB m B)otimes idA) (idB otimes f otimes idA) (idB otimes (A ηA)) (230)

The following lemma is immediate from the definition of (middot)lowast and the properties ofFrobenius algebras We omit the proof

Lemma 218 Let C be a tensor category let A BC be Frobenius algebras in C andlet f A rarr B and g B rarr C be morphisms

(i) (g f )lowast = f lowast glowast(ii) f is a monomorphism iff f lowast is an epimorphism

(iii) f is an algebra map iff f lowast is a coalgebra map(iv) If f is a homomorphism of Frobenius algebras then f lowast f = idA and f f lowast =

idB(v) If C is sovereign and if A and B are symmetric then f lowastlowast = f

Let C and D be tensor categories and let F C rarr D be a Frobenius functor GivenFrobenius algebras A B in C and a morphism f A rarr B the next lemma shows how(middot)lowast behaves under F

Lemma 219 F( f lowast) = F( f )lowast

Proof The definition of the structure morphisms of the Frobenius algebra F(A) is givenin (217) and (218) Substituting these definitions gives

F( f )lowast =[(ψ F

0 F(εB) F(m B) φF2

)otimes idF(A)

] [

idF(B) otimes F( f )otimes idF(A)]

[idF(B) otimes

(ψ F

2 F(A) F(ηA) φF0

)]

= (ψ F0 otimes idF(A)) [

F (εB m B (idB otimes f ))otimes idF(A)]

(φF2 otimes idF(A)) (idF(B) otimes ψ F

2 )

[idF(B) otimes F(A ηA)

] (idF(B) otimes φF0 ) (231)

884 L Kong I Runkel

In the middle line of the last expression we can use the defining property (25) of F namely we substitute (φF

2 otimes idF(A)) (idF(B) otimes ψ F2 ) = ψ F

2 φF2 Then ψ F

2 can bemoved to the left and φF

2 to the right until they can be omitted against ψ F0 and φF

0 respectively using (22) and (24) This results in

F( f )lowast = F ((εB m B (idB otimes f ))otimes idA) F (idB otimes (A ηA)) (232)

which is nothing but F( f lowast)

23 Modular tensor categories Let C be a modular tensor category [TBK] ie anabelian semi-simple finite C-linear ribbon category with simple tensor unit 1 and a non-degeneracy condition on the braiding (to be stated in a moment) We denote the set ofequivalence classes of simple objects in C by I elements in I by i j k isin I and theirrepresentatives by Ui U j Uk We also set U0 = 1 and for an index k isin I we define kby Uk

sim= Uork

Since the tensor unit is simple we shall for modular tensor categories identifyHom(1 1) sim= C (cf footnote 2) Define numbers si j isin C by4

si j = UiU j (233)

They obey si j = s ji and s0i = dim Ui see eg [BK Sect 31] (In a ribbon cate-gory the left and right dimension (214) of Ui coincide and are denoted by dim Ui )The non-degeneracy condition on the braiding of a modular tensor category is that the|I|times|I|-matrix s should be invertible In fact [BK Thm 317]

sumkisinI

sik sk j = Dim C δij (234)

where Dim C = sumiisinI(dim Ui )

2 One can show (even in the weaker context of fusioncategories over C) that Dim C ge 1 [ENO1 Thm 23] In particular Dim C = 0 We fixonce and for all a square root

radicDim C of Dim C

Let us fix a basis λα(i j)kN k

i jα=1 in HomC(Ui otimesU j Uk) and the dual basis ϒ(i j)k

α N ki j

α=1

in HomC(UkUi otimesU j ) The duality of the bases means that λα(i j)k ϒ(i j)kβ = δαβ idUk

We also fix λ(0i)i = λ(i0)i = idUi We denote the basis vectors graphically as follows

λα(i j)k = α

Uk

Ui U j

ϒ(i j)kα = α

Uk

Ui U j

(235)

4 In the graphical notation used below we have given an orientation to the ribbons indicated by the arrowsFor example it is understood that this orientation determines which of the duality morphisms in (213) to use

Cardy Algebras and Sewing Constraints I 885

For V isin C we also choose a basis b(iα)V of HomC(VUi ) and the dual basis bV(iβ)

of HomC(Ui V ) for i isin I such that b(iα)V bV(iβ) = δαβ idUi We use the graphical

notation

b(iα)V = α

Ui

V

bV(iα) = α

Ui

V

(236)

Given two modular tensor categories C and D by C D we mean the tensor productof additive categories over C [BK Def 1115] ie the category whose objects are directsums of pairs V times W of objects V isin C and W isin D and whose morphism spaces are

HomCD(V times W V prime times W prime) = HomC(V V prime)otimesC HomC(WW prime) (237)

for pairs and direct sums of these if the objects are direct sums of pairsIf we replace the braiding and the twist in C by the antibraiding cminus1 and the antitwist

θminus1 respectively we obtain another ribbon category structure on C In order to distin-guish these two distinct structures we denote (C c θ) and (C cminus1 θminus1) by C+ and Cminusrespectively As in the Introduction we will abbreviate

C2plusmn = C+ Cminus (238)

Note that a set of representatives of the simple objects in C2plusmn is given by Ui times U j fori j isin I

For the remainder of Sect 2 we fix a modular tensor category C

24 The functors T and R The tensor product bifunctor otimes can be naturally extendedto a functor T C2plusmn rarr C Namely T (oplusN

i=1Vi times Wi ) = oplusNi=1Vi otimes Wi for all Vi Wi isin C

and N isin N The functor T becomes a tensor functor as follows For φT0 1 rarr T (1 times 1)

take φT0 = id1 (or lminus1

1 in the non-strict case) Next notice that for U VW X isin C

T (U times V )otimes T (W times X)= (U otimes V )otimes (W otimes X)

T ((U times V )otimes (W times X))= (U otimes W )otimes (V otimes X)(239)

We define φT2 T (U times V )otimes T (W times X) rarr T ((U times V )otimes (W times X)) by

φT2 = idU otimes cminus1

W V otimes idX (240)

(In the non-strict case the appropriate associators have to be added) The above definitionof φT

2 can be naturally extended to a morphism φT2 T (M1 otimes M2) rarr T (M1)otimes T (M2)

for any pair of objects M1M2 in C2plusmn The following result can be checked by directcalculation [JS Prop 52]

Lemma 220 The triple (T φT2 φ

T0 ) gives a tensor functor

886 L Kong I Runkel

In particular (T φT2 φ

T0 ψ

T2 ψ

T0 ) where ψT

2 = (φT2 )

minus1 and ψT0 = (φT

0 )minus1 gives

a Frobenius functorDefine the functor R C rarr C2plusmn as follows for A isin C and f isin HomC(A B)

R(A) =oplusiisinI

(A otimes Uori )times Ui R( f ) =

oplusiisinI

( f otimes idUori)times idUi (241)

This functor was also considered in a slightly different context in [ENO2 Prop 23]The family of isomorphisms γ R

A = oplusiisinI DimCdim Ui

id(AotimesUori )timesUi

isin Aut(R(A)) defines a

natural isomorphism γ R R rarr ROur next aim is to show that R is left and right adjoint to T in other words R

and T form an ambidextrous adjunction (see eg [Ld] for a discussion of ambidextrousadjunctions) To this end we introduce two linear isomorphisms for A isin C and M isin C2plusmn

χ HomC(T (M) A) minusrarr HomC2plusmn(M R(A))

χ HomC(A T (M)) minusrarr HomC2plusmn(R(A)M)(242)

If we decompose M as M = oplusNn=1 Ml

n times Mrn then χ and χ are given by

χ Noplus

n=1

fn

Mln Mr

n

A

rarrNoplus

n=1

oplusiisinI

sumα

fn

Mln

Mrn

A

α

Uori

times α

Mrn

Ui

(243)

and

χ Noplus

n=1 gn

Mln Mr

n

A

rarrNoplus

n=1

oplusiisinI

sumα gn

Mln

Mrn

A

α

Uori

times α

Ui

Mrn

DimCdim Ui

(244)

Notice that χ and χ are independent of the choice of basis

Theorem 221 〈T R χ〉 and 〈R T χminus1〉 are adjunctions ie R is both left and rightadjoint of T

Proof Write M as M = oplusNn=1 Ml

n times Mrn The isomorphism χ amounts to the following

composition of natural isomorphisms

HomC(T (M) A) = oplusnHomC(Mln otimes Mr

n A)sim= oplusni HomC(Ml

n otimes Ui A)otimes HomC(Mrn Ui )

sim= oplusni HomC(Mln A otimes Uor

i )otimes HomC(Mrn Ui ) = HomC2plusmn(M R(A)) (245)

Cardy Algebras and Sewing Constraints I 887

Thus χ is natural Let (γ RA )

lowast HomC2plusmn(R(A)M) rarr HomC2plusmn(R(A)M) denote the

pull-back of γ RA The isomorphism χ is equal to the composition of (γ R

A )lowast and the

following sequence of natural isomorphisms

HomC(A T (M)) = oplusn HomC(AMln otimes Mr

n)

sim= oplusni HomC(AMln otimes Ui )otimes HomC(Ui Mr

n)

sim= oplusni HomC(A otimes Uori Ml

n)otimes HomC(Ui Mrn) = HomC2plusmn(R(A)M) (246)

We have proved that both χ and χ are natural isomorphisms There are four natural transformations associated to χ and χ namely

idC2plusmnδminusrarr RT

ρminusrarr idC2plusmn and idCδminusrarr T R

ρminusrarr idC (247)

defined by for A isin C M isin C2plusmn

δM = χ(idT (M)) ρA = χminus1(idR(A))

ρM = χ (idT (M)) δA = χminus1(idR(A))(248)

They can be expressed graphically as follows with M = oplusNn=1 Ml

n times Mrn

δM =oplusni

sumα

Mln

Mln Mr

n Uori

αtimes

Ui

Mrn

α

ρA =oplusiisinI

A

A

Uori Ui

(249)

ρM =oplusni

sumα

Mln

Mln

Mrn Uor

i

α times

Ui

Mrn

α DimCdim Ui

δA =oplusiisinI

A

A Uori Ui

dim UiDimC

Note that

ρM δM = Dim C middot idM and ρA δA = idA (250)

Lemma 222 The functors T and R as maps on the sets of morphisms have left inversesand thus are injective

Proof Let f A rarr B be a morphism in C We define a map Q R HomC2plusmn(R(A)

R(B)) rarr HomC(A B) by f prime rarr ρB T ( f prime) δA Then we have

Q R R( f ) = ρB T R( f ) δA = ρB δB f = f (251)

888 L Kong I Runkel

where we used naturality of δ and (250) in the second and third equalities respectivelySo Q R is a left inverse of R on morphisms Thus R is injective on morphisms Similarlylet g M rarr N be a morphism in C2plusmn We define a map QT HomC(T (M) T (N )) rarrHomC2plusmn(M N ) by gprime rarr (Dim C)minus1 middot ρN R(gprime) δM Then we have

QT T (g) = (Dim C)minus1 middot ρN RT (g) δM = (Dim C)minus1 middot ρN δN g = g (252)

So QT is a left inverse of T on morphisms Thus T is injective on morphisms Using (29) and (250) one can express the two inverse maps χminus1 χminus1 as follows

for f isin HomC2plusmn(M R(A)) and g isin HomC2plusmn(R(A)M)

χminus1( f ) = ρ T ( f ) χminus1(g) = T (g) δ (253)

By Proposition 25 and Lemma 27 R is both a lax and colax tensor functor In particularφR

0 1 times 1 rarr R(1) is given by

φR0 = χ (ψT

0 ) = R(ψT0 ) δ1 times 1 = id1times1 (254)

and φR2 R(A)otimes R(B) rarr R(A otimes B) by φR

2 = R(ρA otimes ρB) R(ψT2 ) δ which can

be expressed graphically as

φR2 =

oplusi jkisinI

sumα

A

A

Uori B

B

Uorj

Uork

α

times

Ui U j

α

Uk

(255)

Similarly ψ R0 R(1) rarr 1 times 1 is given by

ψ R0 = ρ1 R(φT

0 ) = Dim C id1times1 (256)

and ψ R2 R(A otimes B) rarr R(A) otimes R(B) by ψ R

2 = ρ R(φT2 ) R(δA otimes δB) which in

graphical notation reads

ψ R2 =

oplusi jkisinI

sumα

A

A Uori

B

B Uorj

Uork

α

times

Ui U j

α

Uk

dim Ui dim U j

dim Uk Dim C

(257)

If C has more than one simple object then R does not take the tensor unit of C to thetensor unit of C2plusmn and so is clearly not a tensor functor However we will show that R isstill a Frobenius functor This will imply that if A is a Frobenius algebra in C then

R(A) = (R(A)m R(A) ηR(A)R(A) εR(A)) (258)

Cardy Algebras and Sewing Constraints I 889

is a Frobenius algebra in C2plusmn where the structure morphisms were given in (217) and(218) In the case A = 1 it was proved in [Muuml1 Prop 41] (see also [Fr Lem 619]and [K1 Thm 52]) that (258) is a commutative simple symmetric normalised-specialFrobenius algebra in C2plusmn In fact given a Frobenius algebra A in C it is straightforwardto verify that the structure morphisms in (258) are precisely those of (A times 1)otimes R(1)cf Sect 22

Proposition 223 (R φR2 φ

R0 ψ

R2 ψ

R0 ) is a Frobenius functor

Proof Using the explicit graphical expression of φR2 φ

R0 ψ

R2 ψ

R0 it is easy to see that

the commutativity of the diagrams (25) and (26) are equivalent to the statement thatR(1) with structure morphisms as in (258) is a Frobenius algebra in C2plusmn The latterstatement is true by [Muuml1 Prop 41]

From Lemma 220 and Proposition 223 we see that T and R take Frobenius alge-bras to Frobenius algebras The following two propositions show how the properties ofFrobenius algebras are transported

Proposition 224 Let A be a Frobenius algebra in C2plusmn Then T (A) is a Frobenius algebrain C and

(i) A is symmetric iff T (A) is symmetric(ii) A is (normalised-)special iff T (A) is (normalised-)special

Proof For part (i) write A as a direct sum oplusNn=1 Al

n times Arn Then the maps IT (A) I prime

T (A) T (Aor) rarr T (A)or defined in (220) are given by

IT (A) = I primeT (A) = oplusN

n=1c(Aln)

or(Arn)

or (259)

Therefore by (221) T (A) = primeT (A) is equivalent to T (A) = T (prime

A) Since byLemma 222 T is injective on morphisms this proves part (i) Part (ii) can be checkedin the same way for example the condition mT (A) T (A) = ζ idT (A) is easily checkedto be equivalent to T (m A A) = ζ T (idA) Proposition 225 Let A be a Frobenius algebra A in C Then R(A) is a Frobeniusalgebra in C2plusmn and

(i) A is symmetric iff R(A) is symmetric(ii) A is (normalised-)special iff R(A) is (normalised-)special

Proof Recall that the structure morphisms of the Frobenius algebra R(A) are equal tothose of (A times 1) otimes R(1) Using this equality part (i) and (ii) follow because R(1) issymmetric and normalised-special For example

m R(A) R(A) = [(m A A)times id1] otimes idR(1) = R(m A A) (260)

so that m R(A) R(A) = ζ idR(A) is equivalent to R(m A A) = ζ R(idA) which byLemma 222 is equivalent to m A A = ζ idA

The functor R has one additional property not shared by T namely R takes absolutelysimple algebras to absolutely simple algebras We will see explicitly in Sect 33 thatthis is not true for T

890 L Kong I Runkel

Lemma 226 For a C-algebra A the map

R HomA|A(A A) rarr HomR(A)|R(A)(R(A) R(A)) (261)

given by f rarr R( f ) is well-defined and an isomorphism

Proof Since R is a lax tensor functor R(A) is naturally a R(A)-bimodule It is easy tosee that R in (261) is a well-defined map R(A) is also naturally a R(1)-bimodule whichcan be identified with the induced R(1)-bimodule structure on (A times 1)otimes R(1) wherethe left R(1) action on (Atimes1)otimes R(1) is given by (idAtimes1 otimesm R(1))(cminus1

Atimes1R(1)otimes idR(1))We have the following natural isomorphisms

HomR(1)|R(1)(R(A) R(A))sim=minusrarr HomC2plusmn(A times 1 R(A))

χminus1

minusminusrarr HomC(A A) (262)

which by (253) are given by for f isin HomR(1)|R(1)(R(A) R(A))

f rarr f prime = f (idAtimes1 otimes ηR(1)) rarr f primeprime = ρ T ( f prime) (263)

and its inverse is given by for g isin HomC(A A)

g rarr gprime = R(g) δ rarr gprimeprime = (idAtimes1 otimes m R(1)) (gprime otimes idR(1)) (264)

where gprimeprime is indeed a R(1)-bimodule map due to the commutativity of R(1) It is easy tocheck that gprimeprime = R(g) in (264) Therefore R gives an isomorphism from HomC(A A)to HomR(1)|R(1)(R(A) R(A)) Moreover one verifies that R(g) is an R(A)-bimodulemap iff g is an A-bimodule map In other words R g rarr R(g) gives an isomorphism

HomA|A(A A)sim=minusrarr HomR(A)|R(A)(R(A) R(A))

Corollary 227 Let A be a C-algebra

(i) A is absolutely simple iff R(A) is absolutely simple(ii) Let A be in addition Frobenius Then A is simple and special iff R(A) is simple and

special

Proof Part (i) immediately follows from Lemma 226 The statement of part (ii) withoutthe qualifier lsquosimplersquo is proved in Proposition 225 But as in the proof of (iii)rArr(i) inLemma 211 a special Frobenius algebra in a semi-simple category has a semi-simplecategory of bimodules and for a semi-simple C-linear category simple and absolutelysimple are equivalent Part (ii) then follows from part (i)

The following lemma will be needed in Sect 32 below to discuss the properties ofCardy algebras

Lemma 228 Let A be a Frobenius algebra in C Then (δA)lowast = ρA

Proof Recall from (250) that δA is a morphism A rarr T R(A) Since T and R are bothFrobenius functors T R(A) is a Frobenius algebra in C Substituting the definitions aftera short calculation one finds

Cardy Algebras and Sewing Constraints I 891

εT R(A) mT R(A) = Dim(C)oplusiisinI

A Uori Ui A U or

ı Uı

(265)

Substituting this in the definition of (δA)lowast gives again after a short calculation the mor-

phism ρA At an intermediate step one uses that the part of the morphism (δA)lowast which

is made up of Ui and Uı ribbons their duals and the basis morphisms λ(iı)0 andϒ(iı)0can be replaced by 1

dim Uimiddot dUi

3 Cardy Algebras

In this section we start by investigating the properties of Frobenius algebras which sat-isfy the so-called modular invariance condition We then give two definitions of a Cardyalgebra and prove their equivalence Finally in Sect 33 we study the properties of thesealgebras and state our main results

We fix a modular tensor category C Recall that C2plusmn is an abbreviation for C+ Cminus

31 Modular invariance In C2plusmn we define the object K and the morphism ω K rarr Kas

K =oplus

i jisinIUi times U j ω =

sumi jisinI

dim Ui dim U j

Dim C idUi timesU j (31)

They have the property (see eg [BK Cor 3111])

ω

K

(Uk times Ul )or

(Uk times Ul )or

Ui times U j

Ui times U j

= δik δ jlDim C

dim Ui dim U jbUi timesU j dUi timesU j (32)

892 L Kong I Runkel

Definition 31

(i) Let A B be objects of C2plusmn A morphism f A otimes B rarr B is called S-invariant iff

A

B

W

W

f =

A

B

W

W

f

K

ω(33)

holds for all for W isin C2plusmn(ii) A C2plusmn-algebra (Aclmcl ηcl) is called modular invariant iff θAcl = idAcl and mcl

is S-invariant

Lemma 32 The morphism f A otimes B rarr B is S-invariant if and only if

A

B

Ui times U j

Ui times U j

f = Dim Cdim Ui dim U j

sumα

A

B

B

f

Ui times U j

Ui times U j

α

α

(34)

holds for all i j isin I

Proof Condition (33) holds for all W iff it holds for all W = Ui times U j i j isin I so itis enough to show that the right-hand side of (33) with W = Ui times U j is equal to theright-hand side of (34) Recall the notation for basis morphisms in (236) Starting from(33) write

idBor otimes idUi timesU j =⎛⎝sum

klα

(b(ktimeslα)

B

)or (

bB(ktimeslα)

)or⎞⎠ otimes idUi timesU j (35)

and then apply (32) The graphical representation of the resulting morphism can bedeformed to give (34) Remark 33 As shown in [K3 Sect 61] the modular invariance condition of a C2plusmn-alge-bra exactly coincides with the modular invariance condition for torus 1-point correlationfunctions of a genus-01 closed CFT In particular the condition θAcl = idAcl is equiv-alent to invariance under the modular transformation T τ rarr τ + 1 and the condition(34) with f = mcl is equivalent to invariance under S τ rarr minus 1

τ Combining the

modular invariance condition with the genus zero properties of a genus-01 closed CFTresults in a modular invariant commutative symmetric Frobenius algebra in C2plusmn

Cardy Algebras and Sewing Constraints I 893

Let Acl be a modular invariant C2plusmn-algebra Evaluating (34) for f = mcl composingit with ηcl otimes idUi timesU j and taking the trace implies the following identity

Zi j = 1

DimCAclUi timesU j where

(36)

Zi j = dimC HomC2plusmn(UitimesU j Acl)

Decomposing Acl into simple objects this gives

Zi j =sum

klisinISik Zkl Sminus1

l j where Si j = si jradic

DimC (37)

which in CFT terms is of course nothing but the invariance of the torus partition functionunder the modular S-transformation

The following theorem gives a simple criterion for modular invariance

Theorem 34 Let Acl be a haploid commutative symmetric Frobenius algebra in C2plusmn

(i) If Acl is modular invariant then dim Acl = Dim C(ii) If dim Acl = Dim C then Acl is special and modular invariant

Proof Part (i) Since Acl is haploid for i = j = 0 Equation (36) reduces to 1 =dim AclDimCPart (ii) By the same reasoning as in the proof of (iii)rArr(i) in Lemma 211 one showsthat Acl is special Thus mcl cl = ζclidAcl for some ζcl = 0

By [KO Thm 45] the category (C2plusmn)locAcl

of local Acl-modules is again a mod-

ular tensor category and Dim (C2plusmn)locAcl

= (Dim C dim Acl)2 (see [Fr Prop 321 amp

Rem 323] for the same statement in the notation used here) Thus by assumption wehave Dim(C2plusmn)loc

Acl= 1 It then follows from [ENO1 Thm 23] that up to isomorphism

(C2plusmn)locAcl

has a unique simple object (namely the tensor unit) In other words every simplelocal Acl-module is isomorphic to Acl (seen as a left-module over itself)

We have the following isomorphisms between morphism spaces [FS Prop 47 amp 411]

HomAcl(Acl otimes (Ui times U j ) Acl) sim= HomC2plusmn(Ui times U j Acl)

HomAcl(Acl Acl otimes (Ui times U j )) sim= HomC2plusmn(AclUi times U j )

Using these to transport the bases (236) from the right to the left we obtain bases bα(i j)αof HomAcl(Acl otimes (Ui times U j ) Acl) and b(i j)

β β of HomAcl(Acl Acl otimes (Ui times U j )) Thesecan be expressed graphically as

894 L Kong I Runkel

bα(i j) =

Acl

Acl

Acl

Ui times U j

α

b(i j)β = DimC

dim Ui dim U j

1

ζcl

Acl

Ui times U jAcl

Acl

β

(38)

where the nonzero factor in b(i j)β is included for convenience Notice that bα(i j) b(i j)

β isa left Acl-module map Since Acl is simple as a left module over itself we have

bα(i j) b(i j)β = λαβ idAcl (39)

for some λαβ isin C By computing tr(bα(i j) b(i j)β ) it is easy to verify that λαβ = δαβ

We will now prove the following identity

1

ζcl

Acl

Acl

Ui times U j

Ui times U j

Acl

=sumα

DimCdim Ui dim U j

1

ζcl

Acl

Ui times U j

Ui times U j

Acl

Acl

Acl

α

α

(310)

One checks that the left-hand side of this equation is an idempotent which we denote byζminus1

cl PlAcl(Ui timesU j ) cf [Fr Sect 31] By [Fr Prop 41] the image Im(ζminus1

cl PlAcl(Ui timesU j ))

is a local Acl-module and hence isomorphic to AoplusNcl for some N isin Zge0

All left-module morphisms from Acl to Aclotimes(Ui timesU j ) are linear combinations of the

b(i j)β Furthermore one verifies that ζminus1

cl PlAcl(Ui timesU j )b(i j)

β = b(i j)β Therefore the b(i j)

β

describe precisely the image of the idempotent ie ζminus1cl Pl

Acl(Ui timesU j ) = sum

α b(i j)α bα(i j)

which is nothing but (310) Composing (310) with ζcl middot εcl otimes idUi timesU j from the left (iefrom the top) produces (34)

In addition since Acl is commutative symmetric Frobenius it satisfies θAcl = idAcl

[Fr Prop 225] Altogether this shows that Acl is modular invariant Remark 35 As we were writing this paper we heard that the results in Theorem 34were obtained independently by Kitaev and Muumlger [Ki]

Remark 36 Setting i = j = 0 in (36) gives the identity dim Acl = Z00 Dim C [KRProp 23] Combining this with Theorem 34 (ii) one may wonder if a general modularinvariant commutative symmetric Frobenius algebra Acl in C2plusmn is isomorphic to a directsum of simple such algebras However this is not so For example one can take the com-mutative symmetric Frobenius algebra Acl = C[x]x2 in the category of vector spacesequipped with the non-degenerate trace ε(ax + b) = a In this case the modular invari-ance condition holds automatically but Acl is clearly not a direct sum of two algebras

Cardy Algebras and Sewing Constraints I 895

For a general modular tensor category C the algebra C[x]x2 R(1) understood as analgebra in C2plusmn via the braided monoidal isomorphism Vect f (C) C2plusmn rarr C2plusmn providesanother counter-example

32 Two definitions Define a morphism PlA A rarr A for a Frobenius algebra A in C

or C2plusmn as follows [Fr Sect 24]

PlA =

A

A

A

m

(311)

If A is also commutative and obeys m A A = ζA idA we have PlA = ζA idA In partic-

ular this holds if A is commutative and special Using the fact that the Frobenius algebraR(1) is commutative and normalised-special one can check that Pl

R(A) R(A) rarr R(A)takes the following form

PlR(A) =

oplusiisinI

A

A

Uori

Uori

A

m

times

Ui

Ui

(312)

With these ingredients we can now give the first definition of a Cardy C|C2plusmn-algebrawhich was introduced in [K3 Def 514] cf Remark 315 below

Definition 37 (CardyC|C2plusmn-algebra I) A CardyC|C2plusmn-algebra is a triple (Aop|Acl ιcl-op)where (Aclmcl ηclcl εcl) is a modular invariant commutative symmetric Frobe-nius C2plusmn-algebra (Aopmop ηopop εop) is a symmetric Frobenius C-algebra andιcl-op Acl rarr R(Aop) an algebra homomorphism such that the following conditionsare satisfied

(i) Centre condition

Acl R(Aop)

R(Aop)

R(Aop)

ιcl-op

m R(Aop)

=

Acl

ιcl-op

R(Aop)

R(Aop)

R(Aop)

m R(Aop)

(313)

896 L Kong I Runkel

(ii) Cardy condition

ιcl-op ιlowastcl-op =

R(Aop)

R(Aop)

R(Aop)

(314)

Remark 38

(i) The name ldquoCardy C|C2plusmn-algebrardquo in Definition 37 was chosen because many ofthe important ingredients were first studied by Cardy the modular invariance ofthe closed theory [C1] the consistency of the annulus amplitude [C2] and thebulk-boundary OPE [CL] On the other hand the boundary-boundary OPE andthe OPE analogue of the centre condition were first considered in [Lw]

(ii) One can easily see that in the special case that C is the category Vect f (C) of finite-dimensional C-vector spaces a Cardy C|C2plusmn-algebra gives exactly the algebraicformulation of two-dimensional open-closed topological field theory over C (cfRemark 614 in [K3]) see [Lz Sect 48] [Mo Thm 11] [AN Thm 45] [LPCor 43] [MS Sect 22] When passing to a general modular tensor category Cthere are two important differences to the two-dimensional topological field the-ory Firstly the algebras Acl and Aop now live in different categories which inparticular affects the formulation of the centre condition and the Cardy conditionSecondly the modular invariance condition has to be imposed on Acl In the caseC = Vect f (C) modular invariance holds automatically

Definition 39 A homomorphism of Cardy C|C2plusmn-algebras (A(1)op |A(1)cl ι(1)cl-op) rarr (A(2)op

|A(2)cl ι(2)cl-op) is a pair ( fop fcl) of Frobenius algebra homomorphisms fop A(1)op rarr A(2)op

and fcl A(1)cl rarr A(2)cl such that the diagram

A(1)cl

fcl

ι(1)cl-op

A(2)cl

ι(2)cl-op

R(A(1)op )

R( fop) R(A(2)op )

(315)

commutes

Remark 310 Since a homomorphism of Frobenius algebras is invertible (cf Lemma218 (iv)) a homomorphism of Cardy algebras is always an isomorphism

For a homomorphism ( fop fcl) of Cardy C|C2plusmn-algebras using the commutativity of(315) and the fact that fcl and fop are both algebra and coalgebra homomorphisms itis easy to show that (315) commutes iff

R(A(1)op )R( fop)

(ι(1)cl-op)

lowast

R(A(2)op )

(ι(2)cl-op)

lowast

A(1)cl

fcl A(2)cl

(316)

Cardy Algebras and Sewing Constraints I 897

commutesLet (Aop|Acl ιcl-op) be a Cardy C|C2plusmn-algebra Define the morphism

ιcl-op = χminus1(ιcl-op) T (Acl) minusrarr Aop (317)

Decompose Acl as Acl = oplusNn=1Cl

n times Crn such that Cl

1 times Cr1 = ηcl(1 times 1) We use ι(n)cl-op

to denote the restriction of ιcl-op to Cln times Cr

n and ι(n)cl-op to denote the restriction of ιcl-op

to Cln otimes Cr

n We introduce the following graphical notation

ι(n)cl-op =

Cln Cr

n

Aop

(318)

By (243) ιcl-op can be expressed in terms of ιcl-op as follows

ιcl-op =Noplus

n=1

oplusiisinI

sumα

Cln

Crn

Aop

α

Uori

times α

Crn

Ui

(319)

Lemma 311 The centre condition (313) is equivalent to the following condition in C

Cln Cr

n

Aop

Aop

Aop

=

Cln Cr

n

Aop

Aop

Aop

for n = 1 N (320)

Proof First insert (319) and the definition (217) of m R(Aop) into (313) Then apply thecommutativity of R(1) to the left hand side of (313) The equivalence between (313)and (320) follows immediately Remark 312 The centre condition (320) is very natural from the open-closed confor-mal field theory point of view Correlators on the upper half plane are expressed in termsof conformal blocks on the full complex plane The objects Cl

n and Crn are associated to

the field insertion at a point z in the upper half plane and at the complex conjugate pointz in the lower half plane respectively The object Aop corresponds to a field inserted ata point r on the real axis The centre condition (320) simply says that the correlationfunctions in the disjoint domains |z| gt r gt 0 and r gt |z| gt 0 are analytic continuationsof each other see [K2 Prop 118]

898 L Kong I Runkel

Recall that we define ιlowastcl-op Aop rarr T (Acl) as in (230) We introduce the graphicalnotation

ιlowastcl-op =Noplus

n=1

Cln Cr

n

Aop

=Noplus

n=1

Cln Cr

n

Aop

Aop T (minus1Acl) (321)

where the second equality follows from (215) (221) and (259)

Lemma 313 The Cardy condition (314) is equivalent to the following identity in C

Noplusn=1

DimCdim Ui

sumα

Cln

Crn

Crn

α

α

Aop

Aop Uori

Uori

=Aop

Aop

Aop

Uori

Uori

op

mop for all i isin I (322)

Proof By (244) (312) and (319) it is easy to see that (322) is equivalent to thefollowing identity

ιcl-op χ(ιlowastcl-op) = PlR(Aop)

(323)

Therefore it is enough to show that

χ(ιlowastcl-op) = ιlowastcl-op (324)

We have

χminus1(ιlowastcl-op)(1)= T (ιlowastcl-op) δ (2)= T (ιcl-op)

lowast δlowastlowast (3)= (ρ T (ιcl-op)

)lowast (4)= (ιcl-op

)lowast (325)

In step (1) we use the expression (253) for χminus1 step (2) follows from Lemma 218 (v)and Lemma 219 Step (3) is Lemma 218 (i) and Lemma 228 and finally step (4)amounts to substituting (253) and (317) Acting with χ on both sides of the aboveequality produces (324)

Combining Lemmas 311 and 313 and Proposition 216 we obtain the followingequivalent definition of Cardy C|C2plusmn-algebra (recall the graphical notation (318) forιcl-op and (321) for ιlowastcl-op)

Cardy Algebras and Sewing Constraints I 899

Definition 314 (CardyC|C2plusmn-algebra II) A CardyC-algebra is a triple (Aop|Acl ιcl-op)where Acl is a commutative symmetric Frobenius C2plusmn-algebra satisfying property (34)with f = mcl Aop is a symmetric Frobenius C-algebra and ιcl-op T (Acl) rarr Aop isan algebra homomorphism satisfying the conditions (320) and (322)

Remark 315 Up to a choice of normalisation Definition 314 is the same as the orig-inal one in [K3 Def 613] The difference between the two definitions is the factorDim C dim Ui on the left hand side of (322) which in [K3 Def 613] is given byradic

Dim C dim Ui The two definitions are related by rescaling the coproduct cl andcounit εcl of Acl by 1

radicDim C and

radicDim C respectively We chose the convention in

(322) to remove all dimension factors from the expression (314) for the Cardy condition

33 Uniqueness and existence theorems In this subsection we investigate the structureof Cardy algebras We start with the following proposition which when combined withthe results of part II provides an alternative proof of [Fj Prop 422]

Proposition 316 Let (Aop|Acl ιcl-op) be a Cardy C|C2plusmn-algebra If Acl is simple anddim Aop = 0 then Aop is simple and special

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis therefore haploid Restricting the Cardy condition (322) to the case Ui = 1 andcomposing both sides with εop from the left we see that εop kills all terms associated toU j times 1 isin Acl in the sum except for a single 1 times 1 term Thus we obtain the followingidentity

β εop = dAop (mop otimes idAorop) (idAop otimes bAop) (326)

where β isin C Composing with ηop from the right in turn implies that βεop ηop =dim Aop which is nonzero by assumption Thus also β = 0 and εop is a nonzero multipleof the morphism on the right hand side of (326) By [FRS Lem 311] Aop is special

Since Aop is a special Frobenius algebra Aop is semi-simple as an Aop-bimodule(apply [FS Prop 524] to Aop tensored with its opposite algebra) Suppose Aop is not

simple so that we can write Aop = A(1)op oplus A(2)op for nonzero Aop-bimodules A(1)op and A(2)op We denote the canonical embeddings and projections associated to this decompositionas ι12 and π12 We have the identities

mop (ι1 otimes ι2) = 0 εop ηop =2sum

i=1

εop ιi πi ηop (327)

The first identity follows since π1 mop (ι1 otimes ι2) = 0 (as mop gives the left action of

Aop on Aop and hence it preserves A(2)op ) and similarly π2 mop (ι1 otimes ι2) = 0 Thesecond identity is just the completeness of ι12 π12

Since εop ηop = 0 without losing generality we can assume εop ι1 π1 ηop = 0Using that π2 is a bimodule map we compute

π2 [LHS of (322)]Ui =1 ι1 π1 ηop = π2 [RHS of (322)]Ui =1 ι1 π1 ηop

= PlA(2)op

π2 ι1 π1 ηop = 0 (328)

900 L Kong I Runkel

On the other hand using that Acl is haploid that ιcl-op is an algebra map and thatι lowastcl-op is a coalgebra map one can check that the left-hand side of (328) is equal toλ(εop ι1 π1 ηop) π2 ηop for some λ = 0 This implies that π2 ηop = 0 Thusηop = sum

i ιi πi ηop = ι1 π1 ηop Hence we have

0 = π2 ι2 = π2 mop (ηop otimes ι2) = π2 mop ((ι1 π1 ηop)otimes ι2) (329)

However the right-hand side is zero by (327) This is a contradiction and hence Aopmust be simple

To formulate the next theorem we need the notion of the full centre of an algebra [FjDef 49] Recall that an algebra A in a braided tensor category has a left centre and aright centre [VZO] both of which are sub-algebras of A Of these two we will onlyneed the left centre The following definition is [Fr Def 231] which in our setting isequivalent to that of [VZO]

Definition 317 Let A be a symmetric special Frobenius algebra such that m A A =ζA idA

(i) The left centre Cl(A) of A is the image of the idempotent ζminus1A Pl

A(ii) The full centre Z(A) is Cl(R(A))

That ζminus1A Pl

A is an idempotent follows from [FRS Lem 52] when keeping track ofthe factors ζA ([FRS] assumes normalised-special ie ζA = 1) Note that Cl(A) is againan object of C while Z(A) is an object of C2plusmn Let el

A Cl(A) rarr A be the embeddingof Cl(A) into A The left centre is in fact the maximal subobject of A such that

m A cAA (elA otimes idA) = m A (330)

see [Fr Lem 232] This observation explains the name left centre and also makes theconnection to [O Def 15]

The full centre is by definition the image of the idempotent ζminus1A Pl

R(A) R(A) rarrR(A) Since C2plusmn is abelian the idempotent splits and we obtain the embedding andrestriction morphisms

e Z(A) rarr R(A) and r R(A) Z(A) (331)

which obey r e = idZ(A) and e r = ζminus1A Pl

R(A) It follows from Proposition 225 and

[Fr Prop 237] that Z(A) is a commutative symmetric Frobenius algebra in C2plusmn withstructure morphisms5

m Z(A) = r m R(A) (e otimes e) ηZ(A) = r ηR(A)

Z(A) = ζA middot (r otimes r) R(A) e εZ(A) = ζminus1A middot εR(A) e

(332)

Moreover if A is simple then Z(A) is simple and if A is simple and dim A = 0 thenZ(A) is simple and special The normalisation of the counit is such that

εZ(A) ηZ(A) = ζminus2A dim A DimC (333)

5 The normalisation of product and unit is the standard one The factors in the coproduct and counit haveto be included in order for (A|Z(A) e) to be a Cardy algebra see Theorem 318 below The normalisation ofthe counit enters the Cardy condition (314) through the definition of ( middot )lowast

Cardy Algebras and Sewing Constraints I 901

Theorem 318 Let A be a special symmetric Frobenius C-algebra Then (A|Z(A) e)is a Cardy C|C2plusmn-algebra

The proof of this theorem makes use the following two lemmas

Lemma 319 e Z(A) rarr R(A) is an algebra map and elowast = ζA middot r

Proof It follows from [Fr Lem 229] (or by direct calculation using in particularm R(A) R(A) = ζAidR(A)) that

m R(A) (e otimes e) = ζminus1A middot Pl

R(A) m R(A) (e otimes e) (334)

Substituting e r = ζminus1A Pl

R(A) shows that e is compatible with multiplication For theunit one finds

e ηZ(A) = e r ηR(A) = ζminus1A Pl

R(A) ηR(A) = ηR(A) (335)

Thus e is an algebra map For the second statement one computes

elowast (1)= ζA

R(A)

r

R(A)

R(A)

Z(A)

R(A)

Z(A)

r

e

e

(2)=

R(A)

r

R(A)

R(A)

R(A)

Z(A)

(3)=

R(A)

r

R(A)

Z(A)

(4)= ζA middot r (336)

where in (1) the definitions (230) and (332) have been substituted step (2) is e r = ζminus1

A PlR(A) step (3) uses that R(A) is symmetric Frobenius and step (4) is again

e r = ζminus1A Pl

R(A) Lemma 320 Let A be a symmetric Frobenius algebra in C The morphism

PlR(A) m R(A)

(Pl

R(A) otimes PlR(A)

) R(A)otimes R(A) minusrarr R(A) (337)

is S-invariant

The proof of this lemma is a slightly lengthy explicit calculation and has been deferredto Appendix A2

902 L Kong I Runkel

Proof of Theorem 318 That e is an algebra map was proved in Lemma 319 The centrecondition (313) holds by property (330) of the left centre The Cardy condition (314)also is an immediate consequence of Lemma 319

ιcl-op ιlowastcl-op = e (ζA r) = PlR(A) (338)

The full centre Z(A) is a commutative symmetric Frobenius algebra It remains to provemodular invariance That θZ(A) = idZ(A) is implied by commutativity and symmetryof Z(A) [Fr Prop 225] The S-invariance condition (33) follows from Lemma 320In (337) substitute Pl

R(A) = ζA e r and then put the resulting morphism into (33)Compose the resulting equation with e otimes idW from the right (ie from the bottom) andsubstitute the definition (332) of m Z(A) This results in the statement that m Z(A) isS-invariant

The following theorem is analogous to [LR Prop 29] and [Fj Thm 426] whichroughly speaking answer the question under which circumstances the restriction of atwo-dimensional conformal field theory to the boundary already determines the entireconformal field theory The first work is set in Minkowski space and uses operator alge-bras and subfactors while the second work is set in Euclidean space and uses modulartensor categories

Theorem 321 Let (A|Acl ιcl-op) be a Cardy C|C2plusmn-algebra such that dim A = 0 andAcl is simple Then A is special and (A|Acl ιcl-op) sim= (A|Z(A) e) as Cardy algebras

Proof By Proposition 316 A is simple and special Since Acl is simple the algebramap ιcl-op Acl rarr R(A) is either zero or a monomorphism But ιcl-op ηcl = ηR(A)and so ιcl-op = 0 Thus ιcl-op is monic By Lemma 218 (ii) ζminus1

A ιlowastcl-op is epi The Cardycondition (314) implies

ιcl-op ζminus1A ιlowastcl-op = ζminus1

A PlR(A) = e r (339)

Composing this with e r from the left yields e r ιcl-op ζminus1A ιlowastcl-op = e r =

ιcl-op ζminus1A ιlowastcl-op Since ζminus1

A ιlowastcl-op is epi we have

e r ιcl-op = ιcl-op (340)

Actually (340) also follows from (313) and specialness of R(A) We will prove that

( fop fcl) (A|Acl ιcl-op) minusrarr (A|Z(A) e) where fop = idA fcl = r ιcl-op

(341)

is an isomorphism of Cardy algebrasfcl is an algebra map Compatibility with the units follows since ιcl-op is an algebramap

fcl ηcl = r ιcl-op ηcl = r ηR(A) = ηZ(A) (342)

Compatibility with the multiplication also follows since ιcl-op is an algebra map

m Z(A) ( fcl otimes fcl) = r m R(A) (e otimes e) (r otimes r) (ιcl-op otimes ιcl-op)

= r m R(A) (ιcl-op otimes ιcl-op) = r ιcl-op mcl = fcl mcl (343)

where in the second step we used (340)

Cardy Algebras and Sewing Constraints I 903

fcl is an isomorphism As above since fcl is an algebra map and since Acl is simple fcl

has to be monic By Lemma 319 rlowast = ζminus1A e Thus f lowast

cl = ιlowastcl-op rlowast = ζminus1A ιlowastcl-op e and

fcl f lowastcl = r ιcl-op ζminus1

A ιlowastcl-op e = r e r e = idZ(A) (344)

and so fcl is also epi and hence isofcl is a coalgebra map Since fcl is an algebra map so is f minus1

cl By (344) f minus1cl = f lowast

cland by Lemma 218 (iii) this implies that fcl is a also coalgebra mapThe diagram (315) commutes Commutativity of (315) is equivalent to e fcl = ιcl-opwhich holds by (340)

Let A be a special symmetric Frobenius algebra So far we have seen that (A|Z(A) e)is a Cardy algebra and that all Cardy algebras with Aop = A and simple Acl are of thisform It is now natural to ask if every simple Acl does occur as part of a Cardy algebraThe following theorem provides an affirmative answer Recall that for an A-left moduleM the object Mor otimesA M is an algebra (see eg [KR Lem 42])

Theorem 322 If Acl is a simple modular invariant commutative symmetric FrobeniusC2plusmn-algebra then there exist a simple special symmetric Frobenius C-algebra A and amorphism ιcl-op Acl rarr R(A) such that

(i) Acl sim= Z(A) as Frobenius algebras(ii) (A|Acl ιcl-op) is a Cardy C|C2plusmn-algebra

(iii) T (Acl) sim= oplusκisinJ Morκ otimesA Mκ as algebras where MκκisinJ is a set of representa-

tives of the isomorphism classes of simple A-left modules

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis haploid It then follows from Theorem 34 that Acl is special By Proposition 224T (Acl) is a special symmetric Frobenius algebra in C Thus T (Acl) = oplusi Ai where theAi are simple symmetric Frobenius algebras We will show that at least one of the Aiis special Since T (Acl) is special we have mT (Acl) T (Acl) = ζ idT (Acl) for someζ isin C

times Restricting this to the summand Ai shows mi i = ζ idAi FurthermoreεT (Acl) ηT (Acl) = ξ id1 for some ξ isin C

times But εT (Acl) ηT (Acl) = sumi εi ηi and so

at least one of the εi ηi has to be nonzero Therefore at least one of the Ai is speciallet A equiv Ai be this summand We denote the embedding A rarr T (Acl) by e0 and therestriction T (Acl) A by r0 Notice that r0 is an algebra homomorphism Define

ιcl-op = χ (r0) Acl minusrarr R(A) (345)

By Proposition 216 ιcl-op is an algebra homomorphism Next we verify the centre con-dition (313) or rather its equivalent form (320) By substituting the definitions onecan convince oneself that the commutativity mcl cAclAcl = mcl of Acl in C2plusmn implies thecondition mT (Acl) = mT (Acl) in C see [K2 Prop 36] Here T (Acl)otimes T (Acl) rarrT (Acl)otimes T (Acl) is given by

=oplusmn

(idCl

notimes cCl

m Crnotimes idCr

m

)

(cCl

m Clnotimes cminus1

Crn C

rm

)

(idCl

motimes cminus1

Cln C

rm

otimes idCrn

)

(346)

and we decomposed Acl as Acl = oplusnCln times Cr

n As a consequence we obtain the identity

r0 mT (Acl) (idAcl otimes e0) = r0 mT (Acl) (idAcl otimes e0) (347)

904 L Kong I Runkel

Using that r0 is an algebra map and that by definition ιcl-op = r0 we obtain (320)In order to show that (A|Acl ιcl-op) is a Cardy algebra it remains to show that the

Cardy condition (314) is satisfied We will demonstrate this via a detour by first provingthat Acl sim= Z(A) as Frobenius algebras

Recall the notations e and r given in (331) Using the centre condition (320) onecan check that Pl

R(A) ιcl-op = m R(A) R(A) ιcl-op By specialness of A we have

m A A = ζA idA and so together with e r = ζminus1A Pl

R(A) we get

e r ιcl-op = ιcl-op (348)

Next consider the morphism

fcl = r ιcl-op Acl minusrarr Z(A) (349)

By the same derivation as in (342) and (343) one sees that fcl is an algebra map Inparticular fcl ηcl = ηZ(A) = 0 and so fcl = 0 Since Acl is simple fcl has to be amonomorphism

By the same argument as used in the proof of Theorem 34 (ii) up to isomorphismAcl is the unique simple local Acl-(left-)module The algebra monomorphism fcl turnsZ(A) into an Acl-module Since Z(A) is commutative it is local as an Acl-module andso Z(A) sim= AoplusN

cl for some N ge 1 By construction A is a simple special symmetricFrobenius algebra Proposition 225 and Corollary 227 show that R(A) inherits all theseproperties and thus Z(A) is simple (see the comment below Eq (332)) By Theorem318 Z(A) is modular invariant and then by Theorem 34 (i) dim Z(A) = Dim C Thisimplies that N = 1 in Z(A) sim= AoplusN

cl and so fcl is in fact an isomorphismSince Acl and Z(A) are both haploid we have εZ(A) fcl = ξ εcl for some ξ isin C

timesThe counit uniquely determines the Frobenius structure on Acl and Z(A) (see eg [FRSLemma 37]) so that fcl is a coalgebra isomorphism iff ξ = 1 To compute ξ we composethe above identity with ηcl from the right Defining ζcl via εcl ηcl = ζminus1

cl Dim C middot id1 andusing (333) gives ξ = dim A ζclζ

2A By rescaling the comultiplication and the counit

of A and consequently changing ζA we can always achieve ξ = 1 This proves part (i)of the theorem

Equation (348) implies that ιcl-op = e fcl Since fcl is an isomorphism of Frobeniusalgebras by Lemmas 218 and 319 we have

ιcl-op ιlowastcl-op = e fcl f lowastcl elowast = ζA e r = Pl

R(A) (350)

Thus (A|Acl ιcl-op) is a Cardy algebra This proves part (ii) of the theoremPart (iii) can be seen as follows By [KR Prop 43] T Z(A) sim= oplusκisinJ Mor

κ otimesA Mκ asalgebras Together with the observation that T ( fcl) T (Acl) rarr T Z(A) is an isomor-phism of algebras this proves part (iii) Remark 323 Part (i) of Theorem 322 was announced by Muumlger [Muuml2] We provide anindependent proof in the setting of Cardy algebras

The above theorem together with Lemma 211 and Theorem 34 shows that a sim-ple commutative symmetric Frobenius C2plusmn-algebra Acl with dim Acl = Dim C is alwayspart of a Cardy algebra (Aop|Acl ιcl-op) for some simple special symmetric Frobeniusalgebra Aop in C However the above proof also illustrates that Aop is not unique Thisraises the question how two Cardy algebras with a given Acl can differ This question isanswered by [KR Thm 11] which in the present framework can be restated as follows

Cardy Algebras and Sewing Constraints I 905

Theorem 324 If (A(i)op |A(i)cl ι(i)cl-op) i = 1 2 are two Cardy C|C2plusmn-algebras such that

A(i)cl is simple and dim A(i)op = 0 for i = 1 2 then A(1)clsim= A(2)cl as algebras if and only

if A(1)op and A(2)op are Morita equivalent

Proof Theorem 11 in [KR] is stated for A(i)op being non-degenerate algebras and A(i)cl =Z(A(i)op ) for i = 1 2 By Proposition 316 A(i)op are simple and special for i = 1 2

Then by [KR Lem 21] A(i)op are non-degenerate algebras By Theorem 321 we have

A(i)clsim= Z(A(i)op ) as Frobenius algebras Finally by [KR Thm 11] Z(A(1)op ) sim= Z(A(2)op )

as algebras iff A(1)op and A(2)op are Morita equivalent Let Cmax(C2plusmn) be the set of equivalence classes [B] of simple modular invariant

commutative symmetric Frobenius algebras B in C2plusmn Two such algebras B and B prime areequivalent if B and B prime are isomorphic as algebras (but not necessarily as Frobenius alge-bras) Let Msimp(C) be the set of Morita classes of simple special symmetric Frobeniusalgebras in C Define the map z Msimp(C) rarr Cmax(C2plusmn) by z A rarr [Z(A)] whereA denotes the Morita class of A From Theorem 322 (i) and [KR Thm 11] we learn

Corollary 325 The map z Msimp(C) rarr Cmax(C2plusmn) is a bijection

A Appendix

A1 Proof of Lemma 27 We will show that if (F ψ F2 ψ

F0 ) is a colax tensor functor

from C1 to C2 then (G φG2 φ

G0 ) is a lax tensor functor from C2 to C1 Applying this

result to the opposed categories then gives the converse statementWe need to show that φG

0 and φG2 make the diagrams (21) and (22) commute We

first prove the commutativity of (21) Consider the following diagram

G(A)otimes (G(B)otimes G(C))

GψF2 δ

idG(A)otimesφG2 G(A)otimes G(B otimes C)

GψF2 δ

G(FG(A)otimes F(G(B)otimes G(C)))

G(F(idG(A))otimesF(φG2 ))

G(idFG(A)otimesψ F2 )

G(FG(A)otimes FG(B otimes C))

G(ρAotimesρBotimesC )

G(FG(A)otimes (FG(B)otimes FG(C)))

G(ρAotimes(ρBotimesρC )) G(A otimes (B otimes C))

(A1)

The top subdiagram is commutative because of the naturality of Gψ F2 δ The commu-

tativity of the bottom subdiagram follows from the following identities

(ρB otimes ρC ) ψ F2 = (ρB otimes ρC ) ψ F

2 ρF Fδ

= ρBotimesC FG(ρB otimes ρC ) FG(ψ F2 ) Fδ

= ρBotimesC F(φG2 ) (A2)

as a map F(G(B) otimes G(C)) rarr B otimes C The commutativity of (A1) implies that thecomposition of maps in the left column in (21) can be replaced by

G(ρA otimes (ρB otimes ρC )) G((idFG(A) otimes ψ F

2 ) ψ F2

) δ (A3)

906 L Kong I Runkel

Similarly we can show that the composition of maps in the right column in (21) can bereplaced by

G((ρA otimes ρB)otimes ρC ) G((ψ F

2 otimes idFG(C)) ψ F2

) δ (A4)

Using the commutativity of (23) it is easy to see that (21) with the left and right col-umns of (21) replaced by (A3) and (A4) respectively is commutative Hence (21) iscommutative

Now we prove the commutativity of the first diagram in (22)

φG2 (φG

0 otimes idG(A))

(1)= G(ρ12 otimes ρA) Gψ F2 δ

[(Gψ F

0 δ11)otimes idG(A)

]

(2)= G(ρ12 otimes ρA) Gψ F2 G F(Gψ0 otimes idG(A)) G F(δ11 otimes idG(A)) δ

(3)= G(ρ12 otimes ρA) G(FG(ψ F0 )otimes idFG(A)) Gψ F

2 G F(δ11 otimes idG(A)) δ(4)= G(id12 otimes ρA) G

([ψ F

0 ρF(11) (Fδ)11] otimes idFG(A)

) Gψ F

2 δ(5)= G(id12 otimes ρA) G(ψ F

0 otimes idFG(A)) Gψ F2 δ

(6)= G(id12 otimes ρA) G(lminus1FG(A)) G F(lG(A)) δ

(7)= G(lminus1A ) GρA δG lG(A)

(8)= G(lminus1A ) lG(A) (A5)

where in step (1) we substituted the definition of φG0 φ

G2 given in (210) in step (2)

we used the naturality of δ in step (3) we used the naturality of Gψ F2 in step (4) we

switched the position between Gρ12 and G FG(ψ F0 ) and the position between Gψ2 and

G F(δ12 otimes idG(A)) using the naturality of ρ and FψG2 respectively in step (5) we applied

the second identity in (28) in step (6) we used (24) in step (7) we used the naturalityof lminus1 and δ in step (8) we used the first identity in (28)

The proof of the commutativity of the second diagram in (22) is similar Thus wehave shown that G is a lax tensor functor

A2 Proof of Lemma 320 To prepare the proof recall that for a given object B isin Cthe modular group P SL(2Z) acts on the space oplusi HomC(B otimes Ui Ui ) see eg [BKSect 31] and [K3 Eq (455)] We will only need the action of S and Sminus1 Let f isinoplusi HomC(B otimes Ui Ui ) Then

S oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A6)

Cardy Algebras and Sewing Constraints I 907

Sminus1 oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A7)

By Lemma 32 to establish that (337) is S-invariant it is enough to prove the identity(34) when f is given by (337) Using (A6) and (A7) we can see that Eq (34) simplysays that oplusi j [RHS of (34)] is invariant under the action of S times S Consider the elementg of oplus jkisinIHomC2plusmn(R(A)otimes (Uor

j times Uk)Uorj times Uk) given by

g =oplusjkisinI

sumα

R(A)

R(A)

R(A)

R(A)

Uorj times Uk

Uorj times Uk

α

α

(A8)

By the above arguments proving S-invariance of (337) is equivalent to proving invari-ance of g under the action of S times S

For i isin I we denote by gi the component of g in(oplus jisinIHomC(A otimes Uor

i otimes Uorj U

orj )

)otimes (opluskisinIHomC(Ui otimes UkUk))

We view the second Hom-space in the above tensor product as a Hom-space in C+ insteadof Cminus It is enough to show that gi is invariant under the action of S times Sminus1 Note thatthe action of Sminus1 in C+ is equivalent to that of S in Cminus

The morphism gi can be canonically identified with a bilinear pairing

( middot middot )i ⎛⎝oplus

jisinIHomC(Uor

j A otimes Uori otimes Uor

j )

⎞⎠

times(oplus

kisinIHomC(UkUi otimes Uk)

)minusrarr C (A9)

as follows For h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) we set

(h1 h2)i = (dim U j dim Uk)minus1 trUor

j timesUk[gi (h1 times h2)] (A10)

908 L Kong I Runkel

When substituting the explicit form of the product m R(A) of R(A) = (A times 1) otimes R(1)after a short calculation one finds

(h1 h2)i =sumα

1

dim U j

A

A

A

A

A A

A α

α

h1

h2

U jUk

Uorj

Ui

Uk (A11)

Here the top morphism PlR(A) has been simplified with the help of the identity

PR(A) m R(A) (PR(A) otimes PR(A))

=(oplus

iisinI((m A A)otimes idUor

i)times idUi

) m R(A) (PR(A) otimes PR(A)) (A12)

which can be checked by direct calculation along the same lines as in the proof of [FrLem 310]

The action of the modular transformation S on oplusiisinI(B otimes Ui Ui ) for B isin C nat-urally induces an action on oplusiisinI(Ui B otimes Ui ) [K3 Prop 514] which we denote bySlowast In the present case we get an action of Slowast on oplus jisinIHomC(Uor

j A otimes Uori otimes Uor

j ) and

opluskisinIHomC(UkUi otimes Uk) Then to show gi is invariant under the action of S times Sminus1

amounts to showing that

(h1 h2)i = ((Sminus1)lowast h1 Slowast h2)i (A13)

Cardy Algebras and Sewing Constraints I 909

for all h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) We have

((Sminus1)lowast h1 Slowast h2)i =sum

mnα

dim Un

DimC

A

A

A

A

A A

A

Uk

α

α

h1

h2

Ui

U j

UmUn

(A14)

Now drag the upper vertex indexed by α in the above graph along its Uorm -leg until it

meets the lower vertex also indexed by α then sum over α and m This gives

((Sminus1)lowast h1 Slowast h2)i =sum

n

dim Un

DimC

A

A

A

A

A A

A

Uk

Un

h1

h2

Ui

U j

(A15)

910 L Kong I Runkel

If we just look at the neighbourhood of the Un-loop in the above graph we see thefollowing subgraphs

sumn

dim Un

DimC

A Aor

Uorj U j

Uk Uork

Un=

sumα

1

dim U j

A Aor Uk Uork

Uorj U j

αα

(A16)

where we have applied [BK Cor 3111] Substituting this subgraph back to the originalgraph in (A15) we obtain

((Sminus1)lowast h1 Slowast h2)i =sumα

1

dim U j

A

A

A

A

A A

A

A

A

Uorj

α

α

h1

h2m A

Ui

U jUk

Uk

(A17)

The graph in (A17) is equal to that in (A11) In order to see this we first drag theldquobubblerdquo (m A A) along A lines and through the m A vertex (because m A A is abimodule map) until it reaches the lower-left leg of the upper vertex indexed by α Thendrag the m A vertex along the (red) dotted line in the above graph Finally we applythe associativity of A (A12) and [Fr Lem 311] Then we see that the graph in (A17)exactly matches with the one in (A11)

Acknowledgements We would like to thank the organisers of the Oberwolfach Arbeitsgemeinschaft ldquoAlge-braic structures in conformal field theoriesrdquo (April 2007) where this work was started for an inspiring meetingWe would further like to thank the Hausdorff Institute for Mathematics in Bonn and the organisers of the stim-ulating meeting ldquoGeometry and Physicsrdquo (May 2008) We are indebted to Alexei Davydov Jens FjelstadJuumlrgen Fuchs Yi-Zhi Huang Alexei Kitaev Urs Schreiber Christoph Schweigert Stephan Stolz and PeterTeichner for helpful discussions andor comments on a draft of this paper The research of IR was partiallysupported by the EPSRC First Grant EPE0050471 the PPARC rolling grant PPC5071451 and the MarieCurie network lsquoSuperstring Theoryrsquo (MRTN-CT-2004-512194)

Cardy Algebras and Sewing Constraints I 911

References

[AN] Alexeevski A Natanzon SM Noncommutative two-dimensional topological field theories andhurwitz numbers for real algebraic curves Sel Math New Ser 12 307ndash377 (2006)

[Bi] Bichon J Cosovereign hopf algebras J Pure Appl Alg 157 121ndash133 (2001)[BK] Bakalov B Kirillov AA Lectures on Tensor Categories and Modular Functors Providence

RI Amer Math Soc 2001[C1] Cardy JL Operator content of two-dimensional conformal invariant theories Nucl Phys

B 270 186ndash204 (1986)[C2] Cardy JL Boundary conditions fusion rules and the verlinde formula Nucl Phys B 324

581ndash596 (1989)[CL] Cardy JL Lewellen DC Bulk and boundary operators in conformal field theory Phys Lett

B 259 274ndash278 (1991)[DMZ] Dong C-Y Mason G Zhu Y-C Discrete series of the Virasoro algebra and the moonshine

module In Algebraic Groups and Their Generalizations Quantum and infinite-dimensional Meth-ods Proc Symp Pure Math 56 Part 2 Providence RI Amer MathSoc 1994 pp 295ndash316

[DP] Day B Pastro C Note on frobenius monoidal functors New York J Math 14 733ndash742 (2008)[ENO1] Etingof PI Nikshych D Ostrik V On fusion categories Ann Math 162 581ndash642 (2005)[ENO2] Etingof PI Nikshych D Ostrik V An analogue of radfordrsquos s4 formula for finite tensor

categories Int Math Research Notices 54 2915ndash2933 (2004)[Fe] Felder G Froumlhlich J Fuchs J Schweigert C Correlation functions and boundary conditions in

rational conformal field theory and three-dimensional topology Comp Math 131 189ndash237 (2002)[FHL] Frenkel IB Huang Y-Z Lepowsky J On axiomatic approaches to vertex operator algebras

and modules Mem Amer Math Soc 104 (1993)[Fj] Fjelstad J Fuchs J Runkel I Schweigert C Uniqueness of openclosed rational cft with

given algebra of open states Adv Theor Math Phys 12 1283ndash1375 (2008)[FK] Froumlhlich J King C The chern-simons theory and knot polynomials Commun Math

Phys 126 167ndash199 (1989)[Fr] Froumlhlich J Fuchs J Runkel I Schweigert C Correspondences of ribbon categories Adv

Math 199 192ndash329 (2006)[FRS] Fuchs J Runkel I Schweigert C Tft construction of rcft correlators I partition functions Nucl

Phys B 646 353ndash497 (2002)[FS] Fuchs J Schweigert C Category theory for conformal boundary conditions Fields Inst Com-

mun 39 25ndash71 (2003)[H1] Huang Y-Z Two-dimensional conformal geometry and vertex operator algebras Progress in

Mathematics Vol 148 Boston Birkhaumluser 1997[H2] Huang Y-Z Rigidity and modularity of vertex tensor categories Comm Contemp Math 10

871ndash911 (2008)[HK1] Huang Y-Z Kong L Open-string vertex algebra category and operads Commun Math

Phys 250 433ndash471 (2004)[HK2] Huang Y-Z Kong L Full field algebras Commun Math Phys 272 345ndash396 (2007)[HK3] Huang Y-Z Kong L Modular invariance for conformal full field algebras httparxivorgabs

math0609570v2[mathQA] 2006[HL] Huang Y-Z Lepowsky J Tensor products of modules for a vertex operator algebra and vertex

tensor categories In Lie Theory and Geometry in honor of Bertram Kostant ed R BrylinskiJ-L Brylinski V Guillemin V Kac Boston Birkhaumluser 1994 pp 349ndash383

[JS] Joyal A Street R Braided tensor categories Adv Math 102 20ndash78 (1993)[K1] Kong L Full field algebras operads and tensor categories Adv Math 213 271ndash340 (2007)[K2] Kong L Open-closed field algebras Commun Math Phys 280 207ndash261 (2008)[K3] Kong L Cardy condition for open-closed field algebras Commun Math Phys 283 25ndash92 (2008)[Ki] Kitaev A Private communication[KO] Kirillov AA Ostrik V On q-analog of mckay correspondence and ade classification of sl(2)

conformal field theories Adv Math 171 183ndash227 (2002)[KR] Kong L Runkel I Morita classes of algebras in modular tensor categories Adv Math 219

1548ndash1576 (2008)[Ld] Lauda AD Frobenius algebras and ambidextrous adjunctions Theo Appl Cat 16 84ndash

122 (2006)[Li1] Li H-S Regular representations of vertex operator algebras Commun Contemp Math 4

639ndash683 (2002)[Li2] Li H-S Regular representations and huang-lepowsky tensor functors for vertex operator

algebras J Alge 255 423ndash462 (2002)

Cardy Algebras and Sewing Constraints I 883

where (1) is the adjunction property (28) (2) is naturality of δ (3) functoriality of Gand (4) amounts to (29) and functoriality of F

On the other hand we see that the first equality in (224) is equivalent to

m B (ρ otimes ρ) (F( f )otimes F( f )) = ρ F( f ) F(m A) φF2 (227)

Using that φF2 is invertible with inverse (φF

2 )minus1 = ψ F

2 and that χ is an isomorphismit follows that the statement that (225) is equal to (226) is equivalent to the identity(227)

Now we prove that the second identity in (223) is equivalent to the second identity in(224) Using (217) and (210) we can write ηG(B) = G(ηB)φF

0 = G(ηB)G(ψ F0 )δ1

Together with (29) this shows that the second identity in (223) is equivalent to

f ηA = χ(ηB ψ F0 ) (228)

On the other hand the second identity in (224) is equivalent to

ρ F( f ) F(ηA) φF0 = ηB (229)

which by φF0 = (ψ F

0 )minus1 and (29) is further equivalent to (228)

Definition 217 Let (Am A ηAA εA) and (Bm B ηBB εB) be two Frobeniusalgebras in a tensor category C For f A rarr B we define f lowast B rarr A by

f lowast = ((εB m B)otimes idA) (idB otimes f otimes idA) (idB otimes (A ηA)) (230)

The following lemma is immediate from the definition of (middot)lowast and the properties ofFrobenius algebras We omit the proof

Lemma 218 Let C be a tensor category let A BC be Frobenius algebras in C andlet f A rarr B and g B rarr C be morphisms

(i) (g f )lowast = f lowast glowast(ii) f is a monomorphism iff f lowast is an epimorphism

(iii) f is an algebra map iff f lowast is a coalgebra map(iv) If f is a homomorphism of Frobenius algebras then f lowast f = idA and f f lowast =

idB(v) If C is sovereign and if A and B are symmetric then f lowastlowast = f

Let C and D be tensor categories and let F C rarr D be a Frobenius functor GivenFrobenius algebras A B in C and a morphism f A rarr B the next lemma shows how(middot)lowast behaves under F

Lemma 219 F( f lowast) = F( f )lowast

Proof The definition of the structure morphisms of the Frobenius algebra F(A) is givenin (217) and (218) Substituting these definitions gives

F( f )lowast =[(ψ F

0 F(εB) F(m B) φF2

)otimes idF(A)

] [

idF(B) otimes F( f )otimes idF(A)]

[idF(B) otimes

(ψ F

2 F(A) F(ηA) φF0

)]

= (ψ F0 otimes idF(A)) [

F (εB m B (idB otimes f ))otimes idF(A)]

(φF2 otimes idF(A)) (idF(B) otimes ψ F

2 )

[idF(B) otimes F(A ηA)

] (idF(B) otimes φF0 ) (231)

884 L Kong I Runkel

In the middle line of the last expression we can use the defining property (25) of F namely we substitute (φF

2 otimes idF(A)) (idF(B) otimes ψ F2 ) = ψ F

2 φF2 Then ψ F

2 can bemoved to the left and φF

2 to the right until they can be omitted against ψ F0 and φF

0 respectively using (22) and (24) This results in

F( f )lowast = F ((εB m B (idB otimes f ))otimes idA) F (idB otimes (A ηA)) (232)

which is nothing but F( f lowast)

23 Modular tensor categories Let C be a modular tensor category [TBK] ie anabelian semi-simple finite C-linear ribbon category with simple tensor unit 1 and a non-degeneracy condition on the braiding (to be stated in a moment) We denote the set ofequivalence classes of simple objects in C by I elements in I by i j k isin I and theirrepresentatives by Ui U j Uk We also set U0 = 1 and for an index k isin I we define kby Uk

sim= Uork

Since the tensor unit is simple we shall for modular tensor categories identifyHom(1 1) sim= C (cf footnote 2) Define numbers si j isin C by4

si j = UiU j (233)

They obey si j = s ji and s0i = dim Ui see eg [BK Sect 31] (In a ribbon cate-gory the left and right dimension (214) of Ui coincide and are denoted by dim Ui )The non-degeneracy condition on the braiding of a modular tensor category is that the|I|times|I|-matrix s should be invertible In fact [BK Thm 317]

sumkisinI

sik sk j = Dim C δij (234)

where Dim C = sumiisinI(dim Ui )

2 One can show (even in the weaker context of fusioncategories over C) that Dim C ge 1 [ENO1 Thm 23] In particular Dim C = 0 We fixonce and for all a square root

radicDim C of Dim C

Let us fix a basis λα(i j)kN k

i jα=1 in HomC(Ui otimesU j Uk) and the dual basis ϒ(i j)k

α N ki j

α=1

in HomC(UkUi otimesU j ) The duality of the bases means that λα(i j)k ϒ(i j)kβ = δαβ idUk

We also fix λ(0i)i = λ(i0)i = idUi We denote the basis vectors graphically as follows

λα(i j)k = α

Uk

Ui U j

ϒ(i j)kα = α

Uk

Ui U j

(235)

4 In the graphical notation used below we have given an orientation to the ribbons indicated by the arrowsFor example it is understood that this orientation determines which of the duality morphisms in (213) to use

Cardy Algebras and Sewing Constraints I 885

For V isin C we also choose a basis b(iα)V of HomC(VUi ) and the dual basis bV(iβ)

of HomC(Ui V ) for i isin I such that b(iα)V bV(iβ) = δαβ idUi We use the graphical

notation

b(iα)V = α

Ui

V

bV(iα) = α

Ui

V

(236)

Given two modular tensor categories C and D by C D we mean the tensor productof additive categories over C [BK Def 1115] ie the category whose objects are directsums of pairs V times W of objects V isin C and W isin D and whose morphism spaces are

HomCD(V times W V prime times W prime) = HomC(V V prime)otimesC HomC(WW prime) (237)

for pairs and direct sums of these if the objects are direct sums of pairsIf we replace the braiding and the twist in C by the antibraiding cminus1 and the antitwist

θminus1 respectively we obtain another ribbon category structure on C In order to distin-guish these two distinct structures we denote (C c θ) and (C cminus1 θminus1) by C+ and Cminusrespectively As in the Introduction we will abbreviate

C2plusmn = C+ Cminus (238)

Note that a set of representatives of the simple objects in C2plusmn is given by Ui times U j fori j isin I

For the remainder of Sect 2 we fix a modular tensor category C

24 The functors T and R The tensor product bifunctor otimes can be naturally extendedto a functor T C2plusmn rarr C Namely T (oplusN

i=1Vi times Wi ) = oplusNi=1Vi otimes Wi for all Vi Wi isin C

and N isin N The functor T becomes a tensor functor as follows For φT0 1 rarr T (1 times 1)

take φT0 = id1 (or lminus1

1 in the non-strict case) Next notice that for U VW X isin C

T (U times V )otimes T (W times X)= (U otimes V )otimes (W otimes X)

T ((U times V )otimes (W times X))= (U otimes W )otimes (V otimes X)(239)

We define φT2 T (U times V )otimes T (W times X) rarr T ((U times V )otimes (W times X)) by

φT2 = idU otimes cminus1

W V otimes idX (240)

(In the non-strict case the appropriate associators have to be added) The above definitionof φT

2 can be naturally extended to a morphism φT2 T (M1 otimes M2) rarr T (M1)otimes T (M2)

for any pair of objects M1M2 in C2plusmn The following result can be checked by directcalculation [JS Prop 52]

Lemma 220 The triple (T φT2 φ

T0 ) gives a tensor functor

886 L Kong I Runkel

In particular (T φT2 φ

T0 ψ

T2 ψ

T0 ) where ψT

2 = (φT2 )

minus1 and ψT0 = (φT

0 )minus1 gives

a Frobenius functorDefine the functor R C rarr C2plusmn as follows for A isin C and f isin HomC(A B)

R(A) =oplusiisinI

(A otimes Uori )times Ui R( f ) =

oplusiisinI

( f otimes idUori)times idUi (241)

This functor was also considered in a slightly different context in [ENO2 Prop 23]The family of isomorphisms γ R

A = oplusiisinI DimCdim Ui

id(AotimesUori )timesUi

isin Aut(R(A)) defines a

natural isomorphism γ R R rarr ROur next aim is to show that R is left and right adjoint to T in other words R

and T form an ambidextrous adjunction (see eg [Ld] for a discussion of ambidextrousadjunctions) To this end we introduce two linear isomorphisms for A isin C and M isin C2plusmn

χ HomC(T (M) A) minusrarr HomC2plusmn(M R(A))

χ HomC(A T (M)) minusrarr HomC2plusmn(R(A)M)(242)

If we decompose M as M = oplusNn=1 Ml

n times Mrn then χ and χ are given by

χ Noplus

n=1

fn

Mln Mr

n

A

rarrNoplus

n=1

oplusiisinI

sumα

fn

Mln

Mrn

A

α

Uori

times α

Mrn

Ui

(243)

and

χ Noplus

n=1 gn

Mln Mr

n

A

rarrNoplus

n=1

oplusiisinI

sumα gn

Mln

Mrn

A

α

Uori

times α

Ui

Mrn

DimCdim Ui

(244)

Notice that χ and χ are independent of the choice of basis

Theorem 221 〈T R χ〉 and 〈R T χminus1〉 are adjunctions ie R is both left and rightadjoint of T

Proof Write M as M = oplusNn=1 Ml

n times Mrn The isomorphism χ amounts to the following

composition of natural isomorphisms

HomC(T (M) A) = oplusnHomC(Mln otimes Mr

n A)sim= oplusni HomC(Ml

n otimes Ui A)otimes HomC(Mrn Ui )

sim= oplusni HomC(Mln A otimes Uor

i )otimes HomC(Mrn Ui ) = HomC2plusmn(M R(A)) (245)

Cardy Algebras and Sewing Constraints I 887

Thus χ is natural Let (γ RA )

lowast HomC2plusmn(R(A)M) rarr HomC2plusmn(R(A)M) denote the

pull-back of γ RA The isomorphism χ is equal to the composition of (γ R

A )lowast and the

following sequence of natural isomorphisms

HomC(A T (M)) = oplusn HomC(AMln otimes Mr

n)

sim= oplusni HomC(AMln otimes Ui )otimes HomC(Ui Mr

n)

sim= oplusni HomC(A otimes Uori Ml

n)otimes HomC(Ui Mrn) = HomC2plusmn(R(A)M) (246)

We have proved that both χ and χ are natural isomorphisms There are four natural transformations associated to χ and χ namely

idC2plusmnδminusrarr RT

ρminusrarr idC2plusmn and idCδminusrarr T R

ρminusrarr idC (247)

defined by for A isin C M isin C2plusmn

δM = χ(idT (M)) ρA = χminus1(idR(A))

ρM = χ (idT (M)) δA = χminus1(idR(A))(248)

They can be expressed graphically as follows with M = oplusNn=1 Ml

n times Mrn

δM =oplusni

sumα

Mln

Mln Mr

n Uori

αtimes

Ui

Mrn

α

ρA =oplusiisinI

A

A

Uori Ui

(249)

ρM =oplusni

sumα

Mln

Mln

Mrn Uor

i

α times

Ui

Mrn

α DimCdim Ui

δA =oplusiisinI

A

A Uori Ui

dim UiDimC

Note that

ρM δM = Dim C middot idM and ρA δA = idA (250)

Lemma 222 The functors T and R as maps on the sets of morphisms have left inversesand thus are injective

Proof Let f A rarr B be a morphism in C We define a map Q R HomC2plusmn(R(A)

R(B)) rarr HomC(A B) by f prime rarr ρB T ( f prime) δA Then we have

Q R R( f ) = ρB T R( f ) δA = ρB δB f = f (251)

888 L Kong I Runkel

where we used naturality of δ and (250) in the second and third equalities respectivelySo Q R is a left inverse of R on morphisms Thus R is injective on morphisms Similarlylet g M rarr N be a morphism in C2plusmn We define a map QT HomC(T (M) T (N )) rarrHomC2plusmn(M N ) by gprime rarr (Dim C)minus1 middot ρN R(gprime) δM Then we have

QT T (g) = (Dim C)minus1 middot ρN RT (g) δM = (Dim C)minus1 middot ρN δN g = g (252)

So QT is a left inverse of T on morphisms Thus T is injective on morphisms Using (29) and (250) one can express the two inverse maps χminus1 χminus1 as follows

for f isin HomC2plusmn(M R(A)) and g isin HomC2plusmn(R(A)M)

χminus1( f ) = ρ T ( f ) χminus1(g) = T (g) δ (253)

By Proposition 25 and Lemma 27 R is both a lax and colax tensor functor In particularφR

0 1 times 1 rarr R(1) is given by

φR0 = χ (ψT

0 ) = R(ψT0 ) δ1 times 1 = id1times1 (254)

and φR2 R(A)otimes R(B) rarr R(A otimes B) by φR

2 = R(ρA otimes ρB) R(ψT2 ) δ which can

be expressed graphically as

φR2 =

oplusi jkisinI

sumα

A

A

Uori B

B

Uorj

Uork

α

times

Ui U j

α

Uk

(255)

Similarly ψ R0 R(1) rarr 1 times 1 is given by

ψ R0 = ρ1 R(φT

0 ) = Dim C id1times1 (256)

and ψ R2 R(A otimes B) rarr R(A) otimes R(B) by ψ R

2 = ρ R(φT2 ) R(δA otimes δB) which in

graphical notation reads

ψ R2 =

oplusi jkisinI

sumα

A

A Uori

B

B Uorj

Uork

α

times

Ui U j

α

Uk

dim Ui dim U j

dim Uk Dim C

(257)

If C has more than one simple object then R does not take the tensor unit of C to thetensor unit of C2plusmn and so is clearly not a tensor functor However we will show that R isstill a Frobenius functor This will imply that if A is a Frobenius algebra in C then

R(A) = (R(A)m R(A) ηR(A)R(A) εR(A)) (258)

Cardy Algebras and Sewing Constraints I 889

is a Frobenius algebra in C2plusmn where the structure morphisms were given in (217) and(218) In the case A = 1 it was proved in [Muuml1 Prop 41] (see also [Fr Lem 619]and [K1 Thm 52]) that (258) is a commutative simple symmetric normalised-specialFrobenius algebra in C2plusmn In fact given a Frobenius algebra A in C it is straightforwardto verify that the structure morphisms in (258) are precisely those of (A times 1)otimes R(1)cf Sect 22

Proposition 223 (R φR2 φ

R0 ψ

R2 ψ

R0 ) is a Frobenius functor

Proof Using the explicit graphical expression of φR2 φ

R0 ψ

R2 ψ

R0 it is easy to see that

the commutativity of the diagrams (25) and (26) are equivalent to the statement thatR(1) with structure morphisms as in (258) is a Frobenius algebra in C2plusmn The latterstatement is true by [Muuml1 Prop 41]

From Lemma 220 and Proposition 223 we see that T and R take Frobenius alge-bras to Frobenius algebras The following two propositions show how the properties ofFrobenius algebras are transported

Proposition 224 Let A be a Frobenius algebra in C2plusmn Then T (A) is a Frobenius algebrain C and

(i) A is symmetric iff T (A) is symmetric(ii) A is (normalised-)special iff T (A) is (normalised-)special

Proof For part (i) write A as a direct sum oplusNn=1 Al

n times Arn Then the maps IT (A) I prime

T (A) T (Aor) rarr T (A)or defined in (220) are given by

IT (A) = I primeT (A) = oplusN

n=1c(Aln)

or(Arn)

or (259)

Therefore by (221) T (A) = primeT (A) is equivalent to T (A) = T (prime

A) Since byLemma 222 T is injective on morphisms this proves part (i) Part (ii) can be checkedin the same way for example the condition mT (A) T (A) = ζ idT (A) is easily checkedto be equivalent to T (m A A) = ζ T (idA) Proposition 225 Let A be a Frobenius algebra A in C Then R(A) is a Frobeniusalgebra in C2plusmn and

(i) A is symmetric iff R(A) is symmetric(ii) A is (normalised-)special iff R(A) is (normalised-)special

Proof Recall that the structure morphisms of the Frobenius algebra R(A) are equal tothose of (A times 1) otimes R(1) Using this equality part (i) and (ii) follow because R(1) issymmetric and normalised-special For example

m R(A) R(A) = [(m A A)times id1] otimes idR(1) = R(m A A) (260)

so that m R(A) R(A) = ζ idR(A) is equivalent to R(m A A) = ζ R(idA) which byLemma 222 is equivalent to m A A = ζ idA

The functor R has one additional property not shared by T namely R takes absolutelysimple algebras to absolutely simple algebras We will see explicitly in Sect 33 thatthis is not true for T

890 L Kong I Runkel

Lemma 226 For a C-algebra A the map

R HomA|A(A A) rarr HomR(A)|R(A)(R(A) R(A)) (261)

given by f rarr R( f ) is well-defined and an isomorphism

Proof Since R is a lax tensor functor R(A) is naturally a R(A)-bimodule It is easy tosee that R in (261) is a well-defined map R(A) is also naturally a R(1)-bimodule whichcan be identified with the induced R(1)-bimodule structure on (A times 1)otimes R(1) wherethe left R(1) action on (Atimes1)otimes R(1) is given by (idAtimes1 otimesm R(1))(cminus1

Atimes1R(1)otimes idR(1))We have the following natural isomorphisms

HomR(1)|R(1)(R(A) R(A))sim=minusrarr HomC2plusmn(A times 1 R(A))

χminus1

minusminusrarr HomC(A A) (262)

which by (253) are given by for f isin HomR(1)|R(1)(R(A) R(A))

f rarr f prime = f (idAtimes1 otimes ηR(1)) rarr f primeprime = ρ T ( f prime) (263)

and its inverse is given by for g isin HomC(A A)

g rarr gprime = R(g) δ rarr gprimeprime = (idAtimes1 otimes m R(1)) (gprime otimes idR(1)) (264)

where gprimeprime is indeed a R(1)-bimodule map due to the commutativity of R(1) It is easy tocheck that gprimeprime = R(g) in (264) Therefore R gives an isomorphism from HomC(A A)to HomR(1)|R(1)(R(A) R(A)) Moreover one verifies that R(g) is an R(A)-bimodulemap iff g is an A-bimodule map In other words R g rarr R(g) gives an isomorphism

HomA|A(A A)sim=minusrarr HomR(A)|R(A)(R(A) R(A))

Corollary 227 Let A be a C-algebra

(i) A is absolutely simple iff R(A) is absolutely simple(ii) Let A be in addition Frobenius Then A is simple and special iff R(A) is simple and

special

Proof Part (i) immediately follows from Lemma 226 The statement of part (ii) withoutthe qualifier lsquosimplersquo is proved in Proposition 225 But as in the proof of (iii)rArr(i) inLemma 211 a special Frobenius algebra in a semi-simple category has a semi-simplecategory of bimodules and for a semi-simple C-linear category simple and absolutelysimple are equivalent Part (ii) then follows from part (i)

The following lemma will be needed in Sect 32 below to discuss the properties ofCardy algebras

Lemma 228 Let A be a Frobenius algebra in C Then (δA)lowast = ρA

Proof Recall from (250) that δA is a morphism A rarr T R(A) Since T and R are bothFrobenius functors T R(A) is a Frobenius algebra in C Substituting the definitions aftera short calculation one finds

Cardy Algebras and Sewing Constraints I 891

εT R(A) mT R(A) = Dim(C)oplusiisinI

A Uori Ui A U or

ı Uı

(265)

Substituting this in the definition of (δA)lowast gives again after a short calculation the mor-

phism ρA At an intermediate step one uses that the part of the morphism (δA)lowast which

is made up of Ui and Uı ribbons their duals and the basis morphisms λ(iı)0 andϒ(iı)0can be replaced by 1

dim Uimiddot dUi

3 Cardy Algebras

In this section we start by investigating the properties of Frobenius algebras which sat-isfy the so-called modular invariance condition We then give two definitions of a Cardyalgebra and prove their equivalence Finally in Sect 33 we study the properties of thesealgebras and state our main results

We fix a modular tensor category C Recall that C2plusmn is an abbreviation for C+ Cminus

31 Modular invariance In C2plusmn we define the object K and the morphism ω K rarr Kas

K =oplus

i jisinIUi times U j ω =

sumi jisinI

dim Ui dim U j

Dim C idUi timesU j (31)

They have the property (see eg [BK Cor 3111])

ω

K

(Uk times Ul )or

(Uk times Ul )or

Ui times U j

Ui times U j

= δik δ jlDim C

dim Ui dim U jbUi timesU j dUi timesU j (32)

892 L Kong I Runkel

Definition 31

(i) Let A B be objects of C2plusmn A morphism f A otimes B rarr B is called S-invariant iff

A

B

W

W

f =

A

B

W

W

f

K

ω(33)

holds for all for W isin C2plusmn(ii) A C2plusmn-algebra (Aclmcl ηcl) is called modular invariant iff θAcl = idAcl and mcl

is S-invariant

Lemma 32 The morphism f A otimes B rarr B is S-invariant if and only if

A

B

Ui times U j

Ui times U j

f = Dim Cdim Ui dim U j

sumα

A

B

B

f

Ui times U j

Ui times U j

α

α

(34)

holds for all i j isin I

Proof Condition (33) holds for all W iff it holds for all W = Ui times U j i j isin I so itis enough to show that the right-hand side of (33) with W = Ui times U j is equal to theright-hand side of (34) Recall the notation for basis morphisms in (236) Starting from(33) write

idBor otimes idUi timesU j =⎛⎝sum

klα

(b(ktimeslα)

B

)or (

bB(ktimeslα)

)or⎞⎠ otimes idUi timesU j (35)

and then apply (32) The graphical representation of the resulting morphism can bedeformed to give (34) Remark 33 As shown in [K3 Sect 61] the modular invariance condition of a C2plusmn-alge-bra exactly coincides with the modular invariance condition for torus 1-point correlationfunctions of a genus-01 closed CFT In particular the condition θAcl = idAcl is equiv-alent to invariance under the modular transformation T τ rarr τ + 1 and the condition(34) with f = mcl is equivalent to invariance under S τ rarr minus 1

τ Combining the

modular invariance condition with the genus zero properties of a genus-01 closed CFTresults in a modular invariant commutative symmetric Frobenius algebra in C2plusmn

Cardy Algebras and Sewing Constraints I 893

Let Acl be a modular invariant C2plusmn-algebra Evaluating (34) for f = mcl composingit with ηcl otimes idUi timesU j and taking the trace implies the following identity

Zi j = 1

DimCAclUi timesU j where

(36)

Zi j = dimC HomC2plusmn(UitimesU j Acl)

Decomposing Acl into simple objects this gives

Zi j =sum

klisinISik Zkl Sminus1

l j where Si j = si jradic

DimC (37)

which in CFT terms is of course nothing but the invariance of the torus partition functionunder the modular S-transformation

The following theorem gives a simple criterion for modular invariance

Theorem 34 Let Acl be a haploid commutative symmetric Frobenius algebra in C2plusmn

(i) If Acl is modular invariant then dim Acl = Dim C(ii) If dim Acl = Dim C then Acl is special and modular invariant

Proof Part (i) Since Acl is haploid for i = j = 0 Equation (36) reduces to 1 =dim AclDimCPart (ii) By the same reasoning as in the proof of (iii)rArr(i) in Lemma 211 one showsthat Acl is special Thus mcl cl = ζclidAcl for some ζcl = 0

By [KO Thm 45] the category (C2plusmn)locAcl

of local Acl-modules is again a mod-

ular tensor category and Dim (C2plusmn)locAcl

= (Dim C dim Acl)2 (see [Fr Prop 321 amp

Rem 323] for the same statement in the notation used here) Thus by assumption wehave Dim(C2plusmn)loc

Acl= 1 It then follows from [ENO1 Thm 23] that up to isomorphism

(C2plusmn)locAcl

has a unique simple object (namely the tensor unit) In other words every simplelocal Acl-module is isomorphic to Acl (seen as a left-module over itself)

We have the following isomorphisms between morphism spaces [FS Prop 47 amp 411]

HomAcl(Acl otimes (Ui times U j ) Acl) sim= HomC2plusmn(Ui times U j Acl)

HomAcl(Acl Acl otimes (Ui times U j )) sim= HomC2plusmn(AclUi times U j )

Using these to transport the bases (236) from the right to the left we obtain bases bα(i j)αof HomAcl(Acl otimes (Ui times U j ) Acl) and b(i j)

β β of HomAcl(Acl Acl otimes (Ui times U j )) Thesecan be expressed graphically as

894 L Kong I Runkel

bα(i j) =

Acl

Acl

Acl

Ui times U j

α

b(i j)β = DimC

dim Ui dim U j

1

ζcl

Acl

Ui times U jAcl

Acl

β

(38)

where the nonzero factor in b(i j)β is included for convenience Notice that bα(i j) b(i j)

β isa left Acl-module map Since Acl is simple as a left module over itself we have

bα(i j) b(i j)β = λαβ idAcl (39)

for some λαβ isin C By computing tr(bα(i j) b(i j)β ) it is easy to verify that λαβ = δαβ

We will now prove the following identity

1

ζcl

Acl

Acl

Ui times U j

Ui times U j

Acl

=sumα

DimCdim Ui dim U j

1

ζcl

Acl

Ui times U j

Ui times U j

Acl

Acl

Acl

α

α

(310)

One checks that the left-hand side of this equation is an idempotent which we denote byζminus1

cl PlAcl(Ui timesU j ) cf [Fr Sect 31] By [Fr Prop 41] the image Im(ζminus1

cl PlAcl(Ui timesU j ))

is a local Acl-module and hence isomorphic to AoplusNcl for some N isin Zge0

All left-module morphisms from Acl to Aclotimes(Ui timesU j ) are linear combinations of the

b(i j)β Furthermore one verifies that ζminus1

cl PlAcl(Ui timesU j )b(i j)

β = b(i j)β Therefore the b(i j)

β

describe precisely the image of the idempotent ie ζminus1cl Pl

Acl(Ui timesU j ) = sum

α b(i j)α bα(i j)

which is nothing but (310) Composing (310) with ζcl middot εcl otimes idUi timesU j from the left (iefrom the top) produces (34)

In addition since Acl is commutative symmetric Frobenius it satisfies θAcl = idAcl

[Fr Prop 225] Altogether this shows that Acl is modular invariant Remark 35 As we were writing this paper we heard that the results in Theorem 34were obtained independently by Kitaev and Muumlger [Ki]

Remark 36 Setting i = j = 0 in (36) gives the identity dim Acl = Z00 Dim C [KRProp 23] Combining this with Theorem 34 (ii) one may wonder if a general modularinvariant commutative symmetric Frobenius algebra Acl in C2plusmn is isomorphic to a directsum of simple such algebras However this is not so For example one can take the com-mutative symmetric Frobenius algebra Acl = C[x]x2 in the category of vector spacesequipped with the non-degenerate trace ε(ax + b) = a In this case the modular invari-ance condition holds automatically but Acl is clearly not a direct sum of two algebras

Cardy Algebras and Sewing Constraints I 895

For a general modular tensor category C the algebra C[x]x2 R(1) understood as analgebra in C2plusmn via the braided monoidal isomorphism Vect f (C) C2plusmn rarr C2plusmn providesanother counter-example

32 Two definitions Define a morphism PlA A rarr A for a Frobenius algebra A in C

or C2plusmn as follows [Fr Sect 24]

PlA =

A

A

A

m

(311)

If A is also commutative and obeys m A A = ζA idA we have PlA = ζA idA In partic-

ular this holds if A is commutative and special Using the fact that the Frobenius algebraR(1) is commutative and normalised-special one can check that Pl

R(A) R(A) rarr R(A)takes the following form

PlR(A) =

oplusiisinI

A

A

Uori

Uori

A

m

times

Ui

Ui

(312)

With these ingredients we can now give the first definition of a Cardy C|C2plusmn-algebrawhich was introduced in [K3 Def 514] cf Remark 315 below

Definition 37 (CardyC|C2plusmn-algebra I) A CardyC|C2plusmn-algebra is a triple (Aop|Acl ιcl-op)where (Aclmcl ηclcl εcl) is a modular invariant commutative symmetric Frobe-nius C2plusmn-algebra (Aopmop ηopop εop) is a symmetric Frobenius C-algebra andιcl-op Acl rarr R(Aop) an algebra homomorphism such that the following conditionsare satisfied

(i) Centre condition

Acl R(Aop)

R(Aop)

R(Aop)

ιcl-op

m R(Aop)

=

Acl

ιcl-op

R(Aop)

R(Aop)

R(Aop)

m R(Aop)

(313)

896 L Kong I Runkel

(ii) Cardy condition

ιcl-op ιlowastcl-op =

R(Aop)

R(Aop)

R(Aop)

(314)

Remark 38

(i) The name ldquoCardy C|C2plusmn-algebrardquo in Definition 37 was chosen because many ofthe important ingredients were first studied by Cardy the modular invariance ofthe closed theory [C1] the consistency of the annulus amplitude [C2] and thebulk-boundary OPE [CL] On the other hand the boundary-boundary OPE andthe OPE analogue of the centre condition were first considered in [Lw]

(ii) One can easily see that in the special case that C is the category Vect f (C) of finite-dimensional C-vector spaces a Cardy C|C2plusmn-algebra gives exactly the algebraicformulation of two-dimensional open-closed topological field theory over C (cfRemark 614 in [K3]) see [Lz Sect 48] [Mo Thm 11] [AN Thm 45] [LPCor 43] [MS Sect 22] When passing to a general modular tensor category Cthere are two important differences to the two-dimensional topological field the-ory Firstly the algebras Acl and Aop now live in different categories which inparticular affects the formulation of the centre condition and the Cardy conditionSecondly the modular invariance condition has to be imposed on Acl In the caseC = Vect f (C) modular invariance holds automatically

Definition 39 A homomorphism of Cardy C|C2plusmn-algebras (A(1)op |A(1)cl ι(1)cl-op) rarr (A(2)op

|A(2)cl ι(2)cl-op) is a pair ( fop fcl) of Frobenius algebra homomorphisms fop A(1)op rarr A(2)op

and fcl A(1)cl rarr A(2)cl such that the diagram

A(1)cl

fcl

ι(1)cl-op

A(2)cl

ι(2)cl-op

R(A(1)op )

R( fop) R(A(2)op )

(315)

commutes

Remark 310 Since a homomorphism of Frobenius algebras is invertible (cf Lemma218 (iv)) a homomorphism of Cardy algebras is always an isomorphism

For a homomorphism ( fop fcl) of Cardy C|C2plusmn-algebras using the commutativity of(315) and the fact that fcl and fop are both algebra and coalgebra homomorphisms itis easy to show that (315) commutes iff

R(A(1)op )R( fop)

(ι(1)cl-op)

lowast

R(A(2)op )

(ι(2)cl-op)

lowast

A(1)cl

fcl A(2)cl

(316)

Cardy Algebras and Sewing Constraints I 897

commutesLet (Aop|Acl ιcl-op) be a Cardy C|C2plusmn-algebra Define the morphism

ιcl-op = χminus1(ιcl-op) T (Acl) minusrarr Aop (317)

Decompose Acl as Acl = oplusNn=1Cl

n times Crn such that Cl

1 times Cr1 = ηcl(1 times 1) We use ι(n)cl-op

to denote the restriction of ιcl-op to Cln times Cr

n and ι(n)cl-op to denote the restriction of ιcl-op

to Cln otimes Cr

n We introduce the following graphical notation

ι(n)cl-op =

Cln Cr

n

Aop

(318)

By (243) ιcl-op can be expressed in terms of ιcl-op as follows

ιcl-op =Noplus

n=1

oplusiisinI

sumα

Cln

Crn

Aop

α

Uori

times α

Crn

Ui

(319)

Lemma 311 The centre condition (313) is equivalent to the following condition in C

Cln Cr

n

Aop

Aop

Aop

=

Cln Cr

n

Aop

Aop

Aop

for n = 1 N (320)

Proof First insert (319) and the definition (217) of m R(Aop) into (313) Then apply thecommutativity of R(1) to the left hand side of (313) The equivalence between (313)and (320) follows immediately Remark 312 The centre condition (320) is very natural from the open-closed confor-mal field theory point of view Correlators on the upper half plane are expressed in termsof conformal blocks on the full complex plane The objects Cl

n and Crn are associated to

the field insertion at a point z in the upper half plane and at the complex conjugate pointz in the lower half plane respectively The object Aop corresponds to a field inserted ata point r on the real axis The centre condition (320) simply says that the correlationfunctions in the disjoint domains |z| gt r gt 0 and r gt |z| gt 0 are analytic continuationsof each other see [K2 Prop 118]

898 L Kong I Runkel

Recall that we define ιlowastcl-op Aop rarr T (Acl) as in (230) We introduce the graphicalnotation

ιlowastcl-op =Noplus

n=1

Cln Cr

n

Aop

=Noplus

n=1

Cln Cr

n

Aop

Aop T (minus1Acl) (321)

where the second equality follows from (215) (221) and (259)

Lemma 313 The Cardy condition (314) is equivalent to the following identity in C

Noplusn=1

DimCdim Ui

sumα

Cln

Crn

Crn

α

α

Aop

Aop Uori

Uori

=Aop

Aop

Aop

Uori

Uori

op

mop for all i isin I (322)

Proof By (244) (312) and (319) it is easy to see that (322) is equivalent to thefollowing identity

ιcl-op χ(ιlowastcl-op) = PlR(Aop)

(323)

Therefore it is enough to show that

χ(ιlowastcl-op) = ιlowastcl-op (324)

We have

χminus1(ιlowastcl-op)(1)= T (ιlowastcl-op) δ (2)= T (ιcl-op)

lowast δlowastlowast (3)= (ρ T (ιcl-op)

)lowast (4)= (ιcl-op

)lowast (325)

In step (1) we use the expression (253) for χminus1 step (2) follows from Lemma 218 (v)and Lemma 219 Step (3) is Lemma 218 (i) and Lemma 228 and finally step (4)amounts to substituting (253) and (317) Acting with χ on both sides of the aboveequality produces (324)

Combining Lemmas 311 and 313 and Proposition 216 we obtain the followingequivalent definition of Cardy C|C2plusmn-algebra (recall the graphical notation (318) forιcl-op and (321) for ιlowastcl-op)

Cardy Algebras and Sewing Constraints I 899

Definition 314 (CardyC|C2plusmn-algebra II) A CardyC-algebra is a triple (Aop|Acl ιcl-op)where Acl is a commutative symmetric Frobenius C2plusmn-algebra satisfying property (34)with f = mcl Aop is a symmetric Frobenius C-algebra and ιcl-op T (Acl) rarr Aop isan algebra homomorphism satisfying the conditions (320) and (322)

Remark 315 Up to a choice of normalisation Definition 314 is the same as the orig-inal one in [K3 Def 613] The difference between the two definitions is the factorDim C dim Ui on the left hand side of (322) which in [K3 Def 613] is given byradic

Dim C dim Ui The two definitions are related by rescaling the coproduct cl andcounit εcl of Acl by 1

radicDim C and

radicDim C respectively We chose the convention in

(322) to remove all dimension factors from the expression (314) for the Cardy condition

33 Uniqueness and existence theorems In this subsection we investigate the structureof Cardy algebras We start with the following proposition which when combined withthe results of part II provides an alternative proof of [Fj Prop 422]

Proposition 316 Let (Aop|Acl ιcl-op) be a Cardy C|C2plusmn-algebra If Acl is simple anddim Aop = 0 then Aop is simple and special

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis therefore haploid Restricting the Cardy condition (322) to the case Ui = 1 andcomposing both sides with εop from the left we see that εop kills all terms associated toU j times 1 isin Acl in the sum except for a single 1 times 1 term Thus we obtain the followingidentity

β εop = dAop (mop otimes idAorop) (idAop otimes bAop) (326)

where β isin C Composing with ηop from the right in turn implies that βεop ηop =dim Aop which is nonzero by assumption Thus also β = 0 and εop is a nonzero multipleof the morphism on the right hand side of (326) By [FRS Lem 311] Aop is special

Since Aop is a special Frobenius algebra Aop is semi-simple as an Aop-bimodule(apply [FS Prop 524] to Aop tensored with its opposite algebra) Suppose Aop is not

simple so that we can write Aop = A(1)op oplus A(2)op for nonzero Aop-bimodules A(1)op and A(2)op We denote the canonical embeddings and projections associated to this decompositionas ι12 and π12 We have the identities

mop (ι1 otimes ι2) = 0 εop ηop =2sum

i=1

εop ιi πi ηop (327)

The first identity follows since π1 mop (ι1 otimes ι2) = 0 (as mop gives the left action of

Aop on Aop and hence it preserves A(2)op ) and similarly π2 mop (ι1 otimes ι2) = 0 Thesecond identity is just the completeness of ι12 π12

Since εop ηop = 0 without losing generality we can assume εop ι1 π1 ηop = 0Using that π2 is a bimodule map we compute

π2 [LHS of (322)]Ui =1 ι1 π1 ηop = π2 [RHS of (322)]Ui =1 ι1 π1 ηop

= PlA(2)op

π2 ι1 π1 ηop = 0 (328)

900 L Kong I Runkel

On the other hand using that Acl is haploid that ιcl-op is an algebra map and thatι lowastcl-op is a coalgebra map one can check that the left-hand side of (328) is equal toλ(εop ι1 π1 ηop) π2 ηop for some λ = 0 This implies that π2 ηop = 0 Thusηop = sum

i ιi πi ηop = ι1 π1 ηop Hence we have

0 = π2 ι2 = π2 mop (ηop otimes ι2) = π2 mop ((ι1 π1 ηop)otimes ι2) (329)

However the right-hand side is zero by (327) This is a contradiction and hence Aopmust be simple

To formulate the next theorem we need the notion of the full centre of an algebra [FjDef 49] Recall that an algebra A in a braided tensor category has a left centre and aright centre [VZO] both of which are sub-algebras of A Of these two we will onlyneed the left centre The following definition is [Fr Def 231] which in our setting isequivalent to that of [VZO]

Definition 317 Let A be a symmetric special Frobenius algebra such that m A A =ζA idA

(i) The left centre Cl(A) of A is the image of the idempotent ζminus1A Pl

A(ii) The full centre Z(A) is Cl(R(A))

That ζminus1A Pl

A is an idempotent follows from [FRS Lem 52] when keeping track ofthe factors ζA ([FRS] assumes normalised-special ie ζA = 1) Note that Cl(A) is againan object of C while Z(A) is an object of C2plusmn Let el

A Cl(A) rarr A be the embeddingof Cl(A) into A The left centre is in fact the maximal subobject of A such that

m A cAA (elA otimes idA) = m A (330)

see [Fr Lem 232] This observation explains the name left centre and also makes theconnection to [O Def 15]

The full centre is by definition the image of the idempotent ζminus1A Pl

R(A) R(A) rarrR(A) Since C2plusmn is abelian the idempotent splits and we obtain the embedding andrestriction morphisms

e Z(A) rarr R(A) and r R(A) Z(A) (331)

which obey r e = idZ(A) and e r = ζminus1A Pl

R(A) It follows from Proposition 225 and

[Fr Prop 237] that Z(A) is a commutative symmetric Frobenius algebra in C2plusmn withstructure morphisms5

m Z(A) = r m R(A) (e otimes e) ηZ(A) = r ηR(A)

Z(A) = ζA middot (r otimes r) R(A) e εZ(A) = ζminus1A middot εR(A) e

(332)

Moreover if A is simple then Z(A) is simple and if A is simple and dim A = 0 thenZ(A) is simple and special The normalisation of the counit is such that

εZ(A) ηZ(A) = ζminus2A dim A DimC (333)

5 The normalisation of product and unit is the standard one The factors in the coproduct and counit haveto be included in order for (A|Z(A) e) to be a Cardy algebra see Theorem 318 below The normalisation ofthe counit enters the Cardy condition (314) through the definition of ( middot )lowast

Cardy Algebras and Sewing Constraints I 901

Theorem 318 Let A be a special symmetric Frobenius C-algebra Then (A|Z(A) e)is a Cardy C|C2plusmn-algebra

The proof of this theorem makes use the following two lemmas

Lemma 319 e Z(A) rarr R(A) is an algebra map and elowast = ζA middot r

Proof It follows from [Fr Lem 229] (or by direct calculation using in particularm R(A) R(A) = ζAidR(A)) that

m R(A) (e otimes e) = ζminus1A middot Pl

R(A) m R(A) (e otimes e) (334)

Substituting e r = ζminus1A Pl

R(A) shows that e is compatible with multiplication For theunit one finds

e ηZ(A) = e r ηR(A) = ζminus1A Pl

R(A) ηR(A) = ηR(A) (335)

Thus e is an algebra map For the second statement one computes

elowast (1)= ζA

R(A)

r

R(A)

R(A)

Z(A)

R(A)

Z(A)

r

e

e

(2)=

R(A)

r

R(A)

R(A)

R(A)

Z(A)

(3)=

R(A)

r

R(A)

Z(A)

(4)= ζA middot r (336)

where in (1) the definitions (230) and (332) have been substituted step (2) is e r = ζminus1

A PlR(A) step (3) uses that R(A) is symmetric Frobenius and step (4) is again

e r = ζminus1A Pl

R(A) Lemma 320 Let A be a symmetric Frobenius algebra in C The morphism

PlR(A) m R(A)

(Pl

R(A) otimes PlR(A)

) R(A)otimes R(A) minusrarr R(A) (337)

is S-invariant

The proof of this lemma is a slightly lengthy explicit calculation and has been deferredto Appendix A2

902 L Kong I Runkel

Proof of Theorem 318 That e is an algebra map was proved in Lemma 319 The centrecondition (313) holds by property (330) of the left centre The Cardy condition (314)also is an immediate consequence of Lemma 319

ιcl-op ιlowastcl-op = e (ζA r) = PlR(A) (338)

The full centre Z(A) is a commutative symmetric Frobenius algebra It remains to provemodular invariance That θZ(A) = idZ(A) is implied by commutativity and symmetryof Z(A) [Fr Prop 225] The S-invariance condition (33) follows from Lemma 320In (337) substitute Pl

R(A) = ζA e r and then put the resulting morphism into (33)Compose the resulting equation with e otimes idW from the right (ie from the bottom) andsubstitute the definition (332) of m Z(A) This results in the statement that m Z(A) isS-invariant

The following theorem is analogous to [LR Prop 29] and [Fj Thm 426] whichroughly speaking answer the question under which circumstances the restriction of atwo-dimensional conformal field theory to the boundary already determines the entireconformal field theory The first work is set in Minkowski space and uses operator alge-bras and subfactors while the second work is set in Euclidean space and uses modulartensor categories

Theorem 321 Let (A|Acl ιcl-op) be a Cardy C|C2plusmn-algebra such that dim A = 0 andAcl is simple Then A is special and (A|Acl ιcl-op) sim= (A|Z(A) e) as Cardy algebras

Proof By Proposition 316 A is simple and special Since Acl is simple the algebramap ιcl-op Acl rarr R(A) is either zero or a monomorphism But ιcl-op ηcl = ηR(A)and so ιcl-op = 0 Thus ιcl-op is monic By Lemma 218 (ii) ζminus1

A ιlowastcl-op is epi The Cardycondition (314) implies

ιcl-op ζminus1A ιlowastcl-op = ζminus1

A PlR(A) = e r (339)

Composing this with e r from the left yields e r ιcl-op ζminus1A ιlowastcl-op = e r =

ιcl-op ζminus1A ιlowastcl-op Since ζminus1

A ιlowastcl-op is epi we have

e r ιcl-op = ιcl-op (340)

Actually (340) also follows from (313) and specialness of R(A) We will prove that

( fop fcl) (A|Acl ιcl-op) minusrarr (A|Z(A) e) where fop = idA fcl = r ιcl-op

(341)

is an isomorphism of Cardy algebrasfcl is an algebra map Compatibility with the units follows since ιcl-op is an algebramap

fcl ηcl = r ιcl-op ηcl = r ηR(A) = ηZ(A) (342)

Compatibility with the multiplication also follows since ιcl-op is an algebra map

m Z(A) ( fcl otimes fcl) = r m R(A) (e otimes e) (r otimes r) (ιcl-op otimes ιcl-op)

= r m R(A) (ιcl-op otimes ιcl-op) = r ιcl-op mcl = fcl mcl (343)

where in the second step we used (340)

Cardy Algebras and Sewing Constraints I 903

fcl is an isomorphism As above since fcl is an algebra map and since Acl is simple fcl

has to be monic By Lemma 319 rlowast = ζminus1A e Thus f lowast

cl = ιlowastcl-op rlowast = ζminus1A ιlowastcl-op e and

fcl f lowastcl = r ιcl-op ζminus1

A ιlowastcl-op e = r e r e = idZ(A) (344)

and so fcl is also epi and hence isofcl is a coalgebra map Since fcl is an algebra map so is f minus1

cl By (344) f minus1cl = f lowast

cland by Lemma 218 (iii) this implies that fcl is a also coalgebra mapThe diagram (315) commutes Commutativity of (315) is equivalent to e fcl = ιcl-opwhich holds by (340)

Let A be a special symmetric Frobenius algebra So far we have seen that (A|Z(A) e)is a Cardy algebra and that all Cardy algebras with Aop = A and simple Acl are of thisform It is now natural to ask if every simple Acl does occur as part of a Cardy algebraThe following theorem provides an affirmative answer Recall that for an A-left moduleM the object Mor otimesA M is an algebra (see eg [KR Lem 42])

Theorem 322 If Acl is a simple modular invariant commutative symmetric FrobeniusC2plusmn-algebra then there exist a simple special symmetric Frobenius C-algebra A and amorphism ιcl-op Acl rarr R(A) such that

(i) Acl sim= Z(A) as Frobenius algebras(ii) (A|Acl ιcl-op) is a Cardy C|C2plusmn-algebra

(iii) T (Acl) sim= oplusκisinJ Morκ otimesA Mκ as algebras where MκκisinJ is a set of representa-

tives of the isomorphism classes of simple A-left modules

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis haploid It then follows from Theorem 34 that Acl is special By Proposition 224T (Acl) is a special symmetric Frobenius algebra in C Thus T (Acl) = oplusi Ai where theAi are simple symmetric Frobenius algebras We will show that at least one of the Aiis special Since T (Acl) is special we have mT (Acl) T (Acl) = ζ idT (Acl) for someζ isin C

times Restricting this to the summand Ai shows mi i = ζ idAi FurthermoreεT (Acl) ηT (Acl) = ξ id1 for some ξ isin C

times But εT (Acl) ηT (Acl) = sumi εi ηi and so

at least one of the εi ηi has to be nonzero Therefore at least one of the Ai is speciallet A equiv Ai be this summand We denote the embedding A rarr T (Acl) by e0 and therestriction T (Acl) A by r0 Notice that r0 is an algebra homomorphism Define

ιcl-op = χ (r0) Acl minusrarr R(A) (345)

By Proposition 216 ιcl-op is an algebra homomorphism Next we verify the centre con-dition (313) or rather its equivalent form (320) By substituting the definitions onecan convince oneself that the commutativity mcl cAclAcl = mcl of Acl in C2plusmn implies thecondition mT (Acl) = mT (Acl) in C see [K2 Prop 36] Here T (Acl)otimes T (Acl) rarrT (Acl)otimes T (Acl) is given by

=oplusmn

(idCl

notimes cCl

m Crnotimes idCr

m

)

(cCl

m Clnotimes cminus1

Crn C

rm

)

(idCl

motimes cminus1

Cln C

rm

otimes idCrn

)

(346)

and we decomposed Acl as Acl = oplusnCln times Cr

n As a consequence we obtain the identity

r0 mT (Acl) (idAcl otimes e0) = r0 mT (Acl) (idAcl otimes e0) (347)

904 L Kong I Runkel

Using that r0 is an algebra map and that by definition ιcl-op = r0 we obtain (320)In order to show that (A|Acl ιcl-op) is a Cardy algebra it remains to show that the

Cardy condition (314) is satisfied We will demonstrate this via a detour by first provingthat Acl sim= Z(A) as Frobenius algebras

Recall the notations e and r given in (331) Using the centre condition (320) onecan check that Pl

R(A) ιcl-op = m R(A) R(A) ιcl-op By specialness of A we have

m A A = ζA idA and so together with e r = ζminus1A Pl

R(A) we get

e r ιcl-op = ιcl-op (348)

Next consider the morphism

fcl = r ιcl-op Acl minusrarr Z(A) (349)

By the same derivation as in (342) and (343) one sees that fcl is an algebra map Inparticular fcl ηcl = ηZ(A) = 0 and so fcl = 0 Since Acl is simple fcl has to be amonomorphism

By the same argument as used in the proof of Theorem 34 (ii) up to isomorphismAcl is the unique simple local Acl-(left-)module The algebra monomorphism fcl turnsZ(A) into an Acl-module Since Z(A) is commutative it is local as an Acl-module andso Z(A) sim= AoplusN

cl for some N ge 1 By construction A is a simple special symmetricFrobenius algebra Proposition 225 and Corollary 227 show that R(A) inherits all theseproperties and thus Z(A) is simple (see the comment below Eq (332)) By Theorem318 Z(A) is modular invariant and then by Theorem 34 (i) dim Z(A) = Dim C Thisimplies that N = 1 in Z(A) sim= AoplusN

cl and so fcl is in fact an isomorphismSince Acl and Z(A) are both haploid we have εZ(A) fcl = ξ εcl for some ξ isin C

timesThe counit uniquely determines the Frobenius structure on Acl and Z(A) (see eg [FRSLemma 37]) so that fcl is a coalgebra isomorphism iff ξ = 1 To compute ξ we composethe above identity with ηcl from the right Defining ζcl via εcl ηcl = ζminus1

cl Dim C middot id1 andusing (333) gives ξ = dim A ζclζ

2A By rescaling the comultiplication and the counit

of A and consequently changing ζA we can always achieve ξ = 1 This proves part (i)of the theorem

Equation (348) implies that ιcl-op = e fcl Since fcl is an isomorphism of Frobeniusalgebras by Lemmas 218 and 319 we have

ιcl-op ιlowastcl-op = e fcl f lowastcl elowast = ζA e r = Pl

R(A) (350)

Thus (A|Acl ιcl-op) is a Cardy algebra This proves part (ii) of the theoremPart (iii) can be seen as follows By [KR Prop 43] T Z(A) sim= oplusκisinJ Mor

κ otimesA Mκ asalgebras Together with the observation that T ( fcl) T (Acl) rarr T Z(A) is an isomor-phism of algebras this proves part (iii) Remark 323 Part (i) of Theorem 322 was announced by Muumlger [Muuml2] We provide anindependent proof in the setting of Cardy algebras

The above theorem together with Lemma 211 and Theorem 34 shows that a sim-ple commutative symmetric Frobenius C2plusmn-algebra Acl with dim Acl = Dim C is alwayspart of a Cardy algebra (Aop|Acl ιcl-op) for some simple special symmetric Frobeniusalgebra Aop in C However the above proof also illustrates that Aop is not unique Thisraises the question how two Cardy algebras with a given Acl can differ This question isanswered by [KR Thm 11] which in the present framework can be restated as follows

Cardy Algebras and Sewing Constraints I 905

Theorem 324 If (A(i)op |A(i)cl ι(i)cl-op) i = 1 2 are two Cardy C|C2plusmn-algebras such that

A(i)cl is simple and dim A(i)op = 0 for i = 1 2 then A(1)clsim= A(2)cl as algebras if and only

if A(1)op and A(2)op are Morita equivalent

Proof Theorem 11 in [KR] is stated for A(i)op being non-degenerate algebras and A(i)cl =Z(A(i)op ) for i = 1 2 By Proposition 316 A(i)op are simple and special for i = 1 2

Then by [KR Lem 21] A(i)op are non-degenerate algebras By Theorem 321 we have

A(i)clsim= Z(A(i)op ) as Frobenius algebras Finally by [KR Thm 11] Z(A(1)op ) sim= Z(A(2)op )

as algebras iff A(1)op and A(2)op are Morita equivalent Let Cmax(C2plusmn) be the set of equivalence classes [B] of simple modular invariant

commutative symmetric Frobenius algebras B in C2plusmn Two such algebras B and B prime areequivalent if B and B prime are isomorphic as algebras (but not necessarily as Frobenius alge-bras) Let Msimp(C) be the set of Morita classes of simple special symmetric Frobeniusalgebras in C Define the map z Msimp(C) rarr Cmax(C2plusmn) by z A rarr [Z(A)] whereA denotes the Morita class of A From Theorem 322 (i) and [KR Thm 11] we learn

Corollary 325 The map z Msimp(C) rarr Cmax(C2plusmn) is a bijection

A Appendix

A1 Proof of Lemma 27 We will show that if (F ψ F2 ψ

F0 ) is a colax tensor functor

from C1 to C2 then (G φG2 φ

G0 ) is a lax tensor functor from C2 to C1 Applying this

result to the opposed categories then gives the converse statementWe need to show that φG

0 and φG2 make the diagrams (21) and (22) commute We

first prove the commutativity of (21) Consider the following diagram

G(A)otimes (G(B)otimes G(C))

GψF2 δ

idG(A)otimesφG2 G(A)otimes G(B otimes C)

GψF2 δ

G(FG(A)otimes F(G(B)otimes G(C)))

G(F(idG(A))otimesF(φG2 ))

G(idFG(A)otimesψ F2 )

G(FG(A)otimes FG(B otimes C))

G(ρAotimesρBotimesC )

G(FG(A)otimes (FG(B)otimes FG(C)))

G(ρAotimes(ρBotimesρC )) G(A otimes (B otimes C))

(A1)

The top subdiagram is commutative because of the naturality of Gψ F2 δ The commu-

tativity of the bottom subdiagram follows from the following identities

(ρB otimes ρC ) ψ F2 = (ρB otimes ρC ) ψ F

2 ρF Fδ

= ρBotimesC FG(ρB otimes ρC ) FG(ψ F2 ) Fδ

= ρBotimesC F(φG2 ) (A2)

as a map F(G(B) otimes G(C)) rarr B otimes C The commutativity of (A1) implies that thecomposition of maps in the left column in (21) can be replaced by

G(ρA otimes (ρB otimes ρC )) G((idFG(A) otimes ψ F

2 ) ψ F2

) δ (A3)

906 L Kong I Runkel

Similarly we can show that the composition of maps in the right column in (21) can bereplaced by

G((ρA otimes ρB)otimes ρC ) G((ψ F

2 otimes idFG(C)) ψ F2

) δ (A4)

Using the commutativity of (23) it is easy to see that (21) with the left and right col-umns of (21) replaced by (A3) and (A4) respectively is commutative Hence (21) iscommutative

Now we prove the commutativity of the first diagram in (22)

φG2 (φG

0 otimes idG(A))

(1)= G(ρ12 otimes ρA) Gψ F2 δ

[(Gψ F

0 δ11)otimes idG(A)

]

(2)= G(ρ12 otimes ρA) Gψ F2 G F(Gψ0 otimes idG(A)) G F(δ11 otimes idG(A)) δ

(3)= G(ρ12 otimes ρA) G(FG(ψ F0 )otimes idFG(A)) Gψ F

2 G F(δ11 otimes idG(A)) δ(4)= G(id12 otimes ρA) G

([ψ F

0 ρF(11) (Fδ)11] otimes idFG(A)

) Gψ F

2 δ(5)= G(id12 otimes ρA) G(ψ F

0 otimes idFG(A)) Gψ F2 δ

(6)= G(id12 otimes ρA) G(lminus1FG(A)) G F(lG(A)) δ

(7)= G(lminus1A ) GρA δG lG(A)

(8)= G(lminus1A ) lG(A) (A5)

where in step (1) we substituted the definition of φG0 φ

G2 given in (210) in step (2)

we used the naturality of δ in step (3) we used the naturality of Gψ F2 in step (4) we

switched the position between Gρ12 and G FG(ψ F0 ) and the position between Gψ2 and

G F(δ12 otimes idG(A)) using the naturality of ρ and FψG2 respectively in step (5) we applied

the second identity in (28) in step (6) we used (24) in step (7) we used the naturalityof lminus1 and δ in step (8) we used the first identity in (28)

The proof of the commutativity of the second diagram in (22) is similar Thus wehave shown that G is a lax tensor functor

A2 Proof of Lemma 320 To prepare the proof recall that for a given object B isin Cthe modular group P SL(2Z) acts on the space oplusi HomC(B otimes Ui Ui ) see eg [BKSect 31] and [K3 Eq (455)] We will only need the action of S and Sminus1 Let f isinoplusi HomC(B otimes Ui Ui ) Then

S oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A6)

Cardy Algebras and Sewing Constraints I 907

Sminus1 oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A7)

By Lemma 32 to establish that (337) is S-invariant it is enough to prove the identity(34) when f is given by (337) Using (A6) and (A7) we can see that Eq (34) simplysays that oplusi j [RHS of (34)] is invariant under the action of S times S Consider the elementg of oplus jkisinIHomC2plusmn(R(A)otimes (Uor

j times Uk)Uorj times Uk) given by

g =oplusjkisinI

sumα

R(A)

R(A)

R(A)

R(A)

Uorj times Uk

Uorj times Uk

α

α

(A8)

By the above arguments proving S-invariance of (337) is equivalent to proving invari-ance of g under the action of S times S

For i isin I we denote by gi the component of g in(oplus jisinIHomC(A otimes Uor

i otimes Uorj U

orj )

)otimes (opluskisinIHomC(Ui otimes UkUk))

We view the second Hom-space in the above tensor product as a Hom-space in C+ insteadof Cminus It is enough to show that gi is invariant under the action of S times Sminus1 Note thatthe action of Sminus1 in C+ is equivalent to that of S in Cminus

The morphism gi can be canonically identified with a bilinear pairing

( middot middot )i ⎛⎝oplus

jisinIHomC(Uor

j A otimes Uori otimes Uor

j )

⎞⎠

times(oplus

kisinIHomC(UkUi otimes Uk)

)minusrarr C (A9)

as follows For h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) we set

(h1 h2)i = (dim U j dim Uk)minus1 trUor

j timesUk[gi (h1 times h2)] (A10)

908 L Kong I Runkel

When substituting the explicit form of the product m R(A) of R(A) = (A times 1) otimes R(1)after a short calculation one finds

(h1 h2)i =sumα

1

dim U j

A

A

A

A

A A

A α

α

h1

h2

U jUk

Uorj

Ui

Uk (A11)

Here the top morphism PlR(A) has been simplified with the help of the identity

PR(A) m R(A) (PR(A) otimes PR(A))

=(oplus

iisinI((m A A)otimes idUor

i)times idUi

) m R(A) (PR(A) otimes PR(A)) (A12)

which can be checked by direct calculation along the same lines as in the proof of [FrLem 310]

The action of the modular transformation S on oplusiisinI(B otimes Ui Ui ) for B isin C nat-urally induces an action on oplusiisinI(Ui B otimes Ui ) [K3 Prop 514] which we denote bySlowast In the present case we get an action of Slowast on oplus jisinIHomC(Uor

j A otimes Uori otimes Uor

j ) and

opluskisinIHomC(UkUi otimes Uk) Then to show gi is invariant under the action of S times Sminus1

amounts to showing that

(h1 h2)i = ((Sminus1)lowast h1 Slowast h2)i (A13)

Cardy Algebras and Sewing Constraints I 909

for all h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) We have

((Sminus1)lowast h1 Slowast h2)i =sum

mnα

dim Un

DimC

A

A

A

A

A A

A

Uk

α

α

h1

h2

Ui

U j

UmUn

(A14)

Now drag the upper vertex indexed by α in the above graph along its Uorm -leg until it

meets the lower vertex also indexed by α then sum over α and m This gives

((Sminus1)lowast h1 Slowast h2)i =sum

n

dim Un

DimC

A

A

A

A

A A

A

Uk

Un

h1

h2

Ui

U j

(A15)

910 L Kong I Runkel

If we just look at the neighbourhood of the Un-loop in the above graph we see thefollowing subgraphs

sumn

dim Un

DimC

A Aor

Uorj U j

Uk Uork

Un=

sumα

1

dim U j

A Aor Uk Uork

Uorj U j

αα

(A16)

where we have applied [BK Cor 3111] Substituting this subgraph back to the originalgraph in (A15) we obtain

((Sminus1)lowast h1 Slowast h2)i =sumα

1

dim U j

A

A

A

A

A A

A

A

A

Uorj

α

α

h1

h2m A

Ui

U jUk

Uk

(A17)

The graph in (A17) is equal to that in (A11) In order to see this we first drag theldquobubblerdquo (m A A) along A lines and through the m A vertex (because m A A is abimodule map) until it reaches the lower-left leg of the upper vertex indexed by α Thendrag the m A vertex along the (red) dotted line in the above graph Finally we applythe associativity of A (A12) and [Fr Lem 311] Then we see that the graph in (A17)exactly matches with the one in (A11)

Acknowledgements We would like to thank the organisers of the Oberwolfach Arbeitsgemeinschaft ldquoAlge-braic structures in conformal field theoriesrdquo (April 2007) where this work was started for an inspiring meetingWe would further like to thank the Hausdorff Institute for Mathematics in Bonn and the organisers of the stim-ulating meeting ldquoGeometry and Physicsrdquo (May 2008) We are indebted to Alexei Davydov Jens FjelstadJuumlrgen Fuchs Yi-Zhi Huang Alexei Kitaev Urs Schreiber Christoph Schweigert Stephan Stolz and PeterTeichner for helpful discussions andor comments on a draft of this paper The research of IR was partiallysupported by the EPSRC First Grant EPE0050471 the PPARC rolling grant PPC5071451 and the MarieCurie network lsquoSuperstring Theoryrsquo (MRTN-CT-2004-512194)

Cardy Algebras and Sewing Constraints I 911

References

[AN] Alexeevski A Natanzon SM Noncommutative two-dimensional topological field theories andhurwitz numbers for real algebraic curves Sel Math New Ser 12 307ndash377 (2006)

[Bi] Bichon J Cosovereign hopf algebras J Pure Appl Alg 157 121ndash133 (2001)[BK] Bakalov B Kirillov AA Lectures on Tensor Categories and Modular Functors Providence

RI Amer Math Soc 2001[C1] Cardy JL Operator content of two-dimensional conformal invariant theories Nucl Phys

B 270 186ndash204 (1986)[C2] Cardy JL Boundary conditions fusion rules and the verlinde formula Nucl Phys B 324

581ndash596 (1989)[CL] Cardy JL Lewellen DC Bulk and boundary operators in conformal field theory Phys Lett

B 259 274ndash278 (1991)[DMZ] Dong C-Y Mason G Zhu Y-C Discrete series of the Virasoro algebra and the moonshine

module In Algebraic Groups and Their Generalizations Quantum and infinite-dimensional Meth-ods Proc Symp Pure Math 56 Part 2 Providence RI Amer MathSoc 1994 pp 295ndash316

[DP] Day B Pastro C Note on frobenius monoidal functors New York J Math 14 733ndash742 (2008)[ENO1] Etingof PI Nikshych D Ostrik V On fusion categories Ann Math 162 581ndash642 (2005)[ENO2] Etingof PI Nikshych D Ostrik V An analogue of radfordrsquos s4 formula for finite tensor

categories Int Math Research Notices 54 2915ndash2933 (2004)[Fe] Felder G Froumlhlich J Fuchs J Schweigert C Correlation functions and boundary conditions in

rational conformal field theory and three-dimensional topology Comp Math 131 189ndash237 (2002)[FHL] Frenkel IB Huang Y-Z Lepowsky J On axiomatic approaches to vertex operator algebras

and modules Mem Amer Math Soc 104 (1993)[Fj] Fjelstad J Fuchs J Runkel I Schweigert C Uniqueness of openclosed rational cft with

given algebra of open states Adv Theor Math Phys 12 1283ndash1375 (2008)[FK] Froumlhlich J King C The chern-simons theory and knot polynomials Commun Math

Phys 126 167ndash199 (1989)[Fr] Froumlhlich J Fuchs J Runkel I Schweigert C Correspondences of ribbon categories Adv

Math 199 192ndash329 (2006)[FRS] Fuchs J Runkel I Schweigert C Tft construction of rcft correlators I partition functions Nucl

Phys B 646 353ndash497 (2002)[FS] Fuchs J Schweigert C Category theory for conformal boundary conditions Fields Inst Com-

mun 39 25ndash71 (2003)[H1] Huang Y-Z Two-dimensional conformal geometry and vertex operator algebras Progress in

Mathematics Vol 148 Boston Birkhaumluser 1997[H2] Huang Y-Z Rigidity and modularity of vertex tensor categories Comm Contemp Math 10

871ndash911 (2008)[HK1] Huang Y-Z Kong L Open-string vertex algebra category and operads Commun Math

Phys 250 433ndash471 (2004)[HK2] Huang Y-Z Kong L Full field algebras Commun Math Phys 272 345ndash396 (2007)[HK3] Huang Y-Z Kong L Modular invariance for conformal full field algebras httparxivorgabs

math0609570v2[mathQA] 2006[HL] Huang Y-Z Lepowsky J Tensor products of modules for a vertex operator algebra and vertex

tensor categories In Lie Theory and Geometry in honor of Bertram Kostant ed R BrylinskiJ-L Brylinski V Guillemin V Kac Boston Birkhaumluser 1994 pp 349ndash383

[JS] Joyal A Street R Braided tensor categories Adv Math 102 20ndash78 (1993)[K1] Kong L Full field algebras operads and tensor categories Adv Math 213 271ndash340 (2007)[K2] Kong L Open-closed field algebras Commun Math Phys 280 207ndash261 (2008)[K3] Kong L Cardy condition for open-closed field algebras Commun Math Phys 283 25ndash92 (2008)[Ki] Kitaev A Private communication[KO] Kirillov AA Ostrik V On q-analog of mckay correspondence and ade classification of sl(2)

conformal field theories Adv Math 171 183ndash227 (2002)[KR] Kong L Runkel I Morita classes of algebras in modular tensor categories Adv Math 219

1548ndash1576 (2008)[Ld] Lauda AD Frobenius algebras and ambidextrous adjunctions Theo Appl Cat 16 84ndash

122 (2006)[Li1] Li H-S Regular representations of vertex operator algebras Commun Contemp Math 4

639ndash683 (2002)[Li2] Li H-S Regular representations and huang-lepowsky tensor functors for vertex operator

algebras J Alge 255 423ndash462 (2002)

884 L Kong I Runkel

In the middle line of the last expression we can use the defining property (25) of F namely we substitute (φF

2 otimes idF(A)) (idF(B) otimes ψ F2 ) = ψ F

2 φF2 Then ψ F

2 can bemoved to the left and φF

2 to the right until they can be omitted against ψ F0 and φF

0 respectively using (22) and (24) This results in

F( f )lowast = F ((εB m B (idB otimes f ))otimes idA) F (idB otimes (A ηA)) (232)

which is nothing but F( f lowast)

23 Modular tensor categories Let C be a modular tensor category [TBK] ie anabelian semi-simple finite C-linear ribbon category with simple tensor unit 1 and a non-degeneracy condition on the braiding (to be stated in a moment) We denote the set ofequivalence classes of simple objects in C by I elements in I by i j k isin I and theirrepresentatives by Ui U j Uk We also set U0 = 1 and for an index k isin I we define kby Uk

sim= Uork

Since the tensor unit is simple we shall for modular tensor categories identifyHom(1 1) sim= C (cf footnote 2) Define numbers si j isin C by4

si j = UiU j (233)

They obey si j = s ji and s0i = dim Ui see eg [BK Sect 31] (In a ribbon cate-gory the left and right dimension (214) of Ui coincide and are denoted by dim Ui )The non-degeneracy condition on the braiding of a modular tensor category is that the|I|times|I|-matrix s should be invertible In fact [BK Thm 317]

sumkisinI

sik sk j = Dim C δij (234)

where Dim C = sumiisinI(dim Ui )

2 One can show (even in the weaker context of fusioncategories over C) that Dim C ge 1 [ENO1 Thm 23] In particular Dim C = 0 We fixonce and for all a square root

radicDim C of Dim C

Let us fix a basis λα(i j)kN k

i jα=1 in HomC(Ui otimesU j Uk) and the dual basis ϒ(i j)k

α N ki j

α=1

in HomC(UkUi otimesU j ) The duality of the bases means that λα(i j)k ϒ(i j)kβ = δαβ idUk

We also fix λ(0i)i = λ(i0)i = idUi We denote the basis vectors graphically as follows

λα(i j)k = α

Uk

Ui U j

ϒ(i j)kα = α

Uk

Ui U j

(235)

4 In the graphical notation used below we have given an orientation to the ribbons indicated by the arrowsFor example it is understood that this orientation determines which of the duality morphisms in (213) to use

Cardy Algebras and Sewing Constraints I 885

For V isin C we also choose a basis b(iα)V of HomC(VUi ) and the dual basis bV(iβ)

of HomC(Ui V ) for i isin I such that b(iα)V bV(iβ) = δαβ idUi We use the graphical

notation

b(iα)V = α

Ui

V

bV(iα) = α

Ui

V

(236)

Given two modular tensor categories C and D by C D we mean the tensor productof additive categories over C [BK Def 1115] ie the category whose objects are directsums of pairs V times W of objects V isin C and W isin D and whose morphism spaces are

HomCD(V times W V prime times W prime) = HomC(V V prime)otimesC HomC(WW prime) (237)

for pairs and direct sums of these if the objects are direct sums of pairsIf we replace the braiding and the twist in C by the antibraiding cminus1 and the antitwist

θminus1 respectively we obtain another ribbon category structure on C In order to distin-guish these two distinct structures we denote (C c θ) and (C cminus1 θminus1) by C+ and Cminusrespectively As in the Introduction we will abbreviate

C2plusmn = C+ Cminus (238)

Note that a set of representatives of the simple objects in C2plusmn is given by Ui times U j fori j isin I

For the remainder of Sect 2 we fix a modular tensor category C

24 The functors T and R The tensor product bifunctor otimes can be naturally extendedto a functor T C2plusmn rarr C Namely T (oplusN

i=1Vi times Wi ) = oplusNi=1Vi otimes Wi for all Vi Wi isin C

and N isin N The functor T becomes a tensor functor as follows For φT0 1 rarr T (1 times 1)

take φT0 = id1 (or lminus1

1 in the non-strict case) Next notice that for U VW X isin C

T (U times V )otimes T (W times X)= (U otimes V )otimes (W otimes X)

T ((U times V )otimes (W times X))= (U otimes W )otimes (V otimes X)(239)

We define φT2 T (U times V )otimes T (W times X) rarr T ((U times V )otimes (W times X)) by

φT2 = idU otimes cminus1

W V otimes idX (240)

(In the non-strict case the appropriate associators have to be added) The above definitionof φT

2 can be naturally extended to a morphism φT2 T (M1 otimes M2) rarr T (M1)otimes T (M2)

for any pair of objects M1M2 in C2plusmn The following result can be checked by directcalculation [JS Prop 52]

Lemma 220 The triple (T φT2 φ

T0 ) gives a tensor functor

886 L Kong I Runkel

In particular (T φT2 φ

T0 ψ

T2 ψ

T0 ) where ψT

2 = (φT2 )

minus1 and ψT0 = (φT

0 )minus1 gives

a Frobenius functorDefine the functor R C rarr C2plusmn as follows for A isin C and f isin HomC(A B)

R(A) =oplusiisinI

(A otimes Uori )times Ui R( f ) =

oplusiisinI

( f otimes idUori)times idUi (241)

This functor was also considered in a slightly different context in [ENO2 Prop 23]The family of isomorphisms γ R

A = oplusiisinI DimCdim Ui

id(AotimesUori )timesUi

isin Aut(R(A)) defines a

natural isomorphism γ R R rarr ROur next aim is to show that R is left and right adjoint to T in other words R

and T form an ambidextrous adjunction (see eg [Ld] for a discussion of ambidextrousadjunctions) To this end we introduce two linear isomorphisms for A isin C and M isin C2plusmn

χ HomC(T (M) A) minusrarr HomC2plusmn(M R(A))

χ HomC(A T (M)) minusrarr HomC2plusmn(R(A)M)(242)

If we decompose M as M = oplusNn=1 Ml

n times Mrn then χ and χ are given by

χ Noplus

n=1

fn

Mln Mr

n

A

rarrNoplus

n=1

oplusiisinI

sumα

fn

Mln

Mrn

A

α

Uori

times α

Mrn

Ui

(243)

and

χ Noplus

n=1 gn

Mln Mr

n

A

rarrNoplus

n=1

oplusiisinI

sumα gn

Mln

Mrn

A

α

Uori

times α

Ui

Mrn

DimCdim Ui

(244)

Notice that χ and χ are independent of the choice of basis

Theorem 221 〈T R χ〉 and 〈R T χminus1〉 are adjunctions ie R is both left and rightadjoint of T

Proof Write M as M = oplusNn=1 Ml

n times Mrn The isomorphism χ amounts to the following

composition of natural isomorphisms

HomC(T (M) A) = oplusnHomC(Mln otimes Mr

n A)sim= oplusni HomC(Ml

n otimes Ui A)otimes HomC(Mrn Ui )

sim= oplusni HomC(Mln A otimes Uor

i )otimes HomC(Mrn Ui ) = HomC2plusmn(M R(A)) (245)

Cardy Algebras and Sewing Constraints I 887

Thus χ is natural Let (γ RA )

lowast HomC2plusmn(R(A)M) rarr HomC2plusmn(R(A)M) denote the

pull-back of γ RA The isomorphism χ is equal to the composition of (γ R

A )lowast and the

following sequence of natural isomorphisms

HomC(A T (M)) = oplusn HomC(AMln otimes Mr

n)

sim= oplusni HomC(AMln otimes Ui )otimes HomC(Ui Mr

n)

sim= oplusni HomC(A otimes Uori Ml

n)otimes HomC(Ui Mrn) = HomC2plusmn(R(A)M) (246)

We have proved that both χ and χ are natural isomorphisms There are four natural transformations associated to χ and χ namely

idC2plusmnδminusrarr RT

ρminusrarr idC2plusmn and idCδminusrarr T R

ρminusrarr idC (247)

defined by for A isin C M isin C2plusmn

δM = χ(idT (M)) ρA = χminus1(idR(A))

ρM = χ (idT (M)) δA = χminus1(idR(A))(248)

They can be expressed graphically as follows with M = oplusNn=1 Ml

n times Mrn

δM =oplusni

sumα

Mln

Mln Mr

n Uori

αtimes

Ui

Mrn

α

ρA =oplusiisinI

A

A

Uori Ui

(249)

ρM =oplusni

sumα

Mln

Mln

Mrn Uor

i

α times

Ui

Mrn

α DimCdim Ui

δA =oplusiisinI

A

A Uori Ui

dim UiDimC

Note that

ρM δM = Dim C middot idM and ρA δA = idA (250)

Lemma 222 The functors T and R as maps on the sets of morphisms have left inversesand thus are injective

Proof Let f A rarr B be a morphism in C We define a map Q R HomC2plusmn(R(A)

R(B)) rarr HomC(A B) by f prime rarr ρB T ( f prime) δA Then we have

Q R R( f ) = ρB T R( f ) δA = ρB δB f = f (251)

888 L Kong I Runkel

where we used naturality of δ and (250) in the second and third equalities respectivelySo Q R is a left inverse of R on morphisms Thus R is injective on morphisms Similarlylet g M rarr N be a morphism in C2plusmn We define a map QT HomC(T (M) T (N )) rarrHomC2plusmn(M N ) by gprime rarr (Dim C)minus1 middot ρN R(gprime) δM Then we have

QT T (g) = (Dim C)minus1 middot ρN RT (g) δM = (Dim C)minus1 middot ρN δN g = g (252)

So QT is a left inverse of T on morphisms Thus T is injective on morphisms Using (29) and (250) one can express the two inverse maps χminus1 χminus1 as follows

for f isin HomC2plusmn(M R(A)) and g isin HomC2plusmn(R(A)M)

χminus1( f ) = ρ T ( f ) χminus1(g) = T (g) δ (253)

By Proposition 25 and Lemma 27 R is both a lax and colax tensor functor In particularφR

0 1 times 1 rarr R(1) is given by

φR0 = χ (ψT

0 ) = R(ψT0 ) δ1 times 1 = id1times1 (254)

and φR2 R(A)otimes R(B) rarr R(A otimes B) by φR

2 = R(ρA otimes ρB) R(ψT2 ) δ which can

be expressed graphically as

φR2 =

oplusi jkisinI

sumα

A

A

Uori B

B

Uorj

Uork

α

times

Ui U j

α

Uk

(255)

Similarly ψ R0 R(1) rarr 1 times 1 is given by

ψ R0 = ρ1 R(φT

0 ) = Dim C id1times1 (256)

and ψ R2 R(A otimes B) rarr R(A) otimes R(B) by ψ R

2 = ρ R(φT2 ) R(δA otimes δB) which in

graphical notation reads

ψ R2 =

oplusi jkisinI

sumα

A

A Uori

B

B Uorj

Uork

α

times

Ui U j

α

Uk

dim Ui dim U j

dim Uk Dim C

(257)

If C has more than one simple object then R does not take the tensor unit of C to thetensor unit of C2plusmn and so is clearly not a tensor functor However we will show that R isstill a Frobenius functor This will imply that if A is a Frobenius algebra in C then

R(A) = (R(A)m R(A) ηR(A)R(A) εR(A)) (258)

Cardy Algebras and Sewing Constraints I 889

is a Frobenius algebra in C2plusmn where the structure morphisms were given in (217) and(218) In the case A = 1 it was proved in [Muuml1 Prop 41] (see also [Fr Lem 619]and [K1 Thm 52]) that (258) is a commutative simple symmetric normalised-specialFrobenius algebra in C2plusmn In fact given a Frobenius algebra A in C it is straightforwardto verify that the structure morphisms in (258) are precisely those of (A times 1)otimes R(1)cf Sect 22

Proposition 223 (R φR2 φ

R0 ψ

R2 ψ

R0 ) is a Frobenius functor

Proof Using the explicit graphical expression of φR2 φ

R0 ψ

R2 ψ

R0 it is easy to see that

the commutativity of the diagrams (25) and (26) are equivalent to the statement thatR(1) with structure morphisms as in (258) is a Frobenius algebra in C2plusmn The latterstatement is true by [Muuml1 Prop 41]

From Lemma 220 and Proposition 223 we see that T and R take Frobenius alge-bras to Frobenius algebras The following two propositions show how the properties ofFrobenius algebras are transported

Proposition 224 Let A be a Frobenius algebra in C2plusmn Then T (A) is a Frobenius algebrain C and

(i) A is symmetric iff T (A) is symmetric(ii) A is (normalised-)special iff T (A) is (normalised-)special

Proof For part (i) write A as a direct sum oplusNn=1 Al

n times Arn Then the maps IT (A) I prime

T (A) T (Aor) rarr T (A)or defined in (220) are given by

IT (A) = I primeT (A) = oplusN

n=1c(Aln)

or(Arn)

or (259)

Therefore by (221) T (A) = primeT (A) is equivalent to T (A) = T (prime

A) Since byLemma 222 T is injective on morphisms this proves part (i) Part (ii) can be checkedin the same way for example the condition mT (A) T (A) = ζ idT (A) is easily checkedto be equivalent to T (m A A) = ζ T (idA) Proposition 225 Let A be a Frobenius algebra A in C Then R(A) is a Frobeniusalgebra in C2plusmn and

(i) A is symmetric iff R(A) is symmetric(ii) A is (normalised-)special iff R(A) is (normalised-)special

Proof Recall that the structure morphisms of the Frobenius algebra R(A) are equal tothose of (A times 1) otimes R(1) Using this equality part (i) and (ii) follow because R(1) issymmetric and normalised-special For example

m R(A) R(A) = [(m A A)times id1] otimes idR(1) = R(m A A) (260)

so that m R(A) R(A) = ζ idR(A) is equivalent to R(m A A) = ζ R(idA) which byLemma 222 is equivalent to m A A = ζ idA

The functor R has one additional property not shared by T namely R takes absolutelysimple algebras to absolutely simple algebras We will see explicitly in Sect 33 thatthis is not true for T

890 L Kong I Runkel

Lemma 226 For a C-algebra A the map

R HomA|A(A A) rarr HomR(A)|R(A)(R(A) R(A)) (261)

given by f rarr R( f ) is well-defined and an isomorphism

Proof Since R is a lax tensor functor R(A) is naturally a R(A)-bimodule It is easy tosee that R in (261) is a well-defined map R(A) is also naturally a R(1)-bimodule whichcan be identified with the induced R(1)-bimodule structure on (A times 1)otimes R(1) wherethe left R(1) action on (Atimes1)otimes R(1) is given by (idAtimes1 otimesm R(1))(cminus1

Atimes1R(1)otimes idR(1))We have the following natural isomorphisms

HomR(1)|R(1)(R(A) R(A))sim=minusrarr HomC2plusmn(A times 1 R(A))

χminus1

minusminusrarr HomC(A A) (262)

which by (253) are given by for f isin HomR(1)|R(1)(R(A) R(A))

f rarr f prime = f (idAtimes1 otimes ηR(1)) rarr f primeprime = ρ T ( f prime) (263)

and its inverse is given by for g isin HomC(A A)

g rarr gprime = R(g) δ rarr gprimeprime = (idAtimes1 otimes m R(1)) (gprime otimes idR(1)) (264)

where gprimeprime is indeed a R(1)-bimodule map due to the commutativity of R(1) It is easy tocheck that gprimeprime = R(g) in (264) Therefore R gives an isomorphism from HomC(A A)to HomR(1)|R(1)(R(A) R(A)) Moreover one verifies that R(g) is an R(A)-bimodulemap iff g is an A-bimodule map In other words R g rarr R(g) gives an isomorphism

HomA|A(A A)sim=minusrarr HomR(A)|R(A)(R(A) R(A))

Corollary 227 Let A be a C-algebra

(i) A is absolutely simple iff R(A) is absolutely simple(ii) Let A be in addition Frobenius Then A is simple and special iff R(A) is simple and

special

Proof Part (i) immediately follows from Lemma 226 The statement of part (ii) withoutthe qualifier lsquosimplersquo is proved in Proposition 225 But as in the proof of (iii)rArr(i) inLemma 211 a special Frobenius algebra in a semi-simple category has a semi-simplecategory of bimodules and for a semi-simple C-linear category simple and absolutelysimple are equivalent Part (ii) then follows from part (i)

The following lemma will be needed in Sect 32 below to discuss the properties ofCardy algebras

Lemma 228 Let A be a Frobenius algebra in C Then (δA)lowast = ρA

Proof Recall from (250) that δA is a morphism A rarr T R(A) Since T and R are bothFrobenius functors T R(A) is a Frobenius algebra in C Substituting the definitions aftera short calculation one finds

Cardy Algebras and Sewing Constraints I 891

εT R(A) mT R(A) = Dim(C)oplusiisinI

A Uori Ui A U or

ı Uı

(265)

Substituting this in the definition of (δA)lowast gives again after a short calculation the mor-

phism ρA At an intermediate step one uses that the part of the morphism (δA)lowast which

is made up of Ui and Uı ribbons their duals and the basis morphisms λ(iı)0 andϒ(iı)0can be replaced by 1

dim Uimiddot dUi

3 Cardy Algebras

In this section we start by investigating the properties of Frobenius algebras which sat-isfy the so-called modular invariance condition We then give two definitions of a Cardyalgebra and prove their equivalence Finally in Sect 33 we study the properties of thesealgebras and state our main results

We fix a modular tensor category C Recall that C2plusmn is an abbreviation for C+ Cminus

31 Modular invariance In C2plusmn we define the object K and the morphism ω K rarr Kas

K =oplus

i jisinIUi times U j ω =

sumi jisinI

dim Ui dim U j

Dim C idUi timesU j (31)

They have the property (see eg [BK Cor 3111])

ω

K

(Uk times Ul )or

(Uk times Ul )or

Ui times U j

Ui times U j

= δik δ jlDim C

dim Ui dim U jbUi timesU j dUi timesU j (32)

892 L Kong I Runkel

Definition 31

(i) Let A B be objects of C2plusmn A morphism f A otimes B rarr B is called S-invariant iff

A

B

W

W

f =

A

B

W

W

f

K

ω(33)

holds for all for W isin C2plusmn(ii) A C2plusmn-algebra (Aclmcl ηcl) is called modular invariant iff θAcl = idAcl and mcl

is S-invariant

Lemma 32 The morphism f A otimes B rarr B is S-invariant if and only if

A

B

Ui times U j

Ui times U j

f = Dim Cdim Ui dim U j

sumα

A

B

B

f

Ui times U j

Ui times U j

α

α

(34)

holds for all i j isin I

Proof Condition (33) holds for all W iff it holds for all W = Ui times U j i j isin I so itis enough to show that the right-hand side of (33) with W = Ui times U j is equal to theright-hand side of (34) Recall the notation for basis morphisms in (236) Starting from(33) write

idBor otimes idUi timesU j =⎛⎝sum

klα

(b(ktimeslα)

B

)or (

bB(ktimeslα)

)or⎞⎠ otimes idUi timesU j (35)

and then apply (32) The graphical representation of the resulting morphism can bedeformed to give (34) Remark 33 As shown in [K3 Sect 61] the modular invariance condition of a C2plusmn-alge-bra exactly coincides with the modular invariance condition for torus 1-point correlationfunctions of a genus-01 closed CFT In particular the condition θAcl = idAcl is equiv-alent to invariance under the modular transformation T τ rarr τ + 1 and the condition(34) with f = mcl is equivalent to invariance under S τ rarr minus 1

τ Combining the

modular invariance condition with the genus zero properties of a genus-01 closed CFTresults in a modular invariant commutative symmetric Frobenius algebra in C2plusmn

Cardy Algebras and Sewing Constraints I 893

Let Acl be a modular invariant C2plusmn-algebra Evaluating (34) for f = mcl composingit with ηcl otimes idUi timesU j and taking the trace implies the following identity

Zi j = 1

DimCAclUi timesU j where

(36)

Zi j = dimC HomC2plusmn(UitimesU j Acl)

Decomposing Acl into simple objects this gives

Zi j =sum

klisinISik Zkl Sminus1

l j where Si j = si jradic

DimC (37)

which in CFT terms is of course nothing but the invariance of the torus partition functionunder the modular S-transformation

The following theorem gives a simple criterion for modular invariance

Theorem 34 Let Acl be a haploid commutative symmetric Frobenius algebra in C2plusmn

(i) If Acl is modular invariant then dim Acl = Dim C(ii) If dim Acl = Dim C then Acl is special and modular invariant

Proof Part (i) Since Acl is haploid for i = j = 0 Equation (36) reduces to 1 =dim AclDimCPart (ii) By the same reasoning as in the proof of (iii)rArr(i) in Lemma 211 one showsthat Acl is special Thus mcl cl = ζclidAcl for some ζcl = 0

By [KO Thm 45] the category (C2plusmn)locAcl

of local Acl-modules is again a mod-

ular tensor category and Dim (C2plusmn)locAcl

= (Dim C dim Acl)2 (see [Fr Prop 321 amp

Rem 323] for the same statement in the notation used here) Thus by assumption wehave Dim(C2plusmn)loc

Acl= 1 It then follows from [ENO1 Thm 23] that up to isomorphism

(C2plusmn)locAcl

has a unique simple object (namely the tensor unit) In other words every simplelocal Acl-module is isomorphic to Acl (seen as a left-module over itself)

We have the following isomorphisms between morphism spaces [FS Prop 47 amp 411]

HomAcl(Acl otimes (Ui times U j ) Acl) sim= HomC2plusmn(Ui times U j Acl)

HomAcl(Acl Acl otimes (Ui times U j )) sim= HomC2plusmn(AclUi times U j )

Using these to transport the bases (236) from the right to the left we obtain bases bα(i j)αof HomAcl(Acl otimes (Ui times U j ) Acl) and b(i j)

β β of HomAcl(Acl Acl otimes (Ui times U j )) Thesecan be expressed graphically as

894 L Kong I Runkel

bα(i j) =

Acl

Acl

Acl

Ui times U j

α

b(i j)β = DimC

dim Ui dim U j

1

ζcl

Acl

Ui times U jAcl

Acl

β

(38)

where the nonzero factor in b(i j)β is included for convenience Notice that bα(i j) b(i j)

β isa left Acl-module map Since Acl is simple as a left module over itself we have

bα(i j) b(i j)β = λαβ idAcl (39)

for some λαβ isin C By computing tr(bα(i j) b(i j)β ) it is easy to verify that λαβ = δαβ

We will now prove the following identity

1

ζcl

Acl

Acl

Ui times U j

Ui times U j

Acl

=sumα

DimCdim Ui dim U j

1

ζcl

Acl

Ui times U j

Ui times U j

Acl

Acl

Acl

α

α

(310)

One checks that the left-hand side of this equation is an idempotent which we denote byζminus1

cl PlAcl(Ui timesU j ) cf [Fr Sect 31] By [Fr Prop 41] the image Im(ζminus1

cl PlAcl(Ui timesU j ))

is a local Acl-module and hence isomorphic to AoplusNcl for some N isin Zge0

All left-module morphisms from Acl to Aclotimes(Ui timesU j ) are linear combinations of the

b(i j)β Furthermore one verifies that ζminus1

cl PlAcl(Ui timesU j )b(i j)

β = b(i j)β Therefore the b(i j)

β

describe precisely the image of the idempotent ie ζminus1cl Pl

Acl(Ui timesU j ) = sum

α b(i j)α bα(i j)

which is nothing but (310) Composing (310) with ζcl middot εcl otimes idUi timesU j from the left (iefrom the top) produces (34)

In addition since Acl is commutative symmetric Frobenius it satisfies θAcl = idAcl

[Fr Prop 225] Altogether this shows that Acl is modular invariant Remark 35 As we were writing this paper we heard that the results in Theorem 34were obtained independently by Kitaev and Muumlger [Ki]

Remark 36 Setting i = j = 0 in (36) gives the identity dim Acl = Z00 Dim C [KRProp 23] Combining this with Theorem 34 (ii) one may wonder if a general modularinvariant commutative symmetric Frobenius algebra Acl in C2plusmn is isomorphic to a directsum of simple such algebras However this is not so For example one can take the com-mutative symmetric Frobenius algebra Acl = C[x]x2 in the category of vector spacesequipped with the non-degenerate trace ε(ax + b) = a In this case the modular invari-ance condition holds automatically but Acl is clearly not a direct sum of two algebras

Cardy Algebras and Sewing Constraints I 895

For a general modular tensor category C the algebra C[x]x2 R(1) understood as analgebra in C2plusmn via the braided monoidal isomorphism Vect f (C) C2plusmn rarr C2plusmn providesanother counter-example

32 Two definitions Define a morphism PlA A rarr A for a Frobenius algebra A in C

or C2plusmn as follows [Fr Sect 24]

PlA =

A

A

A

m

(311)

If A is also commutative and obeys m A A = ζA idA we have PlA = ζA idA In partic-

ular this holds if A is commutative and special Using the fact that the Frobenius algebraR(1) is commutative and normalised-special one can check that Pl

R(A) R(A) rarr R(A)takes the following form

PlR(A) =

oplusiisinI

A

A

Uori

Uori

A

m

times

Ui

Ui

(312)

With these ingredients we can now give the first definition of a Cardy C|C2plusmn-algebrawhich was introduced in [K3 Def 514] cf Remark 315 below

Definition 37 (CardyC|C2plusmn-algebra I) A CardyC|C2plusmn-algebra is a triple (Aop|Acl ιcl-op)where (Aclmcl ηclcl εcl) is a modular invariant commutative symmetric Frobe-nius C2plusmn-algebra (Aopmop ηopop εop) is a symmetric Frobenius C-algebra andιcl-op Acl rarr R(Aop) an algebra homomorphism such that the following conditionsare satisfied

(i) Centre condition

Acl R(Aop)

R(Aop)

R(Aop)

ιcl-op

m R(Aop)

=

Acl

ιcl-op

R(Aop)

R(Aop)

R(Aop)

m R(Aop)

(313)

896 L Kong I Runkel

(ii) Cardy condition

ιcl-op ιlowastcl-op =

R(Aop)

R(Aop)

R(Aop)

(314)

Remark 38

(i) The name ldquoCardy C|C2plusmn-algebrardquo in Definition 37 was chosen because many ofthe important ingredients were first studied by Cardy the modular invariance ofthe closed theory [C1] the consistency of the annulus amplitude [C2] and thebulk-boundary OPE [CL] On the other hand the boundary-boundary OPE andthe OPE analogue of the centre condition were first considered in [Lw]

(ii) One can easily see that in the special case that C is the category Vect f (C) of finite-dimensional C-vector spaces a Cardy C|C2plusmn-algebra gives exactly the algebraicformulation of two-dimensional open-closed topological field theory over C (cfRemark 614 in [K3]) see [Lz Sect 48] [Mo Thm 11] [AN Thm 45] [LPCor 43] [MS Sect 22] When passing to a general modular tensor category Cthere are two important differences to the two-dimensional topological field the-ory Firstly the algebras Acl and Aop now live in different categories which inparticular affects the formulation of the centre condition and the Cardy conditionSecondly the modular invariance condition has to be imposed on Acl In the caseC = Vect f (C) modular invariance holds automatically

Definition 39 A homomorphism of Cardy C|C2plusmn-algebras (A(1)op |A(1)cl ι(1)cl-op) rarr (A(2)op

|A(2)cl ι(2)cl-op) is a pair ( fop fcl) of Frobenius algebra homomorphisms fop A(1)op rarr A(2)op

and fcl A(1)cl rarr A(2)cl such that the diagram

A(1)cl

fcl

ι(1)cl-op

A(2)cl

ι(2)cl-op

R(A(1)op )

R( fop) R(A(2)op )

(315)

commutes

Remark 310 Since a homomorphism of Frobenius algebras is invertible (cf Lemma218 (iv)) a homomorphism of Cardy algebras is always an isomorphism

For a homomorphism ( fop fcl) of Cardy C|C2plusmn-algebras using the commutativity of(315) and the fact that fcl and fop are both algebra and coalgebra homomorphisms itis easy to show that (315) commutes iff

R(A(1)op )R( fop)

(ι(1)cl-op)

lowast

R(A(2)op )

(ι(2)cl-op)

lowast

A(1)cl

fcl A(2)cl

(316)

Cardy Algebras and Sewing Constraints I 897

commutesLet (Aop|Acl ιcl-op) be a Cardy C|C2plusmn-algebra Define the morphism

ιcl-op = χminus1(ιcl-op) T (Acl) minusrarr Aop (317)

Decompose Acl as Acl = oplusNn=1Cl

n times Crn such that Cl

1 times Cr1 = ηcl(1 times 1) We use ι(n)cl-op

to denote the restriction of ιcl-op to Cln times Cr

n and ι(n)cl-op to denote the restriction of ιcl-op

to Cln otimes Cr

n We introduce the following graphical notation

ι(n)cl-op =

Cln Cr

n

Aop

(318)

By (243) ιcl-op can be expressed in terms of ιcl-op as follows

ιcl-op =Noplus

n=1

oplusiisinI

sumα

Cln

Crn

Aop

α

Uori

times α

Crn

Ui

(319)

Lemma 311 The centre condition (313) is equivalent to the following condition in C

Cln Cr

n

Aop

Aop

Aop

=

Cln Cr

n

Aop

Aop

Aop

for n = 1 N (320)

Proof First insert (319) and the definition (217) of m R(Aop) into (313) Then apply thecommutativity of R(1) to the left hand side of (313) The equivalence between (313)and (320) follows immediately Remark 312 The centre condition (320) is very natural from the open-closed confor-mal field theory point of view Correlators on the upper half plane are expressed in termsof conformal blocks on the full complex plane The objects Cl

n and Crn are associated to

the field insertion at a point z in the upper half plane and at the complex conjugate pointz in the lower half plane respectively The object Aop corresponds to a field inserted ata point r on the real axis The centre condition (320) simply says that the correlationfunctions in the disjoint domains |z| gt r gt 0 and r gt |z| gt 0 are analytic continuationsof each other see [K2 Prop 118]

898 L Kong I Runkel

Recall that we define ιlowastcl-op Aop rarr T (Acl) as in (230) We introduce the graphicalnotation

ιlowastcl-op =Noplus

n=1

Cln Cr

n

Aop

=Noplus

n=1

Cln Cr

n

Aop

Aop T (minus1Acl) (321)

where the second equality follows from (215) (221) and (259)

Lemma 313 The Cardy condition (314) is equivalent to the following identity in C

Noplusn=1

DimCdim Ui

sumα

Cln

Crn

Crn

α

α

Aop

Aop Uori

Uori

=Aop

Aop

Aop

Uori

Uori

op

mop for all i isin I (322)

Proof By (244) (312) and (319) it is easy to see that (322) is equivalent to thefollowing identity

ιcl-op χ(ιlowastcl-op) = PlR(Aop)

(323)

Therefore it is enough to show that

χ(ιlowastcl-op) = ιlowastcl-op (324)

We have

χminus1(ιlowastcl-op)(1)= T (ιlowastcl-op) δ (2)= T (ιcl-op)

lowast δlowastlowast (3)= (ρ T (ιcl-op)

)lowast (4)= (ιcl-op

)lowast (325)

In step (1) we use the expression (253) for χminus1 step (2) follows from Lemma 218 (v)and Lemma 219 Step (3) is Lemma 218 (i) and Lemma 228 and finally step (4)amounts to substituting (253) and (317) Acting with χ on both sides of the aboveequality produces (324)

Combining Lemmas 311 and 313 and Proposition 216 we obtain the followingequivalent definition of Cardy C|C2plusmn-algebra (recall the graphical notation (318) forιcl-op and (321) for ιlowastcl-op)

Cardy Algebras and Sewing Constraints I 899

Definition 314 (CardyC|C2plusmn-algebra II) A CardyC-algebra is a triple (Aop|Acl ιcl-op)where Acl is a commutative symmetric Frobenius C2plusmn-algebra satisfying property (34)with f = mcl Aop is a symmetric Frobenius C-algebra and ιcl-op T (Acl) rarr Aop isan algebra homomorphism satisfying the conditions (320) and (322)

Remark 315 Up to a choice of normalisation Definition 314 is the same as the orig-inal one in [K3 Def 613] The difference between the two definitions is the factorDim C dim Ui on the left hand side of (322) which in [K3 Def 613] is given byradic

Dim C dim Ui The two definitions are related by rescaling the coproduct cl andcounit εcl of Acl by 1

radicDim C and

radicDim C respectively We chose the convention in

(322) to remove all dimension factors from the expression (314) for the Cardy condition

33 Uniqueness and existence theorems In this subsection we investigate the structureof Cardy algebras We start with the following proposition which when combined withthe results of part II provides an alternative proof of [Fj Prop 422]

Proposition 316 Let (Aop|Acl ιcl-op) be a Cardy C|C2plusmn-algebra If Acl is simple anddim Aop = 0 then Aop is simple and special

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis therefore haploid Restricting the Cardy condition (322) to the case Ui = 1 andcomposing both sides with εop from the left we see that εop kills all terms associated toU j times 1 isin Acl in the sum except for a single 1 times 1 term Thus we obtain the followingidentity

β εop = dAop (mop otimes idAorop) (idAop otimes bAop) (326)

where β isin C Composing with ηop from the right in turn implies that βεop ηop =dim Aop which is nonzero by assumption Thus also β = 0 and εop is a nonzero multipleof the morphism on the right hand side of (326) By [FRS Lem 311] Aop is special

Since Aop is a special Frobenius algebra Aop is semi-simple as an Aop-bimodule(apply [FS Prop 524] to Aop tensored with its opposite algebra) Suppose Aop is not

simple so that we can write Aop = A(1)op oplus A(2)op for nonzero Aop-bimodules A(1)op and A(2)op We denote the canonical embeddings and projections associated to this decompositionas ι12 and π12 We have the identities

mop (ι1 otimes ι2) = 0 εop ηop =2sum

i=1

εop ιi πi ηop (327)

The first identity follows since π1 mop (ι1 otimes ι2) = 0 (as mop gives the left action of

Aop on Aop and hence it preserves A(2)op ) and similarly π2 mop (ι1 otimes ι2) = 0 Thesecond identity is just the completeness of ι12 π12

Since εop ηop = 0 without losing generality we can assume εop ι1 π1 ηop = 0Using that π2 is a bimodule map we compute

π2 [LHS of (322)]Ui =1 ι1 π1 ηop = π2 [RHS of (322)]Ui =1 ι1 π1 ηop

= PlA(2)op

π2 ι1 π1 ηop = 0 (328)

900 L Kong I Runkel

On the other hand using that Acl is haploid that ιcl-op is an algebra map and thatι lowastcl-op is a coalgebra map one can check that the left-hand side of (328) is equal toλ(εop ι1 π1 ηop) π2 ηop for some λ = 0 This implies that π2 ηop = 0 Thusηop = sum

i ιi πi ηop = ι1 π1 ηop Hence we have

0 = π2 ι2 = π2 mop (ηop otimes ι2) = π2 mop ((ι1 π1 ηop)otimes ι2) (329)

However the right-hand side is zero by (327) This is a contradiction and hence Aopmust be simple

To formulate the next theorem we need the notion of the full centre of an algebra [FjDef 49] Recall that an algebra A in a braided tensor category has a left centre and aright centre [VZO] both of which are sub-algebras of A Of these two we will onlyneed the left centre The following definition is [Fr Def 231] which in our setting isequivalent to that of [VZO]

Definition 317 Let A be a symmetric special Frobenius algebra such that m A A =ζA idA

(i) The left centre Cl(A) of A is the image of the idempotent ζminus1A Pl

A(ii) The full centre Z(A) is Cl(R(A))

That ζminus1A Pl

A is an idempotent follows from [FRS Lem 52] when keeping track ofthe factors ζA ([FRS] assumes normalised-special ie ζA = 1) Note that Cl(A) is againan object of C while Z(A) is an object of C2plusmn Let el

A Cl(A) rarr A be the embeddingof Cl(A) into A The left centre is in fact the maximal subobject of A such that

m A cAA (elA otimes idA) = m A (330)

see [Fr Lem 232] This observation explains the name left centre and also makes theconnection to [O Def 15]

The full centre is by definition the image of the idempotent ζminus1A Pl

R(A) R(A) rarrR(A) Since C2plusmn is abelian the idempotent splits and we obtain the embedding andrestriction morphisms

e Z(A) rarr R(A) and r R(A) Z(A) (331)

which obey r e = idZ(A) and e r = ζminus1A Pl

R(A) It follows from Proposition 225 and

[Fr Prop 237] that Z(A) is a commutative symmetric Frobenius algebra in C2plusmn withstructure morphisms5

m Z(A) = r m R(A) (e otimes e) ηZ(A) = r ηR(A)

Z(A) = ζA middot (r otimes r) R(A) e εZ(A) = ζminus1A middot εR(A) e

(332)

Moreover if A is simple then Z(A) is simple and if A is simple and dim A = 0 thenZ(A) is simple and special The normalisation of the counit is such that

εZ(A) ηZ(A) = ζminus2A dim A DimC (333)

5 The normalisation of product and unit is the standard one The factors in the coproduct and counit haveto be included in order for (A|Z(A) e) to be a Cardy algebra see Theorem 318 below The normalisation ofthe counit enters the Cardy condition (314) through the definition of ( middot )lowast

Cardy Algebras and Sewing Constraints I 901

Theorem 318 Let A be a special symmetric Frobenius C-algebra Then (A|Z(A) e)is a Cardy C|C2plusmn-algebra

The proof of this theorem makes use the following two lemmas

Lemma 319 e Z(A) rarr R(A) is an algebra map and elowast = ζA middot r

Proof It follows from [Fr Lem 229] (or by direct calculation using in particularm R(A) R(A) = ζAidR(A)) that

m R(A) (e otimes e) = ζminus1A middot Pl

R(A) m R(A) (e otimes e) (334)

Substituting e r = ζminus1A Pl

R(A) shows that e is compatible with multiplication For theunit one finds

e ηZ(A) = e r ηR(A) = ζminus1A Pl

R(A) ηR(A) = ηR(A) (335)

Thus e is an algebra map For the second statement one computes

elowast (1)= ζA

R(A)

r

R(A)

R(A)

Z(A)

R(A)

Z(A)

r

e

e

(2)=

R(A)

r

R(A)

R(A)

R(A)

Z(A)

(3)=

R(A)

r

R(A)

Z(A)

(4)= ζA middot r (336)

where in (1) the definitions (230) and (332) have been substituted step (2) is e r = ζminus1

A PlR(A) step (3) uses that R(A) is symmetric Frobenius and step (4) is again

e r = ζminus1A Pl

R(A) Lemma 320 Let A be a symmetric Frobenius algebra in C The morphism

PlR(A) m R(A)

(Pl

R(A) otimes PlR(A)

) R(A)otimes R(A) minusrarr R(A) (337)

is S-invariant

The proof of this lemma is a slightly lengthy explicit calculation and has been deferredto Appendix A2

902 L Kong I Runkel

Proof of Theorem 318 That e is an algebra map was proved in Lemma 319 The centrecondition (313) holds by property (330) of the left centre The Cardy condition (314)also is an immediate consequence of Lemma 319

ιcl-op ιlowastcl-op = e (ζA r) = PlR(A) (338)

The full centre Z(A) is a commutative symmetric Frobenius algebra It remains to provemodular invariance That θZ(A) = idZ(A) is implied by commutativity and symmetryof Z(A) [Fr Prop 225] The S-invariance condition (33) follows from Lemma 320In (337) substitute Pl

R(A) = ζA e r and then put the resulting morphism into (33)Compose the resulting equation with e otimes idW from the right (ie from the bottom) andsubstitute the definition (332) of m Z(A) This results in the statement that m Z(A) isS-invariant

The following theorem is analogous to [LR Prop 29] and [Fj Thm 426] whichroughly speaking answer the question under which circumstances the restriction of atwo-dimensional conformal field theory to the boundary already determines the entireconformal field theory The first work is set in Minkowski space and uses operator alge-bras and subfactors while the second work is set in Euclidean space and uses modulartensor categories

Theorem 321 Let (A|Acl ιcl-op) be a Cardy C|C2plusmn-algebra such that dim A = 0 andAcl is simple Then A is special and (A|Acl ιcl-op) sim= (A|Z(A) e) as Cardy algebras

Proof By Proposition 316 A is simple and special Since Acl is simple the algebramap ιcl-op Acl rarr R(A) is either zero or a monomorphism But ιcl-op ηcl = ηR(A)and so ιcl-op = 0 Thus ιcl-op is monic By Lemma 218 (ii) ζminus1

A ιlowastcl-op is epi The Cardycondition (314) implies

ιcl-op ζminus1A ιlowastcl-op = ζminus1

A PlR(A) = e r (339)

Composing this with e r from the left yields e r ιcl-op ζminus1A ιlowastcl-op = e r =

ιcl-op ζminus1A ιlowastcl-op Since ζminus1

A ιlowastcl-op is epi we have

e r ιcl-op = ιcl-op (340)

Actually (340) also follows from (313) and specialness of R(A) We will prove that

( fop fcl) (A|Acl ιcl-op) minusrarr (A|Z(A) e) where fop = idA fcl = r ιcl-op

(341)

is an isomorphism of Cardy algebrasfcl is an algebra map Compatibility with the units follows since ιcl-op is an algebramap

fcl ηcl = r ιcl-op ηcl = r ηR(A) = ηZ(A) (342)

Compatibility with the multiplication also follows since ιcl-op is an algebra map

m Z(A) ( fcl otimes fcl) = r m R(A) (e otimes e) (r otimes r) (ιcl-op otimes ιcl-op)

= r m R(A) (ιcl-op otimes ιcl-op) = r ιcl-op mcl = fcl mcl (343)

where in the second step we used (340)

Cardy Algebras and Sewing Constraints I 903

fcl is an isomorphism As above since fcl is an algebra map and since Acl is simple fcl

has to be monic By Lemma 319 rlowast = ζminus1A e Thus f lowast

cl = ιlowastcl-op rlowast = ζminus1A ιlowastcl-op e and

fcl f lowastcl = r ιcl-op ζminus1

A ιlowastcl-op e = r e r e = idZ(A) (344)

and so fcl is also epi and hence isofcl is a coalgebra map Since fcl is an algebra map so is f minus1

cl By (344) f minus1cl = f lowast

cland by Lemma 218 (iii) this implies that fcl is a also coalgebra mapThe diagram (315) commutes Commutativity of (315) is equivalent to e fcl = ιcl-opwhich holds by (340)

Let A be a special symmetric Frobenius algebra So far we have seen that (A|Z(A) e)is a Cardy algebra and that all Cardy algebras with Aop = A and simple Acl are of thisform It is now natural to ask if every simple Acl does occur as part of a Cardy algebraThe following theorem provides an affirmative answer Recall that for an A-left moduleM the object Mor otimesA M is an algebra (see eg [KR Lem 42])

Theorem 322 If Acl is a simple modular invariant commutative symmetric FrobeniusC2plusmn-algebra then there exist a simple special symmetric Frobenius C-algebra A and amorphism ιcl-op Acl rarr R(A) such that

(i) Acl sim= Z(A) as Frobenius algebras(ii) (A|Acl ιcl-op) is a Cardy C|C2plusmn-algebra

(iii) T (Acl) sim= oplusκisinJ Morκ otimesA Mκ as algebras where MκκisinJ is a set of representa-

tives of the isomorphism classes of simple A-left modules

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis haploid It then follows from Theorem 34 that Acl is special By Proposition 224T (Acl) is a special symmetric Frobenius algebra in C Thus T (Acl) = oplusi Ai where theAi are simple symmetric Frobenius algebras We will show that at least one of the Aiis special Since T (Acl) is special we have mT (Acl) T (Acl) = ζ idT (Acl) for someζ isin C

times Restricting this to the summand Ai shows mi i = ζ idAi FurthermoreεT (Acl) ηT (Acl) = ξ id1 for some ξ isin C

times But εT (Acl) ηT (Acl) = sumi εi ηi and so

at least one of the εi ηi has to be nonzero Therefore at least one of the Ai is speciallet A equiv Ai be this summand We denote the embedding A rarr T (Acl) by e0 and therestriction T (Acl) A by r0 Notice that r0 is an algebra homomorphism Define

ιcl-op = χ (r0) Acl minusrarr R(A) (345)

By Proposition 216 ιcl-op is an algebra homomorphism Next we verify the centre con-dition (313) or rather its equivalent form (320) By substituting the definitions onecan convince oneself that the commutativity mcl cAclAcl = mcl of Acl in C2plusmn implies thecondition mT (Acl) = mT (Acl) in C see [K2 Prop 36] Here T (Acl)otimes T (Acl) rarrT (Acl)otimes T (Acl) is given by

=oplusmn

(idCl

notimes cCl

m Crnotimes idCr

m

)

(cCl

m Clnotimes cminus1

Crn C

rm

)

(idCl

motimes cminus1

Cln C

rm

otimes idCrn

)

(346)

and we decomposed Acl as Acl = oplusnCln times Cr

n As a consequence we obtain the identity

r0 mT (Acl) (idAcl otimes e0) = r0 mT (Acl) (idAcl otimes e0) (347)

904 L Kong I Runkel

Using that r0 is an algebra map and that by definition ιcl-op = r0 we obtain (320)In order to show that (A|Acl ιcl-op) is a Cardy algebra it remains to show that the

Cardy condition (314) is satisfied We will demonstrate this via a detour by first provingthat Acl sim= Z(A) as Frobenius algebras

Recall the notations e and r given in (331) Using the centre condition (320) onecan check that Pl

R(A) ιcl-op = m R(A) R(A) ιcl-op By specialness of A we have

m A A = ζA idA and so together with e r = ζminus1A Pl

R(A) we get

e r ιcl-op = ιcl-op (348)

Next consider the morphism

fcl = r ιcl-op Acl minusrarr Z(A) (349)

By the same derivation as in (342) and (343) one sees that fcl is an algebra map Inparticular fcl ηcl = ηZ(A) = 0 and so fcl = 0 Since Acl is simple fcl has to be amonomorphism

By the same argument as used in the proof of Theorem 34 (ii) up to isomorphismAcl is the unique simple local Acl-(left-)module The algebra monomorphism fcl turnsZ(A) into an Acl-module Since Z(A) is commutative it is local as an Acl-module andso Z(A) sim= AoplusN

cl for some N ge 1 By construction A is a simple special symmetricFrobenius algebra Proposition 225 and Corollary 227 show that R(A) inherits all theseproperties and thus Z(A) is simple (see the comment below Eq (332)) By Theorem318 Z(A) is modular invariant and then by Theorem 34 (i) dim Z(A) = Dim C Thisimplies that N = 1 in Z(A) sim= AoplusN

cl and so fcl is in fact an isomorphismSince Acl and Z(A) are both haploid we have εZ(A) fcl = ξ εcl for some ξ isin C

timesThe counit uniquely determines the Frobenius structure on Acl and Z(A) (see eg [FRSLemma 37]) so that fcl is a coalgebra isomorphism iff ξ = 1 To compute ξ we composethe above identity with ηcl from the right Defining ζcl via εcl ηcl = ζminus1

cl Dim C middot id1 andusing (333) gives ξ = dim A ζclζ

2A By rescaling the comultiplication and the counit

of A and consequently changing ζA we can always achieve ξ = 1 This proves part (i)of the theorem

Equation (348) implies that ιcl-op = e fcl Since fcl is an isomorphism of Frobeniusalgebras by Lemmas 218 and 319 we have

ιcl-op ιlowastcl-op = e fcl f lowastcl elowast = ζA e r = Pl

R(A) (350)

Thus (A|Acl ιcl-op) is a Cardy algebra This proves part (ii) of the theoremPart (iii) can be seen as follows By [KR Prop 43] T Z(A) sim= oplusκisinJ Mor

κ otimesA Mκ asalgebras Together with the observation that T ( fcl) T (Acl) rarr T Z(A) is an isomor-phism of algebras this proves part (iii) Remark 323 Part (i) of Theorem 322 was announced by Muumlger [Muuml2] We provide anindependent proof in the setting of Cardy algebras

The above theorem together with Lemma 211 and Theorem 34 shows that a sim-ple commutative symmetric Frobenius C2plusmn-algebra Acl with dim Acl = Dim C is alwayspart of a Cardy algebra (Aop|Acl ιcl-op) for some simple special symmetric Frobeniusalgebra Aop in C However the above proof also illustrates that Aop is not unique Thisraises the question how two Cardy algebras with a given Acl can differ This question isanswered by [KR Thm 11] which in the present framework can be restated as follows

Cardy Algebras and Sewing Constraints I 905

Theorem 324 If (A(i)op |A(i)cl ι(i)cl-op) i = 1 2 are two Cardy C|C2plusmn-algebras such that

A(i)cl is simple and dim A(i)op = 0 for i = 1 2 then A(1)clsim= A(2)cl as algebras if and only

if A(1)op and A(2)op are Morita equivalent

Proof Theorem 11 in [KR] is stated for A(i)op being non-degenerate algebras and A(i)cl =Z(A(i)op ) for i = 1 2 By Proposition 316 A(i)op are simple and special for i = 1 2

Then by [KR Lem 21] A(i)op are non-degenerate algebras By Theorem 321 we have

A(i)clsim= Z(A(i)op ) as Frobenius algebras Finally by [KR Thm 11] Z(A(1)op ) sim= Z(A(2)op )

as algebras iff A(1)op and A(2)op are Morita equivalent Let Cmax(C2plusmn) be the set of equivalence classes [B] of simple modular invariant

commutative symmetric Frobenius algebras B in C2plusmn Two such algebras B and B prime areequivalent if B and B prime are isomorphic as algebras (but not necessarily as Frobenius alge-bras) Let Msimp(C) be the set of Morita classes of simple special symmetric Frobeniusalgebras in C Define the map z Msimp(C) rarr Cmax(C2plusmn) by z A rarr [Z(A)] whereA denotes the Morita class of A From Theorem 322 (i) and [KR Thm 11] we learn

Corollary 325 The map z Msimp(C) rarr Cmax(C2plusmn) is a bijection

A Appendix

A1 Proof of Lemma 27 We will show that if (F ψ F2 ψ

F0 ) is a colax tensor functor

from C1 to C2 then (G φG2 φ

G0 ) is a lax tensor functor from C2 to C1 Applying this

result to the opposed categories then gives the converse statementWe need to show that φG

0 and φG2 make the diagrams (21) and (22) commute We

first prove the commutativity of (21) Consider the following diagram

G(A)otimes (G(B)otimes G(C))

GψF2 δ

idG(A)otimesφG2 G(A)otimes G(B otimes C)

GψF2 δ

G(FG(A)otimes F(G(B)otimes G(C)))

G(F(idG(A))otimesF(φG2 ))

G(idFG(A)otimesψ F2 )

G(FG(A)otimes FG(B otimes C))

G(ρAotimesρBotimesC )

G(FG(A)otimes (FG(B)otimes FG(C)))

G(ρAotimes(ρBotimesρC )) G(A otimes (B otimes C))

(A1)

The top subdiagram is commutative because of the naturality of Gψ F2 δ The commu-

tativity of the bottom subdiagram follows from the following identities

(ρB otimes ρC ) ψ F2 = (ρB otimes ρC ) ψ F

2 ρF Fδ

= ρBotimesC FG(ρB otimes ρC ) FG(ψ F2 ) Fδ

= ρBotimesC F(φG2 ) (A2)

as a map F(G(B) otimes G(C)) rarr B otimes C The commutativity of (A1) implies that thecomposition of maps in the left column in (21) can be replaced by

G(ρA otimes (ρB otimes ρC )) G((idFG(A) otimes ψ F

2 ) ψ F2

) δ (A3)

906 L Kong I Runkel

Similarly we can show that the composition of maps in the right column in (21) can bereplaced by

G((ρA otimes ρB)otimes ρC ) G((ψ F

2 otimes idFG(C)) ψ F2

) δ (A4)

Using the commutativity of (23) it is easy to see that (21) with the left and right col-umns of (21) replaced by (A3) and (A4) respectively is commutative Hence (21) iscommutative

Now we prove the commutativity of the first diagram in (22)

φG2 (φG

0 otimes idG(A))

(1)= G(ρ12 otimes ρA) Gψ F2 δ

[(Gψ F

0 δ11)otimes idG(A)

]

(2)= G(ρ12 otimes ρA) Gψ F2 G F(Gψ0 otimes idG(A)) G F(δ11 otimes idG(A)) δ

(3)= G(ρ12 otimes ρA) G(FG(ψ F0 )otimes idFG(A)) Gψ F

2 G F(δ11 otimes idG(A)) δ(4)= G(id12 otimes ρA) G

([ψ F

0 ρF(11) (Fδ)11] otimes idFG(A)

) Gψ F

2 δ(5)= G(id12 otimes ρA) G(ψ F

0 otimes idFG(A)) Gψ F2 δ

(6)= G(id12 otimes ρA) G(lminus1FG(A)) G F(lG(A)) δ

(7)= G(lminus1A ) GρA δG lG(A)

(8)= G(lminus1A ) lG(A) (A5)

where in step (1) we substituted the definition of φG0 φ

G2 given in (210) in step (2)

we used the naturality of δ in step (3) we used the naturality of Gψ F2 in step (4) we

switched the position between Gρ12 and G FG(ψ F0 ) and the position between Gψ2 and

G F(δ12 otimes idG(A)) using the naturality of ρ and FψG2 respectively in step (5) we applied

the second identity in (28) in step (6) we used (24) in step (7) we used the naturalityof lminus1 and δ in step (8) we used the first identity in (28)

The proof of the commutativity of the second diagram in (22) is similar Thus wehave shown that G is a lax tensor functor

A2 Proof of Lemma 320 To prepare the proof recall that for a given object B isin Cthe modular group P SL(2Z) acts on the space oplusi HomC(B otimes Ui Ui ) see eg [BKSect 31] and [K3 Eq (455)] We will only need the action of S and Sminus1 Let f isinoplusi HomC(B otimes Ui Ui ) Then

S oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A6)

Cardy Algebras and Sewing Constraints I 907

Sminus1 oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A7)

By Lemma 32 to establish that (337) is S-invariant it is enough to prove the identity(34) when f is given by (337) Using (A6) and (A7) we can see that Eq (34) simplysays that oplusi j [RHS of (34)] is invariant under the action of S times S Consider the elementg of oplus jkisinIHomC2plusmn(R(A)otimes (Uor

j times Uk)Uorj times Uk) given by

g =oplusjkisinI

sumα

R(A)

R(A)

R(A)

R(A)

Uorj times Uk

Uorj times Uk

α

α

(A8)

By the above arguments proving S-invariance of (337) is equivalent to proving invari-ance of g under the action of S times S

For i isin I we denote by gi the component of g in(oplus jisinIHomC(A otimes Uor

i otimes Uorj U

orj )

)otimes (opluskisinIHomC(Ui otimes UkUk))

We view the second Hom-space in the above tensor product as a Hom-space in C+ insteadof Cminus It is enough to show that gi is invariant under the action of S times Sminus1 Note thatthe action of Sminus1 in C+ is equivalent to that of S in Cminus

The morphism gi can be canonically identified with a bilinear pairing

( middot middot )i ⎛⎝oplus

jisinIHomC(Uor

j A otimes Uori otimes Uor

j )

⎞⎠

times(oplus

kisinIHomC(UkUi otimes Uk)

)minusrarr C (A9)

as follows For h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) we set

(h1 h2)i = (dim U j dim Uk)minus1 trUor

j timesUk[gi (h1 times h2)] (A10)

908 L Kong I Runkel

When substituting the explicit form of the product m R(A) of R(A) = (A times 1) otimes R(1)after a short calculation one finds

(h1 h2)i =sumα

1

dim U j

A

A

A

A

A A

A α

α

h1

h2

U jUk

Uorj

Ui

Uk (A11)

Here the top morphism PlR(A) has been simplified with the help of the identity

PR(A) m R(A) (PR(A) otimes PR(A))

=(oplus

iisinI((m A A)otimes idUor

i)times idUi

) m R(A) (PR(A) otimes PR(A)) (A12)

which can be checked by direct calculation along the same lines as in the proof of [FrLem 310]

The action of the modular transformation S on oplusiisinI(B otimes Ui Ui ) for B isin C nat-urally induces an action on oplusiisinI(Ui B otimes Ui ) [K3 Prop 514] which we denote bySlowast In the present case we get an action of Slowast on oplus jisinIHomC(Uor

j A otimes Uori otimes Uor

j ) and

opluskisinIHomC(UkUi otimes Uk) Then to show gi is invariant under the action of S times Sminus1

amounts to showing that

(h1 h2)i = ((Sminus1)lowast h1 Slowast h2)i (A13)

Cardy Algebras and Sewing Constraints I 909

for all h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) We have

((Sminus1)lowast h1 Slowast h2)i =sum

mnα

dim Un

DimC

A

A

A

A

A A

A

Uk

α

α

h1

h2

Ui

U j

UmUn

(A14)

Now drag the upper vertex indexed by α in the above graph along its Uorm -leg until it

meets the lower vertex also indexed by α then sum over α and m This gives

((Sminus1)lowast h1 Slowast h2)i =sum

n

dim Un

DimC

A

A

A

A

A A

A

Uk

Un

h1

h2

Ui

U j

(A15)

910 L Kong I Runkel

If we just look at the neighbourhood of the Un-loop in the above graph we see thefollowing subgraphs

sumn

dim Un

DimC

A Aor

Uorj U j

Uk Uork

Un=

sumα

1

dim U j

A Aor Uk Uork

Uorj U j

αα

(A16)

where we have applied [BK Cor 3111] Substituting this subgraph back to the originalgraph in (A15) we obtain

((Sminus1)lowast h1 Slowast h2)i =sumα

1

dim U j

A

A

A

A

A A

A

A

A

Uorj

α

α

h1

h2m A

Ui

U jUk

Uk

(A17)

The graph in (A17) is equal to that in (A11) In order to see this we first drag theldquobubblerdquo (m A A) along A lines and through the m A vertex (because m A A is abimodule map) until it reaches the lower-left leg of the upper vertex indexed by α Thendrag the m A vertex along the (red) dotted line in the above graph Finally we applythe associativity of A (A12) and [Fr Lem 311] Then we see that the graph in (A17)exactly matches with the one in (A11)

Acknowledgements We would like to thank the organisers of the Oberwolfach Arbeitsgemeinschaft ldquoAlge-braic structures in conformal field theoriesrdquo (April 2007) where this work was started for an inspiring meetingWe would further like to thank the Hausdorff Institute for Mathematics in Bonn and the organisers of the stim-ulating meeting ldquoGeometry and Physicsrdquo (May 2008) We are indebted to Alexei Davydov Jens FjelstadJuumlrgen Fuchs Yi-Zhi Huang Alexei Kitaev Urs Schreiber Christoph Schweigert Stephan Stolz and PeterTeichner for helpful discussions andor comments on a draft of this paper The research of IR was partiallysupported by the EPSRC First Grant EPE0050471 the PPARC rolling grant PPC5071451 and the MarieCurie network lsquoSuperstring Theoryrsquo (MRTN-CT-2004-512194)

Cardy Algebras and Sewing Constraints I 911

References

[AN] Alexeevski A Natanzon SM Noncommutative two-dimensional topological field theories andhurwitz numbers for real algebraic curves Sel Math New Ser 12 307ndash377 (2006)

[Bi] Bichon J Cosovereign hopf algebras J Pure Appl Alg 157 121ndash133 (2001)[BK] Bakalov B Kirillov AA Lectures on Tensor Categories and Modular Functors Providence

RI Amer Math Soc 2001[C1] Cardy JL Operator content of two-dimensional conformal invariant theories Nucl Phys

B 270 186ndash204 (1986)[C2] Cardy JL Boundary conditions fusion rules and the verlinde formula Nucl Phys B 324

581ndash596 (1989)[CL] Cardy JL Lewellen DC Bulk and boundary operators in conformal field theory Phys Lett

B 259 274ndash278 (1991)[DMZ] Dong C-Y Mason G Zhu Y-C Discrete series of the Virasoro algebra and the moonshine

module In Algebraic Groups and Their Generalizations Quantum and infinite-dimensional Meth-ods Proc Symp Pure Math 56 Part 2 Providence RI Amer MathSoc 1994 pp 295ndash316

[DP] Day B Pastro C Note on frobenius monoidal functors New York J Math 14 733ndash742 (2008)[ENO1] Etingof PI Nikshych D Ostrik V On fusion categories Ann Math 162 581ndash642 (2005)[ENO2] Etingof PI Nikshych D Ostrik V An analogue of radfordrsquos s4 formula for finite tensor

categories Int Math Research Notices 54 2915ndash2933 (2004)[Fe] Felder G Froumlhlich J Fuchs J Schweigert C Correlation functions and boundary conditions in

rational conformal field theory and three-dimensional topology Comp Math 131 189ndash237 (2002)[FHL] Frenkel IB Huang Y-Z Lepowsky J On axiomatic approaches to vertex operator algebras

and modules Mem Amer Math Soc 104 (1993)[Fj] Fjelstad J Fuchs J Runkel I Schweigert C Uniqueness of openclosed rational cft with

given algebra of open states Adv Theor Math Phys 12 1283ndash1375 (2008)[FK] Froumlhlich J King C The chern-simons theory and knot polynomials Commun Math

Phys 126 167ndash199 (1989)[Fr] Froumlhlich J Fuchs J Runkel I Schweigert C Correspondences of ribbon categories Adv

Math 199 192ndash329 (2006)[FRS] Fuchs J Runkel I Schweigert C Tft construction of rcft correlators I partition functions Nucl

Phys B 646 353ndash497 (2002)[FS] Fuchs J Schweigert C Category theory for conformal boundary conditions Fields Inst Com-

mun 39 25ndash71 (2003)[H1] Huang Y-Z Two-dimensional conformal geometry and vertex operator algebras Progress in

Mathematics Vol 148 Boston Birkhaumluser 1997[H2] Huang Y-Z Rigidity and modularity of vertex tensor categories Comm Contemp Math 10

871ndash911 (2008)[HK1] Huang Y-Z Kong L Open-string vertex algebra category and operads Commun Math

Phys 250 433ndash471 (2004)[HK2] Huang Y-Z Kong L Full field algebras Commun Math Phys 272 345ndash396 (2007)[HK3] Huang Y-Z Kong L Modular invariance for conformal full field algebras httparxivorgabs

math0609570v2[mathQA] 2006[HL] Huang Y-Z Lepowsky J Tensor products of modules for a vertex operator algebra and vertex

tensor categories In Lie Theory and Geometry in honor of Bertram Kostant ed R BrylinskiJ-L Brylinski V Guillemin V Kac Boston Birkhaumluser 1994 pp 349ndash383

[JS] Joyal A Street R Braided tensor categories Adv Math 102 20ndash78 (1993)[K1] Kong L Full field algebras operads and tensor categories Adv Math 213 271ndash340 (2007)[K2] Kong L Open-closed field algebras Commun Math Phys 280 207ndash261 (2008)[K3] Kong L Cardy condition for open-closed field algebras Commun Math Phys 283 25ndash92 (2008)[Ki] Kitaev A Private communication[KO] Kirillov AA Ostrik V On q-analog of mckay correspondence and ade classification of sl(2)

conformal field theories Adv Math 171 183ndash227 (2002)[KR] Kong L Runkel I Morita classes of algebras in modular tensor categories Adv Math 219

1548ndash1576 (2008)[Ld] Lauda AD Frobenius algebras and ambidextrous adjunctions Theo Appl Cat 16 84ndash

122 (2006)[Li1] Li H-S Regular representations of vertex operator algebras Commun Contemp Math 4

639ndash683 (2002)[Li2] Li H-S Regular representations and huang-lepowsky tensor functors for vertex operator

algebras J Alge 255 423ndash462 (2002)

Cardy Algebras and Sewing Constraints I 885

For V isin C we also choose a basis b(iα)V of HomC(VUi ) and the dual basis bV(iβ)

of HomC(Ui V ) for i isin I such that b(iα)V bV(iβ) = δαβ idUi We use the graphical

notation

b(iα)V = α

Ui

V

bV(iα) = α

Ui

V

(236)

Given two modular tensor categories C and D by C D we mean the tensor productof additive categories over C [BK Def 1115] ie the category whose objects are directsums of pairs V times W of objects V isin C and W isin D and whose morphism spaces are

HomCD(V times W V prime times W prime) = HomC(V V prime)otimesC HomC(WW prime) (237)

for pairs and direct sums of these if the objects are direct sums of pairsIf we replace the braiding and the twist in C by the antibraiding cminus1 and the antitwist

θminus1 respectively we obtain another ribbon category structure on C In order to distin-guish these two distinct structures we denote (C c θ) and (C cminus1 θminus1) by C+ and Cminusrespectively As in the Introduction we will abbreviate

C2plusmn = C+ Cminus (238)

Note that a set of representatives of the simple objects in C2plusmn is given by Ui times U j fori j isin I

For the remainder of Sect 2 we fix a modular tensor category C

24 The functors T and R The tensor product bifunctor otimes can be naturally extendedto a functor T C2plusmn rarr C Namely T (oplusN

i=1Vi times Wi ) = oplusNi=1Vi otimes Wi for all Vi Wi isin C

and N isin N The functor T becomes a tensor functor as follows For φT0 1 rarr T (1 times 1)

take φT0 = id1 (or lminus1

1 in the non-strict case) Next notice that for U VW X isin C

T (U times V )otimes T (W times X)= (U otimes V )otimes (W otimes X)

T ((U times V )otimes (W times X))= (U otimes W )otimes (V otimes X)(239)

We define φT2 T (U times V )otimes T (W times X) rarr T ((U times V )otimes (W times X)) by

φT2 = idU otimes cminus1

W V otimes idX (240)

(In the non-strict case the appropriate associators have to be added) The above definitionof φT

2 can be naturally extended to a morphism φT2 T (M1 otimes M2) rarr T (M1)otimes T (M2)

for any pair of objects M1M2 in C2plusmn The following result can be checked by directcalculation [JS Prop 52]

Lemma 220 The triple (T φT2 φ

T0 ) gives a tensor functor

886 L Kong I Runkel

In particular (T φT2 φ

T0 ψ

T2 ψ

T0 ) where ψT

2 = (φT2 )

minus1 and ψT0 = (φT

0 )minus1 gives

a Frobenius functorDefine the functor R C rarr C2plusmn as follows for A isin C and f isin HomC(A B)

R(A) =oplusiisinI

(A otimes Uori )times Ui R( f ) =

oplusiisinI

( f otimes idUori)times idUi (241)

This functor was also considered in a slightly different context in [ENO2 Prop 23]The family of isomorphisms γ R

A = oplusiisinI DimCdim Ui

id(AotimesUori )timesUi

isin Aut(R(A)) defines a

natural isomorphism γ R R rarr ROur next aim is to show that R is left and right adjoint to T in other words R

and T form an ambidextrous adjunction (see eg [Ld] for a discussion of ambidextrousadjunctions) To this end we introduce two linear isomorphisms for A isin C and M isin C2plusmn

χ HomC(T (M) A) minusrarr HomC2plusmn(M R(A))

χ HomC(A T (M)) minusrarr HomC2plusmn(R(A)M)(242)

If we decompose M as M = oplusNn=1 Ml

n times Mrn then χ and χ are given by

χ Noplus

n=1

fn

Mln Mr

n

A

rarrNoplus

n=1

oplusiisinI

sumα

fn

Mln

Mrn

A

α

Uori

times α

Mrn

Ui

(243)

and

χ Noplus

n=1 gn

Mln Mr

n

A

rarrNoplus

n=1

oplusiisinI

sumα gn

Mln

Mrn

A

α

Uori

times α

Ui

Mrn

DimCdim Ui

(244)

Notice that χ and χ are independent of the choice of basis

Theorem 221 〈T R χ〉 and 〈R T χminus1〉 are adjunctions ie R is both left and rightadjoint of T

Proof Write M as M = oplusNn=1 Ml

n times Mrn The isomorphism χ amounts to the following

composition of natural isomorphisms

HomC(T (M) A) = oplusnHomC(Mln otimes Mr

n A)sim= oplusni HomC(Ml

n otimes Ui A)otimes HomC(Mrn Ui )

sim= oplusni HomC(Mln A otimes Uor

i )otimes HomC(Mrn Ui ) = HomC2plusmn(M R(A)) (245)

Cardy Algebras and Sewing Constraints I 887

Thus χ is natural Let (γ RA )

lowast HomC2plusmn(R(A)M) rarr HomC2plusmn(R(A)M) denote the

pull-back of γ RA The isomorphism χ is equal to the composition of (γ R

A )lowast and the

following sequence of natural isomorphisms

HomC(A T (M)) = oplusn HomC(AMln otimes Mr

n)

sim= oplusni HomC(AMln otimes Ui )otimes HomC(Ui Mr

n)

sim= oplusni HomC(A otimes Uori Ml

n)otimes HomC(Ui Mrn) = HomC2plusmn(R(A)M) (246)

We have proved that both χ and χ are natural isomorphisms There are four natural transformations associated to χ and χ namely

idC2plusmnδminusrarr RT

ρminusrarr idC2plusmn and idCδminusrarr T R

ρminusrarr idC (247)

defined by for A isin C M isin C2plusmn

δM = χ(idT (M)) ρA = χminus1(idR(A))

ρM = χ (idT (M)) δA = χminus1(idR(A))(248)

They can be expressed graphically as follows with M = oplusNn=1 Ml

n times Mrn

δM =oplusni

sumα

Mln

Mln Mr

n Uori

αtimes

Ui

Mrn

α

ρA =oplusiisinI

A

A

Uori Ui

(249)

ρM =oplusni

sumα

Mln

Mln

Mrn Uor

i

α times

Ui

Mrn

α DimCdim Ui

δA =oplusiisinI

A

A Uori Ui

dim UiDimC

Note that

ρM δM = Dim C middot idM and ρA δA = idA (250)

Lemma 222 The functors T and R as maps on the sets of morphisms have left inversesand thus are injective

Proof Let f A rarr B be a morphism in C We define a map Q R HomC2plusmn(R(A)

R(B)) rarr HomC(A B) by f prime rarr ρB T ( f prime) δA Then we have

Q R R( f ) = ρB T R( f ) δA = ρB δB f = f (251)

888 L Kong I Runkel

where we used naturality of δ and (250) in the second and third equalities respectivelySo Q R is a left inverse of R on morphisms Thus R is injective on morphisms Similarlylet g M rarr N be a morphism in C2plusmn We define a map QT HomC(T (M) T (N )) rarrHomC2plusmn(M N ) by gprime rarr (Dim C)minus1 middot ρN R(gprime) δM Then we have

QT T (g) = (Dim C)minus1 middot ρN RT (g) δM = (Dim C)minus1 middot ρN δN g = g (252)

So QT is a left inverse of T on morphisms Thus T is injective on morphisms Using (29) and (250) one can express the two inverse maps χminus1 χminus1 as follows

for f isin HomC2plusmn(M R(A)) and g isin HomC2plusmn(R(A)M)

χminus1( f ) = ρ T ( f ) χminus1(g) = T (g) δ (253)

By Proposition 25 and Lemma 27 R is both a lax and colax tensor functor In particularφR

0 1 times 1 rarr R(1) is given by

φR0 = χ (ψT

0 ) = R(ψT0 ) δ1 times 1 = id1times1 (254)

and φR2 R(A)otimes R(B) rarr R(A otimes B) by φR

2 = R(ρA otimes ρB) R(ψT2 ) δ which can

be expressed graphically as

φR2 =

oplusi jkisinI

sumα

A

A

Uori B

B

Uorj

Uork

α

times

Ui U j

α

Uk

(255)

Similarly ψ R0 R(1) rarr 1 times 1 is given by

ψ R0 = ρ1 R(φT

0 ) = Dim C id1times1 (256)

and ψ R2 R(A otimes B) rarr R(A) otimes R(B) by ψ R

2 = ρ R(φT2 ) R(δA otimes δB) which in

graphical notation reads

ψ R2 =

oplusi jkisinI

sumα

A

A Uori

B

B Uorj

Uork

α

times

Ui U j

α

Uk

dim Ui dim U j

dim Uk Dim C

(257)

If C has more than one simple object then R does not take the tensor unit of C to thetensor unit of C2plusmn and so is clearly not a tensor functor However we will show that R isstill a Frobenius functor This will imply that if A is a Frobenius algebra in C then

R(A) = (R(A)m R(A) ηR(A)R(A) εR(A)) (258)

Cardy Algebras and Sewing Constraints I 889

is a Frobenius algebra in C2plusmn where the structure morphisms were given in (217) and(218) In the case A = 1 it was proved in [Muuml1 Prop 41] (see also [Fr Lem 619]and [K1 Thm 52]) that (258) is a commutative simple symmetric normalised-specialFrobenius algebra in C2plusmn In fact given a Frobenius algebra A in C it is straightforwardto verify that the structure morphisms in (258) are precisely those of (A times 1)otimes R(1)cf Sect 22

Proposition 223 (R φR2 φ

R0 ψ

R2 ψ

R0 ) is a Frobenius functor

Proof Using the explicit graphical expression of φR2 φ

R0 ψ

R2 ψ

R0 it is easy to see that

the commutativity of the diagrams (25) and (26) are equivalent to the statement thatR(1) with structure morphisms as in (258) is a Frobenius algebra in C2plusmn The latterstatement is true by [Muuml1 Prop 41]

From Lemma 220 and Proposition 223 we see that T and R take Frobenius alge-bras to Frobenius algebras The following two propositions show how the properties ofFrobenius algebras are transported

Proposition 224 Let A be a Frobenius algebra in C2plusmn Then T (A) is a Frobenius algebrain C and

(i) A is symmetric iff T (A) is symmetric(ii) A is (normalised-)special iff T (A) is (normalised-)special

Proof For part (i) write A as a direct sum oplusNn=1 Al

n times Arn Then the maps IT (A) I prime

T (A) T (Aor) rarr T (A)or defined in (220) are given by

IT (A) = I primeT (A) = oplusN

n=1c(Aln)

or(Arn)

or (259)

Therefore by (221) T (A) = primeT (A) is equivalent to T (A) = T (prime

A) Since byLemma 222 T is injective on morphisms this proves part (i) Part (ii) can be checkedin the same way for example the condition mT (A) T (A) = ζ idT (A) is easily checkedto be equivalent to T (m A A) = ζ T (idA) Proposition 225 Let A be a Frobenius algebra A in C Then R(A) is a Frobeniusalgebra in C2plusmn and

(i) A is symmetric iff R(A) is symmetric(ii) A is (normalised-)special iff R(A) is (normalised-)special

Proof Recall that the structure morphisms of the Frobenius algebra R(A) are equal tothose of (A times 1) otimes R(1) Using this equality part (i) and (ii) follow because R(1) issymmetric and normalised-special For example

m R(A) R(A) = [(m A A)times id1] otimes idR(1) = R(m A A) (260)

so that m R(A) R(A) = ζ idR(A) is equivalent to R(m A A) = ζ R(idA) which byLemma 222 is equivalent to m A A = ζ idA

The functor R has one additional property not shared by T namely R takes absolutelysimple algebras to absolutely simple algebras We will see explicitly in Sect 33 thatthis is not true for T

890 L Kong I Runkel

Lemma 226 For a C-algebra A the map

R HomA|A(A A) rarr HomR(A)|R(A)(R(A) R(A)) (261)

given by f rarr R( f ) is well-defined and an isomorphism

Proof Since R is a lax tensor functor R(A) is naturally a R(A)-bimodule It is easy tosee that R in (261) is a well-defined map R(A) is also naturally a R(1)-bimodule whichcan be identified with the induced R(1)-bimodule structure on (A times 1)otimes R(1) wherethe left R(1) action on (Atimes1)otimes R(1) is given by (idAtimes1 otimesm R(1))(cminus1

Atimes1R(1)otimes idR(1))We have the following natural isomorphisms

HomR(1)|R(1)(R(A) R(A))sim=minusrarr HomC2plusmn(A times 1 R(A))

χminus1

minusminusrarr HomC(A A) (262)

which by (253) are given by for f isin HomR(1)|R(1)(R(A) R(A))

f rarr f prime = f (idAtimes1 otimes ηR(1)) rarr f primeprime = ρ T ( f prime) (263)

and its inverse is given by for g isin HomC(A A)

g rarr gprime = R(g) δ rarr gprimeprime = (idAtimes1 otimes m R(1)) (gprime otimes idR(1)) (264)

where gprimeprime is indeed a R(1)-bimodule map due to the commutativity of R(1) It is easy tocheck that gprimeprime = R(g) in (264) Therefore R gives an isomorphism from HomC(A A)to HomR(1)|R(1)(R(A) R(A)) Moreover one verifies that R(g) is an R(A)-bimodulemap iff g is an A-bimodule map In other words R g rarr R(g) gives an isomorphism

HomA|A(A A)sim=minusrarr HomR(A)|R(A)(R(A) R(A))

Corollary 227 Let A be a C-algebra

(i) A is absolutely simple iff R(A) is absolutely simple(ii) Let A be in addition Frobenius Then A is simple and special iff R(A) is simple and

special

Proof Part (i) immediately follows from Lemma 226 The statement of part (ii) withoutthe qualifier lsquosimplersquo is proved in Proposition 225 But as in the proof of (iii)rArr(i) inLemma 211 a special Frobenius algebra in a semi-simple category has a semi-simplecategory of bimodules and for a semi-simple C-linear category simple and absolutelysimple are equivalent Part (ii) then follows from part (i)

The following lemma will be needed in Sect 32 below to discuss the properties ofCardy algebras

Lemma 228 Let A be a Frobenius algebra in C Then (δA)lowast = ρA

Proof Recall from (250) that δA is a morphism A rarr T R(A) Since T and R are bothFrobenius functors T R(A) is a Frobenius algebra in C Substituting the definitions aftera short calculation one finds

Cardy Algebras and Sewing Constraints I 891

εT R(A) mT R(A) = Dim(C)oplusiisinI

A Uori Ui A U or

ı Uı

(265)

Substituting this in the definition of (δA)lowast gives again after a short calculation the mor-

phism ρA At an intermediate step one uses that the part of the morphism (δA)lowast which

is made up of Ui and Uı ribbons their duals and the basis morphisms λ(iı)0 andϒ(iı)0can be replaced by 1

dim Uimiddot dUi

3 Cardy Algebras

In this section we start by investigating the properties of Frobenius algebras which sat-isfy the so-called modular invariance condition We then give two definitions of a Cardyalgebra and prove their equivalence Finally in Sect 33 we study the properties of thesealgebras and state our main results

We fix a modular tensor category C Recall that C2plusmn is an abbreviation for C+ Cminus

31 Modular invariance In C2plusmn we define the object K and the morphism ω K rarr Kas

K =oplus

i jisinIUi times U j ω =

sumi jisinI

dim Ui dim U j

Dim C idUi timesU j (31)

They have the property (see eg [BK Cor 3111])

ω

K

(Uk times Ul )or

(Uk times Ul )or

Ui times U j

Ui times U j

= δik δ jlDim C

dim Ui dim U jbUi timesU j dUi timesU j (32)

892 L Kong I Runkel

Definition 31

(i) Let A B be objects of C2plusmn A morphism f A otimes B rarr B is called S-invariant iff

A

B

W

W

f =

A

B

W

W

f

K

ω(33)

holds for all for W isin C2plusmn(ii) A C2plusmn-algebra (Aclmcl ηcl) is called modular invariant iff θAcl = idAcl and mcl

is S-invariant

Lemma 32 The morphism f A otimes B rarr B is S-invariant if and only if

A

B

Ui times U j

Ui times U j

f = Dim Cdim Ui dim U j

sumα

A

B

B

f

Ui times U j

Ui times U j

α

α

(34)

holds for all i j isin I

Proof Condition (33) holds for all W iff it holds for all W = Ui times U j i j isin I so itis enough to show that the right-hand side of (33) with W = Ui times U j is equal to theright-hand side of (34) Recall the notation for basis morphisms in (236) Starting from(33) write

idBor otimes idUi timesU j =⎛⎝sum

klα

(b(ktimeslα)

B

)or (

bB(ktimeslα)

)or⎞⎠ otimes idUi timesU j (35)

and then apply (32) The graphical representation of the resulting morphism can bedeformed to give (34) Remark 33 As shown in [K3 Sect 61] the modular invariance condition of a C2plusmn-alge-bra exactly coincides with the modular invariance condition for torus 1-point correlationfunctions of a genus-01 closed CFT In particular the condition θAcl = idAcl is equiv-alent to invariance under the modular transformation T τ rarr τ + 1 and the condition(34) with f = mcl is equivalent to invariance under S τ rarr minus 1

τ Combining the

modular invariance condition with the genus zero properties of a genus-01 closed CFTresults in a modular invariant commutative symmetric Frobenius algebra in C2plusmn

Cardy Algebras and Sewing Constraints I 893

Let Acl be a modular invariant C2plusmn-algebra Evaluating (34) for f = mcl composingit with ηcl otimes idUi timesU j and taking the trace implies the following identity

Zi j = 1

DimCAclUi timesU j where

(36)

Zi j = dimC HomC2plusmn(UitimesU j Acl)

Decomposing Acl into simple objects this gives

Zi j =sum

klisinISik Zkl Sminus1

l j where Si j = si jradic

DimC (37)

which in CFT terms is of course nothing but the invariance of the torus partition functionunder the modular S-transformation

The following theorem gives a simple criterion for modular invariance

Theorem 34 Let Acl be a haploid commutative symmetric Frobenius algebra in C2plusmn

(i) If Acl is modular invariant then dim Acl = Dim C(ii) If dim Acl = Dim C then Acl is special and modular invariant

Proof Part (i) Since Acl is haploid for i = j = 0 Equation (36) reduces to 1 =dim AclDimCPart (ii) By the same reasoning as in the proof of (iii)rArr(i) in Lemma 211 one showsthat Acl is special Thus mcl cl = ζclidAcl for some ζcl = 0

By [KO Thm 45] the category (C2plusmn)locAcl

of local Acl-modules is again a mod-

ular tensor category and Dim (C2plusmn)locAcl

= (Dim C dim Acl)2 (see [Fr Prop 321 amp

Rem 323] for the same statement in the notation used here) Thus by assumption wehave Dim(C2plusmn)loc

Acl= 1 It then follows from [ENO1 Thm 23] that up to isomorphism

(C2plusmn)locAcl

has a unique simple object (namely the tensor unit) In other words every simplelocal Acl-module is isomorphic to Acl (seen as a left-module over itself)

We have the following isomorphisms between morphism spaces [FS Prop 47 amp 411]

HomAcl(Acl otimes (Ui times U j ) Acl) sim= HomC2plusmn(Ui times U j Acl)

HomAcl(Acl Acl otimes (Ui times U j )) sim= HomC2plusmn(AclUi times U j )

Using these to transport the bases (236) from the right to the left we obtain bases bα(i j)αof HomAcl(Acl otimes (Ui times U j ) Acl) and b(i j)

β β of HomAcl(Acl Acl otimes (Ui times U j )) Thesecan be expressed graphically as

894 L Kong I Runkel

bα(i j) =

Acl

Acl

Acl

Ui times U j

α

b(i j)β = DimC

dim Ui dim U j

1

ζcl

Acl

Ui times U jAcl

Acl

β

(38)

where the nonzero factor in b(i j)β is included for convenience Notice that bα(i j) b(i j)

β isa left Acl-module map Since Acl is simple as a left module over itself we have

bα(i j) b(i j)β = λαβ idAcl (39)

for some λαβ isin C By computing tr(bα(i j) b(i j)β ) it is easy to verify that λαβ = δαβ

We will now prove the following identity

1

ζcl

Acl

Acl

Ui times U j

Ui times U j

Acl

=sumα

DimCdim Ui dim U j

1

ζcl

Acl

Ui times U j

Ui times U j

Acl

Acl

Acl

α

α

(310)

One checks that the left-hand side of this equation is an idempotent which we denote byζminus1

cl PlAcl(Ui timesU j ) cf [Fr Sect 31] By [Fr Prop 41] the image Im(ζminus1

cl PlAcl(Ui timesU j ))

is a local Acl-module and hence isomorphic to AoplusNcl for some N isin Zge0

All left-module morphisms from Acl to Aclotimes(Ui timesU j ) are linear combinations of the

b(i j)β Furthermore one verifies that ζminus1

cl PlAcl(Ui timesU j )b(i j)

β = b(i j)β Therefore the b(i j)

β

describe precisely the image of the idempotent ie ζminus1cl Pl

Acl(Ui timesU j ) = sum

α b(i j)α bα(i j)

which is nothing but (310) Composing (310) with ζcl middot εcl otimes idUi timesU j from the left (iefrom the top) produces (34)

In addition since Acl is commutative symmetric Frobenius it satisfies θAcl = idAcl

[Fr Prop 225] Altogether this shows that Acl is modular invariant Remark 35 As we were writing this paper we heard that the results in Theorem 34were obtained independently by Kitaev and Muumlger [Ki]

Remark 36 Setting i = j = 0 in (36) gives the identity dim Acl = Z00 Dim C [KRProp 23] Combining this with Theorem 34 (ii) one may wonder if a general modularinvariant commutative symmetric Frobenius algebra Acl in C2plusmn is isomorphic to a directsum of simple such algebras However this is not so For example one can take the com-mutative symmetric Frobenius algebra Acl = C[x]x2 in the category of vector spacesequipped with the non-degenerate trace ε(ax + b) = a In this case the modular invari-ance condition holds automatically but Acl is clearly not a direct sum of two algebras

Cardy Algebras and Sewing Constraints I 895

For a general modular tensor category C the algebra C[x]x2 R(1) understood as analgebra in C2plusmn via the braided monoidal isomorphism Vect f (C) C2plusmn rarr C2plusmn providesanother counter-example

32 Two definitions Define a morphism PlA A rarr A for a Frobenius algebra A in C

or C2plusmn as follows [Fr Sect 24]

PlA =

A

A

A

m

(311)

If A is also commutative and obeys m A A = ζA idA we have PlA = ζA idA In partic-

ular this holds if A is commutative and special Using the fact that the Frobenius algebraR(1) is commutative and normalised-special one can check that Pl

R(A) R(A) rarr R(A)takes the following form

PlR(A) =

oplusiisinI

A

A

Uori

Uori

A

m

times

Ui

Ui

(312)

With these ingredients we can now give the first definition of a Cardy C|C2plusmn-algebrawhich was introduced in [K3 Def 514] cf Remark 315 below

Definition 37 (CardyC|C2plusmn-algebra I) A CardyC|C2plusmn-algebra is a triple (Aop|Acl ιcl-op)where (Aclmcl ηclcl εcl) is a modular invariant commutative symmetric Frobe-nius C2plusmn-algebra (Aopmop ηopop εop) is a symmetric Frobenius C-algebra andιcl-op Acl rarr R(Aop) an algebra homomorphism such that the following conditionsare satisfied

(i) Centre condition

Acl R(Aop)

R(Aop)

R(Aop)

ιcl-op

m R(Aop)

=

Acl

ιcl-op

R(Aop)

R(Aop)

R(Aop)

m R(Aop)

(313)

896 L Kong I Runkel

(ii) Cardy condition

ιcl-op ιlowastcl-op =

R(Aop)

R(Aop)

R(Aop)

(314)

Remark 38

(i) The name ldquoCardy C|C2plusmn-algebrardquo in Definition 37 was chosen because many ofthe important ingredients were first studied by Cardy the modular invariance ofthe closed theory [C1] the consistency of the annulus amplitude [C2] and thebulk-boundary OPE [CL] On the other hand the boundary-boundary OPE andthe OPE analogue of the centre condition were first considered in [Lw]

(ii) One can easily see that in the special case that C is the category Vect f (C) of finite-dimensional C-vector spaces a Cardy C|C2plusmn-algebra gives exactly the algebraicformulation of two-dimensional open-closed topological field theory over C (cfRemark 614 in [K3]) see [Lz Sect 48] [Mo Thm 11] [AN Thm 45] [LPCor 43] [MS Sect 22] When passing to a general modular tensor category Cthere are two important differences to the two-dimensional topological field the-ory Firstly the algebras Acl and Aop now live in different categories which inparticular affects the formulation of the centre condition and the Cardy conditionSecondly the modular invariance condition has to be imposed on Acl In the caseC = Vect f (C) modular invariance holds automatically

Definition 39 A homomorphism of Cardy C|C2plusmn-algebras (A(1)op |A(1)cl ι(1)cl-op) rarr (A(2)op

|A(2)cl ι(2)cl-op) is a pair ( fop fcl) of Frobenius algebra homomorphisms fop A(1)op rarr A(2)op

and fcl A(1)cl rarr A(2)cl such that the diagram

A(1)cl

fcl

ι(1)cl-op

A(2)cl

ι(2)cl-op

R(A(1)op )

R( fop) R(A(2)op )

(315)

commutes

Remark 310 Since a homomorphism of Frobenius algebras is invertible (cf Lemma218 (iv)) a homomorphism of Cardy algebras is always an isomorphism

For a homomorphism ( fop fcl) of Cardy C|C2plusmn-algebras using the commutativity of(315) and the fact that fcl and fop are both algebra and coalgebra homomorphisms itis easy to show that (315) commutes iff

R(A(1)op )R( fop)

(ι(1)cl-op)

lowast

R(A(2)op )

(ι(2)cl-op)

lowast

A(1)cl

fcl A(2)cl

(316)

Cardy Algebras and Sewing Constraints I 897

commutesLet (Aop|Acl ιcl-op) be a Cardy C|C2plusmn-algebra Define the morphism

ιcl-op = χminus1(ιcl-op) T (Acl) minusrarr Aop (317)

Decompose Acl as Acl = oplusNn=1Cl

n times Crn such that Cl

1 times Cr1 = ηcl(1 times 1) We use ι(n)cl-op

to denote the restriction of ιcl-op to Cln times Cr

n and ι(n)cl-op to denote the restriction of ιcl-op

to Cln otimes Cr

n We introduce the following graphical notation

ι(n)cl-op =

Cln Cr

n

Aop

(318)

By (243) ιcl-op can be expressed in terms of ιcl-op as follows

ιcl-op =Noplus

n=1

oplusiisinI

sumα

Cln

Crn

Aop

α

Uori

times α

Crn

Ui

(319)

Lemma 311 The centre condition (313) is equivalent to the following condition in C

Cln Cr

n

Aop

Aop

Aop

=

Cln Cr

n

Aop

Aop

Aop

for n = 1 N (320)

Proof First insert (319) and the definition (217) of m R(Aop) into (313) Then apply thecommutativity of R(1) to the left hand side of (313) The equivalence between (313)and (320) follows immediately Remark 312 The centre condition (320) is very natural from the open-closed confor-mal field theory point of view Correlators on the upper half plane are expressed in termsof conformal blocks on the full complex plane The objects Cl

n and Crn are associated to

the field insertion at a point z in the upper half plane and at the complex conjugate pointz in the lower half plane respectively The object Aop corresponds to a field inserted ata point r on the real axis The centre condition (320) simply says that the correlationfunctions in the disjoint domains |z| gt r gt 0 and r gt |z| gt 0 are analytic continuationsof each other see [K2 Prop 118]

898 L Kong I Runkel

Recall that we define ιlowastcl-op Aop rarr T (Acl) as in (230) We introduce the graphicalnotation

ιlowastcl-op =Noplus

n=1

Cln Cr

n

Aop

=Noplus

n=1

Cln Cr

n

Aop

Aop T (minus1Acl) (321)

where the second equality follows from (215) (221) and (259)

Lemma 313 The Cardy condition (314) is equivalent to the following identity in C

Noplusn=1

DimCdim Ui

sumα

Cln

Crn

Crn

α

α

Aop

Aop Uori

Uori

=Aop

Aop

Aop

Uori

Uori

op

mop for all i isin I (322)

Proof By (244) (312) and (319) it is easy to see that (322) is equivalent to thefollowing identity

ιcl-op χ(ιlowastcl-op) = PlR(Aop)

(323)

Therefore it is enough to show that

χ(ιlowastcl-op) = ιlowastcl-op (324)

We have

χminus1(ιlowastcl-op)(1)= T (ιlowastcl-op) δ (2)= T (ιcl-op)

lowast δlowastlowast (3)= (ρ T (ιcl-op)

)lowast (4)= (ιcl-op

)lowast (325)

In step (1) we use the expression (253) for χminus1 step (2) follows from Lemma 218 (v)and Lemma 219 Step (3) is Lemma 218 (i) and Lemma 228 and finally step (4)amounts to substituting (253) and (317) Acting with χ on both sides of the aboveequality produces (324)

Combining Lemmas 311 and 313 and Proposition 216 we obtain the followingequivalent definition of Cardy C|C2plusmn-algebra (recall the graphical notation (318) forιcl-op and (321) for ιlowastcl-op)

Cardy Algebras and Sewing Constraints I 899

Definition 314 (CardyC|C2plusmn-algebra II) A CardyC-algebra is a triple (Aop|Acl ιcl-op)where Acl is a commutative symmetric Frobenius C2plusmn-algebra satisfying property (34)with f = mcl Aop is a symmetric Frobenius C-algebra and ιcl-op T (Acl) rarr Aop isan algebra homomorphism satisfying the conditions (320) and (322)

Remark 315 Up to a choice of normalisation Definition 314 is the same as the orig-inal one in [K3 Def 613] The difference between the two definitions is the factorDim C dim Ui on the left hand side of (322) which in [K3 Def 613] is given byradic

Dim C dim Ui The two definitions are related by rescaling the coproduct cl andcounit εcl of Acl by 1

radicDim C and

radicDim C respectively We chose the convention in

(322) to remove all dimension factors from the expression (314) for the Cardy condition

33 Uniqueness and existence theorems In this subsection we investigate the structureof Cardy algebras We start with the following proposition which when combined withthe results of part II provides an alternative proof of [Fj Prop 422]

Proposition 316 Let (Aop|Acl ιcl-op) be a Cardy C|C2plusmn-algebra If Acl is simple anddim Aop = 0 then Aop is simple and special

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis therefore haploid Restricting the Cardy condition (322) to the case Ui = 1 andcomposing both sides with εop from the left we see that εop kills all terms associated toU j times 1 isin Acl in the sum except for a single 1 times 1 term Thus we obtain the followingidentity

β εop = dAop (mop otimes idAorop) (idAop otimes bAop) (326)

where β isin C Composing with ηop from the right in turn implies that βεop ηop =dim Aop which is nonzero by assumption Thus also β = 0 and εop is a nonzero multipleof the morphism on the right hand side of (326) By [FRS Lem 311] Aop is special

Since Aop is a special Frobenius algebra Aop is semi-simple as an Aop-bimodule(apply [FS Prop 524] to Aop tensored with its opposite algebra) Suppose Aop is not

simple so that we can write Aop = A(1)op oplus A(2)op for nonzero Aop-bimodules A(1)op and A(2)op We denote the canonical embeddings and projections associated to this decompositionas ι12 and π12 We have the identities

mop (ι1 otimes ι2) = 0 εop ηop =2sum

i=1

εop ιi πi ηop (327)

The first identity follows since π1 mop (ι1 otimes ι2) = 0 (as mop gives the left action of

Aop on Aop and hence it preserves A(2)op ) and similarly π2 mop (ι1 otimes ι2) = 0 Thesecond identity is just the completeness of ι12 π12

Since εop ηop = 0 without losing generality we can assume εop ι1 π1 ηop = 0Using that π2 is a bimodule map we compute

π2 [LHS of (322)]Ui =1 ι1 π1 ηop = π2 [RHS of (322)]Ui =1 ι1 π1 ηop

= PlA(2)op

π2 ι1 π1 ηop = 0 (328)

900 L Kong I Runkel

On the other hand using that Acl is haploid that ιcl-op is an algebra map and thatι lowastcl-op is a coalgebra map one can check that the left-hand side of (328) is equal toλ(εop ι1 π1 ηop) π2 ηop for some λ = 0 This implies that π2 ηop = 0 Thusηop = sum

i ιi πi ηop = ι1 π1 ηop Hence we have

0 = π2 ι2 = π2 mop (ηop otimes ι2) = π2 mop ((ι1 π1 ηop)otimes ι2) (329)

However the right-hand side is zero by (327) This is a contradiction and hence Aopmust be simple

To formulate the next theorem we need the notion of the full centre of an algebra [FjDef 49] Recall that an algebra A in a braided tensor category has a left centre and aright centre [VZO] both of which are sub-algebras of A Of these two we will onlyneed the left centre The following definition is [Fr Def 231] which in our setting isequivalent to that of [VZO]

Definition 317 Let A be a symmetric special Frobenius algebra such that m A A =ζA idA

(i) The left centre Cl(A) of A is the image of the idempotent ζminus1A Pl

A(ii) The full centre Z(A) is Cl(R(A))

That ζminus1A Pl

A is an idempotent follows from [FRS Lem 52] when keeping track ofthe factors ζA ([FRS] assumes normalised-special ie ζA = 1) Note that Cl(A) is againan object of C while Z(A) is an object of C2plusmn Let el

A Cl(A) rarr A be the embeddingof Cl(A) into A The left centre is in fact the maximal subobject of A such that

m A cAA (elA otimes idA) = m A (330)

see [Fr Lem 232] This observation explains the name left centre and also makes theconnection to [O Def 15]

The full centre is by definition the image of the idempotent ζminus1A Pl

R(A) R(A) rarrR(A) Since C2plusmn is abelian the idempotent splits and we obtain the embedding andrestriction morphisms

e Z(A) rarr R(A) and r R(A) Z(A) (331)

which obey r e = idZ(A) and e r = ζminus1A Pl

R(A) It follows from Proposition 225 and

[Fr Prop 237] that Z(A) is a commutative symmetric Frobenius algebra in C2plusmn withstructure morphisms5

m Z(A) = r m R(A) (e otimes e) ηZ(A) = r ηR(A)

Z(A) = ζA middot (r otimes r) R(A) e εZ(A) = ζminus1A middot εR(A) e

(332)

Moreover if A is simple then Z(A) is simple and if A is simple and dim A = 0 thenZ(A) is simple and special The normalisation of the counit is such that

εZ(A) ηZ(A) = ζminus2A dim A DimC (333)

5 The normalisation of product and unit is the standard one The factors in the coproduct and counit haveto be included in order for (A|Z(A) e) to be a Cardy algebra see Theorem 318 below The normalisation ofthe counit enters the Cardy condition (314) through the definition of ( middot )lowast

Cardy Algebras and Sewing Constraints I 901

Theorem 318 Let A be a special symmetric Frobenius C-algebra Then (A|Z(A) e)is a Cardy C|C2plusmn-algebra

The proof of this theorem makes use the following two lemmas

Lemma 319 e Z(A) rarr R(A) is an algebra map and elowast = ζA middot r

Proof It follows from [Fr Lem 229] (or by direct calculation using in particularm R(A) R(A) = ζAidR(A)) that

m R(A) (e otimes e) = ζminus1A middot Pl

R(A) m R(A) (e otimes e) (334)

Substituting e r = ζminus1A Pl

R(A) shows that e is compatible with multiplication For theunit one finds

e ηZ(A) = e r ηR(A) = ζminus1A Pl

R(A) ηR(A) = ηR(A) (335)

Thus e is an algebra map For the second statement one computes

elowast (1)= ζA

R(A)

r

R(A)

R(A)

Z(A)

R(A)

Z(A)

r

e

e

(2)=

R(A)

r

R(A)

R(A)

R(A)

Z(A)

(3)=

R(A)

r

R(A)

Z(A)

(4)= ζA middot r (336)

where in (1) the definitions (230) and (332) have been substituted step (2) is e r = ζminus1

A PlR(A) step (3) uses that R(A) is symmetric Frobenius and step (4) is again

e r = ζminus1A Pl

R(A) Lemma 320 Let A be a symmetric Frobenius algebra in C The morphism

PlR(A) m R(A)

(Pl

R(A) otimes PlR(A)

) R(A)otimes R(A) minusrarr R(A) (337)

is S-invariant

The proof of this lemma is a slightly lengthy explicit calculation and has been deferredto Appendix A2

902 L Kong I Runkel

Proof of Theorem 318 That e is an algebra map was proved in Lemma 319 The centrecondition (313) holds by property (330) of the left centre The Cardy condition (314)also is an immediate consequence of Lemma 319

ιcl-op ιlowastcl-op = e (ζA r) = PlR(A) (338)

The full centre Z(A) is a commutative symmetric Frobenius algebra It remains to provemodular invariance That θZ(A) = idZ(A) is implied by commutativity and symmetryof Z(A) [Fr Prop 225] The S-invariance condition (33) follows from Lemma 320In (337) substitute Pl

R(A) = ζA e r and then put the resulting morphism into (33)Compose the resulting equation with e otimes idW from the right (ie from the bottom) andsubstitute the definition (332) of m Z(A) This results in the statement that m Z(A) isS-invariant

The following theorem is analogous to [LR Prop 29] and [Fj Thm 426] whichroughly speaking answer the question under which circumstances the restriction of atwo-dimensional conformal field theory to the boundary already determines the entireconformal field theory The first work is set in Minkowski space and uses operator alge-bras and subfactors while the second work is set in Euclidean space and uses modulartensor categories

Theorem 321 Let (A|Acl ιcl-op) be a Cardy C|C2plusmn-algebra such that dim A = 0 andAcl is simple Then A is special and (A|Acl ιcl-op) sim= (A|Z(A) e) as Cardy algebras

Proof By Proposition 316 A is simple and special Since Acl is simple the algebramap ιcl-op Acl rarr R(A) is either zero or a monomorphism But ιcl-op ηcl = ηR(A)and so ιcl-op = 0 Thus ιcl-op is monic By Lemma 218 (ii) ζminus1

A ιlowastcl-op is epi The Cardycondition (314) implies

ιcl-op ζminus1A ιlowastcl-op = ζminus1

A PlR(A) = e r (339)

Composing this with e r from the left yields e r ιcl-op ζminus1A ιlowastcl-op = e r =

ιcl-op ζminus1A ιlowastcl-op Since ζminus1

A ιlowastcl-op is epi we have

e r ιcl-op = ιcl-op (340)

Actually (340) also follows from (313) and specialness of R(A) We will prove that

( fop fcl) (A|Acl ιcl-op) minusrarr (A|Z(A) e) where fop = idA fcl = r ιcl-op

(341)

is an isomorphism of Cardy algebrasfcl is an algebra map Compatibility with the units follows since ιcl-op is an algebramap

fcl ηcl = r ιcl-op ηcl = r ηR(A) = ηZ(A) (342)

Compatibility with the multiplication also follows since ιcl-op is an algebra map

m Z(A) ( fcl otimes fcl) = r m R(A) (e otimes e) (r otimes r) (ιcl-op otimes ιcl-op)

= r m R(A) (ιcl-op otimes ιcl-op) = r ιcl-op mcl = fcl mcl (343)

where in the second step we used (340)

Cardy Algebras and Sewing Constraints I 903

fcl is an isomorphism As above since fcl is an algebra map and since Acl is simple fcl

has to be monic By Lemma 319 rlowast = ζminus1A e Thus f lowast

cl = ιlowastcl-op rlowast = ζminus1A ιlowastcl-op e and

fcl f lowastcl = r ιcl-op ζminus1

A ιlowastcl-op e = r e r e = idZ(A) (344)

and so fcl is also epi and hence isofcl is a coalgebra map Since fcl is an algebra map so is f minus1

cl By (344) f minus1cl = f lowast

cland by Lemma 218 (iii) this implies that fcl is a also coalgebra mapThe diagram (315) commutes Commutativity of (315) is equivalent to e fcl = ιcl-opwhich holds by (340)

Let A be a special symmetric Frobenius algebra So far we have seen that (A|Z(A) e)is a Cardy algebra and that all Cardy algebras with Aop = A and simple Acl are of thisform It is now natural to ask if every simple Acl does occur as part of a Cardy algebraThe following theorem provides an affirmative answer Recall that for an A-left moduleM the object Mor otimesA M is an algebra (see eg [KR Lem 42])

Theorem 322 If Acl is a simple modular invariant commutative symmetric FrobeniusC2plusmn-algebra then there exist a simple special symmetric Frobenius C-algebra A and amorphism ιcl-op Acl rarr R(A) such that

(i) Acl sim= Z(A) as Frobenius algebras(ii) (A|Acl ιcl-op) is a Cardy C|C2plusmn-algebra

(iii) T (Acl) sim= oplusκisinJ Morκ otimesA Mκ as algebras where MκκisinJ is a set of representa-

tives of the isomorphism classes of simple A-left modules

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis haploid It then follows from Theorem 34 that Acl is special By Proposition 224T (Acl) is a special symmetric Frobenius algebra in C Thus T (Acl) = oplusi Ai where theAi are simple symmetric Frobenius algebras We will show that at least one of the Aiis special Since T (Acl) is special we have mT (Acl) T (Acl) = ζ idT (Acl) for someζ isin C

times Restricting this to the summand Ai shows mi i = ζ idAi FurthermoreεT (Acl) ηT (Acl) = ξ id1 for some ξ isin C

times But εT (Acl) ηT (Acl) = sumi εi ηi and so

at least one of the εi ηi has to be nonzero Therefore at least one of the Ai is speciallet A equiv Ai be this summand We denote the embedding A rarr T (Acl) by e0 and therestriction T (Acl) A by r0 Notice that r0 is an algebra homomorphism Define

ιcl-op = χ (r0) Acl minusrarr R(A) (345)

By Proposition 216 ιcl-op is an algebra homomorphism Next we verify the centre con-dition (313) or rather its equivalent form (320) By substituting the definitions onecan convince oneself that the commutativity mcl cAclAcl = mcl of Acl in C2plusmn implies thecondition mT (Acl) = mT (Acl) in C see [K2 Prop 36] Here T (Acl)otimes T (Acl) rarrT (Acl)otimes T (Acl) is given by

=oplusmn

(idCl

notimes cCl

m Crnotimes idCr

m

)

(cCl

m Clnotimes cminus1

Crn C

rm

)

(idCl

motimes cminus1

Cln C

rm

otimes idCrn

)

(346)

and we decomposed Acl as Acl = oplusnCln times Cr

n As a consequence we obtain the identity

r0 mT (Acl) (idAcl otimes e0) = r0 mT (Acl) (idAcl otimes e0) (347)

904 L Kong I Runkel

Using that r0 is an algebra map and that by definition ιcl-op = r0 we obtain (320)In order to show that (A|Acl ιcl-op) is a Cardy algebra it remains to show that the

Cardy condition (314) is satisfied We will demonstrate this via a detour by first provingthat Acl sim= Z(A) as Frobenius algebras

Recall the notations e and r given in (331) Using the centre condition (320) onecan check that Pl

R(A) ιcl-op = m R(A) R(A) ιcl-op By specialness of A we have

m A A = ζA idA and so together with e r = ζminus1A Pl

R(A) we get

e r ιcl-op = ιcl-op (348)

Next consider the morphism

fcl = r ιcl-op Acl minusrarr Z(A) (349)

By the same derivation as in (342) and (343) one sees that fcl is an algebra map Inparticular fcl ηcl = ηZ(A) = 0 and so fcl = 0 Since Acl is simple fcl has to be amonomorphism

By the same argument as used in the proof of Theorem 34 (ii) up to isomorphismAcl is the unique simple local Acl-(left-)module The algebra monomorphism fcl turnsZ(A) into an Acl-module Since Z(A) is commutative it is local as an Acl-module andso Z(A) sim= AoplusN

cl for some N ge 1 By construction A is a simple special symmetricFrobenius algebra Proposition 225 and Corollary 227 show that R(A) inherits all theseproperties and thus Z(A) is simple (see the comment below Eq (332)) By Theorem318 Z(A) is modular invariant and then by Theorem 34 (i) dim Z(A) = Dim C Thisimplies that N = 1 in Z(A) sim= AoplusN

cl and so fcl is in fact an isomorphismSince Acl and Z(A) are both haploid we have εZ(A) fcl = ξ εcl for some ξ isin C

timesThe counit uniquely determines the Frobenius structure on Acl and Z(A) (see eg [FRSLemma 37]) so that fcl is a coalgebra isomorphism iff ξ = 1 To compute ξ we composethe above identity with ηcl from the right Defining ζcl via εcl ηcl = ζminus1

cl Dim C middot id1 andusing (333) gives ξ = dim A ζclζ

2A By rescaling the comultiplication and the counit

of A and consequently changing ζA we can always achieve ξ = 1 This proves part (i)of the theorem

Equation (348) implies that ιcl-op = e fcl Since fcl is an isomorphism of Frobeniusalgebras by Lemmas 218 and 319 we have

ιcl-op ιlowastcl-op = e fcl f lowastcl elowast = ζA e r = Pl

R(A) (350)

Thus (A|Acl ιcl-op) is a Cardy algebra This proves part (ii) of the theoremPart (iii) can be seen as follows By [KR Prop 43] T Z(A) sim= oplusκisinJ Mor

κ otimesA Mκ asalgebras Together with the observation that T ( fcl) T (Acl) rarr T Z(A) is an isomor-phism of algebras this proves part (iii) Remark 323 Part (i) of Theorem 322 was announced by Muumlger [Muuml2] We provide anindependent proof in the setting of Cardy algebras

The above theorem together with Lemma 211 and Theorem 34 shows that a sim-ple commutative symmetric Frobenius C2plusmn-algebra Acl with dim Acl = Dim C is alwayspart of a Cardy algebra (Aop|Acl ιcl-op) for some simple special symmetric Frobeniusalgebra Aop in C However the above proof also illustrates that Aop is not unique Thisraises the question how two Cardy algebras with a given Acl can differ This question isanswered by [KR Thm 11] which in the present framework can be restated as follows

Cardy Algebras and Sewing Constraints I 905

Theorem 324 If (A(i)op |A(i)cl ι(i)cl-op) i = 1 2 are two Cardy C|C2plusmn-algebras such that

A(i)cl is simple and dim A(i)op = 0 for i = 1 2 then A(1)clsim= A(2)cl as algebras if and only

if A(1)op and A(2)op are Morita equivalent

Proof Theorem 11 in [KR] is stated for A(i)op being non-degenerate algebras and A(i)cl =Z(A(i)op ) for i = 1 2 By Proposition 316 A(i)op are simple and special for i = 1 2

Then by [KR Lem 21] A(i)op are non-degenerate algebras By Theorem 321 we have

A(i)clsim= Z(A(i)op ) as Frobenius algebras Finally by [KR Thm 11] Z(A(1)op ) sim= Z(A(2)op )

as algebras iff A(1)op and A(2)op are Morita equivalent Let Cmax(C2plusmn) be the set of equivalence classes [B] of simple modular invariant

commutative symmetric Frobenius algebras B in C2plusmn Two such algebras B and B prime areequivalent if B and B prime are isomorphic as algebras (but not necessarily as Frobenius alge-bras) Let Msimp(C) be the set of Morita classes of simple special symmetric Frobeniusalgebras in C Define the map z Msimp(C) rarr Cmax(C2plusmn) by z A rarr [Z(A)] whereA denotes the Morita class of A From Theorem 322 (i) and [KR Thm 11] we learn

Corollary 325 The map z Msimp(C) rarr Cmax(C2plusmn) is a bijection

A Appendix

A1 Proof of Lemma 27 We will show that if (F ψ F2 ψ

F0 ) is a colax tensor functor

from C1 to C2 then (G φG2 φ

G0 ) is a lax tensor functor from C2 to C1 Applying this

result to the opposed categories then gives the converse statementWe need to show that φG

0 and φG2 make the diagrams (21) and (22) commute We

first prove the commutativity of (21) Consider the following diagram

G(A)otimes (G(B)otimes G(C))

GψF2 δ

idG(A)otimesφG2 G(A)otimes G(B otimes C)

GψF2 δ

G(FG(A)otimes F(G(B)otimes G(C)))

G(F(idG(A))otimesF(φG2 ))

G(idFG(A)otimesψ F2 )

G(FG(A)otimes FG(B otimes C))

G(ρAotimesρBotimesC )

G(FG(A)otimes (FG(B)otimes FG(C)))

G(ρAotimes(ρBotimesρC )) G(A otimes (B otimes C))

(A1)

The top subdiagram is commutative because of the naturality of Gψ F2 δ The commu-

tativity of the bottom subdiagram follows from the following identities

(ρB otimes ρC ) ψ F2 = (ρB otimes ρC ) ψ F

2 ρF Fδ

= ρBotimesC FG(ρB otimes ρC ) FG(ψ F2 ) Fδ

= ρBotimesC F(φG2 ) (A2)

as a map F(G(B) otimes G(C)) rarr B otimes C The commutativity of (A1) implies that thecomposition of maps in the left column in (21) can be replaced by

G(ρA otimes (ρB otimes ρC )) G((idFG(A) otimes ψ F

2 ) ψ F2

) δ (A3)

906 L Kong I Runkel

Similarly we can show that the composition of maps in the right column in (21) can bereplaced by

G((ρA otimes ρB)otimes ρC ) G((ψ F

2 otimes idFG(C)) ψ F2

) δ (A4)

Using the commutativity of (23) it is easy to see that (21) with the left and right col-umns of (21) replaced by (A3) and (A4) respectively is commutative Hence (21) iscommutative

Now we prove the commutativity of the first diagram in (22)

φG2 (φG

0 otimes idG(A))

(1)= G(ρ12 otimes ρA) Gψ F2 δ

[(Gψ F

0 δ11)otimes idG(A)

]

(2)= G(ρ12 otimes ρA) Gψ F2 G F(Gψ0 otimes idG(A)) G F(δ11 otimes idG(A)) δ

(3)= G(ρ12 otimes ρA) G(FG(ψ F0 )otimes idFG(A)) Gψ F

2 G F(δ11 otimes idG(A)) δ(4)= G(id12 otimes ρA) G

([ψ F

0 ρF(11) (Fδ)11] otimes idFG(A)

) Gψ F

2 δ(5)= G(id12 otimes ρA) G(ψ F

0 otimes idFG(A)) Gψ F2 δ

(6)= G(id12 otimes ρA) G(lminus1FG(A)) G F(lG(A)) δ

(7)= G(lminus1A ) GρA δG lG(A)

(8)= G(lminus1A ) lG(A) (A5)

where in step (1) we substituted the definition of φG0 φ

G2 given in (210) in step (2)

we used the naturality of δ in step (3) we used the naturality of Gψ F2 in step (4) we

switched the position between Gρ12 and G FG(ψ F0 ) and the position between Gψ2 and

G F(δ12 otimes idG(A)) using the naturality of ρ and FψG2 respectively in step (5) we applied

the second identity in (28) in step (6) we used (24) in step (7) we used the naturalityof lminus1 and δ in step (8) we used the first identity in (28)

The proof of the commutativity of the second diagram in (22) is similar Thus wehave shown that G is a lax tensor functor

A2 Proof of Lemma 320 To prepare the proof recall that for a given object B isin Cthe modular group P SL(2Z) acts on the space oplusi HomC(B otimes Ui Ui ) see eg [BKSect 31] and [K3 Eq (455)] We will only need the action of S and Sminus1 Let f isinoplusi HomC(B otimes Ui Ui ) Then

S oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A6)

Cardy Algebras and Sewing Constraints I 907

Sminus1 oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A7)

By Lemma 32 to establish that (337) is S-invariant it is enough to prove the identity(34) when f is given by (337) Using (A6) and (A7) we can see that Eq (34) simplysays that oplusi j [RHS of (34)] is invariant under the action of S times S Consider the elementg of oplus jkisinIHomC2plusmn(R(A)otimes (Uor

j times Uk)Uorj times Uk) given by

g =oplusjkisinI

sumα

R(A)

R(A)

R(A)

R(A)

Uorj times Uk

Uorj times Uk

α

α

(A8)

By the above arguments proving S-invariance of (337) is equivalent to proving invari-ance of g under the action of S times S

For i isin I we denote by gi the component of g in(oplus jisinIHomC(A otimes Uor

i otimes Uorj U

orj )

)otimes (opluskisinIHomC(Ui otimes UkUk))

We view the second Hom-space in the above tensor product as a Hom-space in C+ insteadof Cminus It is enough to show that gi is invariant under the action of S times Sminus1 Note thatthe action of Sminus1 in C+ is equivalent to that of S in Cminus

The morphism gi can be canonically identified with a bilinear pairing

( middot middot )i ⎛⎝oplus

jisinIHomC(Uor

j A otimes Uori otimes Uor

j )

⎞⎠

times(oplus

kisinIHomC(UkUi otimes Uk)

)minusrarr C (A9)

as follows For h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) we set

(h1 h2)i = (dim U j dim Uk)minus1 trUor

j timesUk[gi (h1 times h2)] (A10)

908 L Kong I Runkel

When substituting the explicit form of the product m R(A) of R(A) = (A times 1) otimes R(1)after a short calculation one finds

(h1 h2)i =sumα

1

dim U j

A

A

A

A

A A

A α

α

h1

h2

U jUk

Uorj

Ui

Uk (A11)

Here the top morphism PlR(A) has been simplified with the help of the identity

PR(A) m R(A) (PR(A) otimes PR(A))

=(oplus

iisinI((m A A)otimes idUor

i)times idUi

) m R(A) (PR(A) otimes PR(A)) (A12)

which can be checked by direct calculation along the same lines as in the proof of [FrLem 310]

The action of the modular transformation S on oplusiisinI(B otimes Ui Ui ) for B isin C nat-urally induces an action on oplusiisinI(Ui B otimes Ui ) [K3 Prop 514] which we denote bySlowast In the present case we get an action of Slowast on oplus jisinIHomC(Uor

j A otimes Uori otimes Uor

j ) and

opluskisinIHomC(UkUi otimes Uk) Then to show gi is invariant under the action of S times Sminus1

amounts to showing that

(h1 h2)i = ((Sminus1)lowast h1 Slowast h2)i (A13)

Cardy Algebras and Sewing Constraints I 909

for all h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) We have

((Sminus1)lowast h1 Slowast h2)i =sum

mnα

dim Un

DimC

A

A

A

A

A A

A

Uk

α

α

h1

h2

Ui

U j

UmUn

(A14)

Now drag the upper vertex indexed by α in the above graph along its Uorm -leg until it

meets the lower vertex also indexed by α then sum over α and m This gives

((Sminus1)lowast h1 Slowast h2)i =sum

n

dim Un

DimC

A

A

A

A

A A

A

Uk

Un

h1

h2

Ui

U j

(A15)

910 L Kong I Runkel

If we just look at the neighbourhood of the Un-loop in the above graph we see thefollowing subgraphs

sumn

dim Un

DimC

A Aor

Uorj U j

Uk Uork

Un=

sumα

1

dim U j

A Aor Uk Uork

Uorj U j

αα

(A16)

where we have applied [BK Cor 3111] Substituting this subgraph back to the originalgraph in (A15) we obtain

((Sminus1)lowast h1 Slowast h2)i =sumα

1

dim U j

A

A

A

A

A A

A

A

A

Uorj

α

α

h1

h2m A

Ui

U jUk

Uk

(A17)

The graph in (A17) is equal to that in (A11) In order to see this we first drag theldquobubblerdquo (m A A) along A lines and through the m A vertex (because m A A is abimodule map) until it reaches the lower-left leg of the upper vertex indexed by α Thendrag the m A vertex along the (red) dotted line in the above graph Finally we applythe associativity of A (A12) and [Fr Lem 311] Then we see that the graph in (A17)exactly matches with the one in (A11)

Acknowledgements We would like to thank the organisers of the Oberwolfach Arbeitsgemeinschaft ldquoAlge-braic structures in conformal field theoriesrdquo (April 2007) where this work was started for an inspiring meetingWe would further like to thank the Hausdorff Institute for Mathematics in Bonn and the organisers of the stim-ulating meeting ldquoGeometry and Physicsrdquo (May 2008) We are indebted to Alexei Davydov Jens FjelstadJuumlrgen Fuchs Yi-Zhi Huang Alexei Kitaev Urs Schreiber Christoph Schweigert Stephan Stolz and PeterTeichner for helpful discussions andor comments on a draft of this paper The research of IR was partiallysupported by the EPSRC First Grant EPE0050471 the PPARC rolling grant PPC5071451 and the MarieCurie network lsquoSuperstring Theoryrsquo (MRTN-CT-2004-512194)

Cardy Algebras and Sewing Constraints I 911

References

[AN] Alexeevski A Natanzon SM Noncommutative two-dimensional topological field theories andhurwitz numbers for real algebraic curves Sel Math New Ser 12 307ndash377 (2006)

[Bi] Bichon J Cosovereign hopf algebras J Pure Appl Alg 157 121ndash133 (2001)[BK] Bakalov B Kirillov AA Lectures on Tensor Categories and Modular Functors Providence

RI Amer Math Soc 2001[C1] Cardy JL Operator content of two-dimensional conformal invariant theories Nucl Phys

B 270 186ndash204 (1986)[C2] Cardy JL Boundary conditions fusion rules and the verlinde formula Nucl Phys B 324

581ndash596 (1989)[CL] Cardy JL Lewellen DC Bulk and boundary operators in conformal field theory Phys Lett

B 259 274ndash278 (1991)[DMZ] Dong C-Y Mason G Zhu Y-C Discrete series of the Virasoro algebra and the moonshine

module In Algebraic Groups and Their Generalizations Quantum and infinite-dimensional Meth-ods Proc Symp Pure Math 56 Part 2 Providence RI Amer MathSoc 1994 pp 295ndash316

[DP] Day B Pastro C Note on frobenius monoidal functors New York J Math 14 733ndash742 (2008)[ENO1] Etingof PI Nikshych D Ostrik V On fusion categories Ann Math 162 581ndash642 (2005)[ENO2] Etingof PI Nikshych D Ostrik V An analogue of radfordrsquos s4 formula for finite tensor

categories Int Math Research Notices 54 2915ndash2933 (2004)[Fe] Felder G Froumlhlich J Fuchs J Schweigert C Correlation functions and boundary conditions in

rational conformal field theory and three-dimensional topology Comp Math 131 189ndash237 (2002)[FHL] Frenkel IB Huang Y-Z Lepowsky J On axiomatic approaches to vertex operator algebras

and modules Mem Amer Math Soc 104 (1993)[Fj] Fjelstad J Fuchs J Runkel I Schweigert C Uniqueness of openclosed rational cft with

given algebra of open states Adv Theor Math Phys 12 1283ndash1375 (2008)[FK] Froumlhlich J King C The chern-simons theory and knot polynomials Commun Math

Phys 126 167ndash199 (1989)[Fr] Froumlhlich J Fuchs J Runkel I Schweigert C Correspondences of ribbon categories Adv

Math 199 192ndash329 (2006)[FRS] Fuchs J Runkel I Schweigert C Tft construction of rcft correlators I partition functions Nucl

Phys B 646 353ndash497 (2002)[FS] Fuchs J Schweigert C Category theory for conformal boundary conditions Fields Inst Com-

mun 39 25ndash71 (2003)[H1] Huang Y-Z Two-dimensional conformal geometry and vertex operator algebras Progress in

Mathematics Vol 148 Boston Birkhaumluser 1997[H2] Huang Y-Z Rigidity and modularity of vertex tensor categories Comm Contemp Math 10

871ndash911 (2008)[HK1] Huang Y-Z Kong L Open-string vertex algebra category and operads Commun Math

Phys 250 433ndash471 (2004)[HK2] Huang Y-Z Kong L Full field algebras Commun Math Phys 272 345ndash396 (2007)[HK3] Huang Y-Z Kong L Modular invariance for conformal full field algebras httparxivorgabs

math0609570v2[mathQA] 2006[HL] Huang Y-Z Lepowsky J Tensor products of modules for a vertex operator algebra and vertex

tensor categories In Lie Theory and Geometry in honor of Bertram Kostant ed R BrylinskiJ-L Brylinski V Guillemin V Kac Boston Birkhaumluser 1994 pp 349ndash383

[JS] Joyal A Street R Braided tensor categories Adv Math 102 20ndash78 (1993)[K1] Kong L Full field algebras operads and tensor categories Adv Math 213 271ndash340 (2007)[K2] Kong L Open-closed field algebras Commun Math Phys 280 207ndash261 (2008)[K3] Kong L Cardy condition for open-closed field algebras Commun Math Phys 283 25ndash92 (2008)[Ki] Kitaev A Private communication[KO] Kirillov AA Ostrik V On q-analog of mckay correspondence and ade classification of sl(2)

conformal field theories Adv Math 171 183ndash227 (2002)[KR] Kong L Runkel I Morita classes of algebras in modular tensor categories Adv Math 219

1548ndash1576 (2008)[Ld] Lauda AD Frobenius algebras and ambidextrous adjunctions Theo Appl Cat 16 84ndash

122 (2006)[Li1] Li H-S Regular representations of vertex operator algebras Commun Contemp Math 4

639ndash683 (2002)[Li2] Li H-S Regular representations and huang-lepowsky tensor functors for vertex operator

algebras J Alge 255 423ndash462 (2002)

886 L Kong I Runkel

In particular (T φT2 φ

T0 ψ

T2 ψ

T0 ) where ψT

2 = (φT2 )

minus1 and ψT0 = (φT

0 )minus1 gives

a Frobenius functorDefine the functor R C rarr C2plusmn as follows for A isin C and f isin HomC(A B)

R(A) =oplusiisinI

(A otimes Uori )times Ui R( f ) =

oplusiisinI

( f otimes idUori)times idUi (241)

This functor was also considered in a slightly different context in [ENO2 Prop 23]The family of isomorphisms γ R

A = oplusiisinI DimCdim Ui

id(AotimesUori )timesUi

isin Aut(R(A)) defines a

natural isomorphism γ R R rarr ROur next aim is to show that R is left and right adjoint to T in other words R

and T form an ambidextrous adjunction (see eg [Ld] for a discussion of ambidextrousadjunctions) To this end we introduce two linear isomorphisms for A isin C and M isin C2plusmn

χ HomC(T (M) A) minusrarr HomC2plusmn(M R(A))

χ HomC(A T (M)) minusrarr HomC2plusmn(R(A)M)(242)

If we decompose M as M = oplusNn=1 Ml

n times Mrn then χ and χ are given by

χ Noplus

n=1

fn

Mln Mr

n

A

rarrNoplus

n=1

oplusiisinI

sumα

fn

Mln

Mrn

A

α

Uori

times α

Mrn

Ui

(243)

and

χ Noplus

n=1 gn

Mln Mr

n

A

rarrNoplus

n=1

oplusiisinI

sumα gn

Mln

Mrn

A

α

Uori

times α

Ui

Mrn

DimCdim Ui

(244)

Notice that χ and χ are independent of the choice of basis

Theorem 221 〈T R χ〉 and 〈R T χminus1〉 are adjunctions ie R is both left and rightadjoint of T

Proof Write M as M = oplusNn=1 Ml

n times Mrn The isomorphism χ amounts to the following

composition of natural isomorphisms

HomC(T (M) A) = oplusnHomC(Mln otimes Mr

n A)sim= oplusni HomC(Ml

n otimes Ui A)otimes HomC(Mrn Ui )

sim= oplusni HomC(Mln A otimes Uor

i )otimes HomC(Mrn Ui ) = HomC2plusmn(M R(A)) (245)

Cardy Algebras and Sewing Constraints I 887

Thus χ is natural Let (γ RA )

lowast HomC2plusmn(R(A)M) rarr HomC2plusmn(R(A)M) denote the

pull-back of γ RA The isomorphism χ is equal to the composition of (γ R

A )lowast and the

following sequence of natural isomorphisms

HomC(A T (M)) = oplusn HomC(AMln otimes Mr

n)

sim= oplusni HomC(AMln otimes Ui )otimes HomC(Ui Mr

n)

sim= oplusni HomC(A otimes Uori Ml

n)otimes HomC(Ui Mrn) = HomC2plusmn(R(A)M) (246)

We have proved that both χ and χ are natural isomorphisms There are four natural transformations associated to χ and χ namely

idC2plusmnδminusrarr RT

ρminusrarr idC2plusmn and idCδminusrarr T R

ρminusrarr idC (247)

defined by for A isin C M isin C2plusmn

δM = χ(idT (M)) ρA = χminus1(idR(A))

ρM = χ (idT (M)) δA = χminus1(idR(A))(248)

They can be expressed graphically as follows with M = oplusNn=1 Ml

n times Mrn

δM =oplusni

sumα

Mln

Mln Mr

n Uori

αtimes

Ui

Mrn

α

ρA =oplusiisinI

A

A

Uori Ui

(249)

ρM =oplusni

sumα

Mln

Mln

Mrn Uor

i

α times

Ui

Mrn

α DimCdim Ui

δA =oplusiisinI

A

A Uori Ui

dim UiDimC

Note that

ρM δM = Dim C middot idM and ρA δA = idA (250)

Lemma 222 The functors T and R as maps on the sets of morphisms have left inversesand thus are injective

Proof Let f A rarr B be a morphism in C We define a map Q R HomC2plusmn(R(A)

R(B)) rarr HomC(A B) by f prime rarr ρB T ( f prime) δA Then we have

Q R R( f ) = ρB T R( f ) δA = ρB δB f = f (251)

888 L Kong I Runkel

where we used naturality of δ and (250) in the second and third equalities respectivelySo Q R is a left inverse of R on morphisms Thus R is injective on morphisms Similarlylet g M rarr N be a morphism in C2plusmn We define a map QT HomC(T (M) T (N )) rarrHomC2plusmn(M N ) by gprime rarr (Dim C)minus1 middot ρN R(gprime) δM Then we have

QT T (g) = (Dim C)minus1 middot ρN RT (g) δM = (Dim C)minus1 middot ρN δN g = g (252)

So QT is a left inverse of T on morphisms Thus T is injective on morphisms Using (29) and (250) one can express the two inverse maps χminus1 χminus1 as follows

for f isin HomC2plusmn(M R(A)) and g isin HomC2plusmn(R(A)M)

χminus1( f ) = ρ T ( f ) χminus1(g) = T (g) δ (253)

By Proposition 25 and Lemma 27 R is both a lax and colax tensor functor In particularφR

0 1 times 1 rarr R(1) is given by

φR0 = χ (ψT

0 ) = R(ψT0 ) δ1 times 1 = id1times1 (254)

and φR2 R(A)otimes R(B) rarr R(A otimes B) by φR

2 = R(ρA otimes ρB) R(ψT2 ) δ which can

be expressed graphically as

φR2 =

oplusi jkisinI

sumα

A

A

Uori B

B

Uorj

Uork

α

times

Ui U j

α

Uk

(255)

Similarly ψ R0 R(1) rarr 1 times 1 is given by

ψ R0 = ρ1 R(φT

0 ) = Dim C id1times1 (256)

and ψ R2 R(A otimes B) rarr R(A) otimes R(B) by ψ R

2 = ρ R(φT2 ) R(δA otimes δB) which in

graphical notation reads

ψ R2 =

oplusi jkisinI

sumα

A

A Uori

B

B Uorj

Uork

α

times

Ui U j

α

Uk

dim Ui dim U j

dim Uk Dim C

(257)

If C has more than one simple object then R does not take the tensor unit of C to thetensor unit of C2plusmn and so is clearly not a tensor functor However we will show that R isstill a Frobenius functor This will imply that if A is a Frobenius algebra in C then

R(A) = (R(A)m R(A) ηR(A)R(A) εR(A)) (258)

Cardy Algebras and Sewing Constraints I 889

is a Frobenius algebra in C2plusmn where the structure morphisms were given in (217) and(218) In the case A = 1 it was proved in [Muuml1 Prop 41] (see also [Fr Lem 619]and [K1 Thm 52]) that (258) is a commutative simple symmetric normalised-specialFrobenius algebra in C2plusmn In fact given a Frobenius algebra A in C it is straightforwardto verify that the structure morphisms in (258) are precisely those of (A times 1)otimes R(1)cf Sect 22

Proposition 223 (R φR2 φ

R0 ψ

R2 ψ

R0 ) is a Frobenius functor

Proof Using the explicit graphical expression of φR2 φ

R0 ψ

R2 ψ

R0 it is easy to see that

the commutativity of the diagrams (25) and (26) are equivalent to the statement thatR(1) with structure morphisms as in (258) is a Frobenius algebra in C2plusmn The latterstatement is true by [Muuml1 Prop 41]

From Lemma 220 and Proposition 223 we see that T and R take Frobenius alge-bras to Frobenius algebras The following two propositions show how the properties ofFrobenius algebras are transported

Proposition 224 Let A be a Frobenius algebra in C2plusmn Then T (A) is a Frobenius algebrain C and

(i) A is symmetric iff T (A) is symmetric(ii) A is (normalised-)special iff T (A) is (normalised-)special

Proof For part (i) write A as a direct sum oplusNn=1 Al

n times Arn Then the maps IT (A) I prime

T (A) T (Aor) rarr T (A)or defined in (220) are given by

IT (A) = I primeT (A) = oplusN

n=1c(Aln)

or(Arn)

or (259)

Therefore by (221) T (A) = primeT (A) is equivalent to T (A) = T (prime

A) Since byLemma 222 T is injective on morphisms this proves part (i) Part (ii) can be checkedin the same way for example the condition mT (A) T (A) = ζ idT (A) is easily checkedto be equivalent to T (m A A) = ζ T (idA) Proposition 225 Let A be a Frobenius algebra A in C Then R(A) is a Frobeniusalgebra in C2plusmn and

(i) A is symmetric iff R(A) is symmetric(ii) A is (normalised-)special iff R(A) is (normalised-)special

Proof Recall that the structure morphisms of the Frobenius algebra R(A) are equal tothose of (A times 1) otimes R(1) Using this equality part (i) and (ii) follow because R(1) issymmetric and normalised-special For example

m R(A) R(A) = [(m A A)times id1] otimes idR(1) = R(m A A) (260)

so that m R(A) R(A) = ζ idR(A) is equivalent to R(m A A) = ζ R(idA) which byLemma 222 is equivalent to m A A = ζ idA

The functor R has one additional property not shared by T namely R takes absolutelysimple algebras to absolutely simple algebras We will see explicitly in Sect 33 thatthis is not true for T

890 L Kong I Runkel

Lemma 226 For a C-algebra A the map

R HomA|A(A A) rarr HomR(A)|R(A)(R(A) R(A)) (261)

given by f rarr R( f ) is well-defined and an isomorphism

Proof Since R is a lax tensor functor R(A) is naturally a R(A)-bimodule It is easy tosee that R in (261) is a well-defined map R(A) is also naturally a R(1)-bimodule whichcan be identified with the induced R(1)-bimodule structure on (A times 1)otimes R(1) wherethe left R(1) action on (Atimes1)otimes R(1) is given by (idAtimes1 otimesm R(1))(cminus1

Atimes1R(1)otimes idR(1))We have the following natural isomorphisms

HomR(1)|R(1)(R(A) R(A))sim=minusrarr HomC2plusmn(A times 1 R(A))

χminus1

minusminusrarr HomC(A A) (262)

which by (253) are given by for f isin HomR(1)|R(1)(R(A) R(A))

f rarr f prime = f (idAtimes1 otimes ηR(1)) rarr f primeprime = ρ T ( f prime) (263)

and its inverse is given by for g isin HomC(A A)

g rarr gprime = R(g) δ rarr gprimeprime = (idAtimes1 otimes m R(1)) (gprime otimes idR(1)) (264)

where gprimeprime is indeed a R(1)-bimodule map due to the commutativity of R(1) It is easy tocheck that gprimeprime = R(g) in (264) Therefore R gives an isomorphism from HomC(A A)to HomR(1)|R(1)(R(A) R(A)) Moreover one verifies that R(g) is an R(A)-bimodulemap iff g is an A-bimodule map In other words R g rarr R(g) gives an isomorphism

HomA|A(A A)sim=minusrarr HomR(A)|R(A)(R(A) R(A))

Corollary 227 Let A be a C-algebra

(i) A is absolutely simple iff R(A) is absolutely simple(ii) Let A be in addition Frobenius Then A is simple and special iff R(A) is simple and

special

Proof Part (i) immediately follows from Lemma 226 The statement of part (ii) withoutthe qualifier lsquosimplersquo is proved in Proposition 225 But as in the proof of (iii)rArr(i) inLemma 211 a special Frobenius algebra in a semi-simple category has a semi-simplecategory of bimodules and for a semi-simple C-linear category simple and absolutelysimple are equivalent Part (ii) then follows from part (i)

The following lemma will be needed in Sect 32 below to discuss the properties ofCardy algebras

Lemma 228 Let A be a Frobenius algebra in C Then (δA)lowast = ρA

Proof Recall from (250) that δA is a morphism A rarr T R(A) Since T and R are bothFrobenius functors T R(A) is a Frobenius algebra in C Substituting the definitions aftera short calculation one finds

Cardy Algebras and Sewing Constraints I 891

εT R(A) mT R(A) = Dim(C)oplusiisinI

A Uori Ui A U or

ı Uı

(265)

Substituting this in the definition of (δA)lowast gives again after a short calculation the mor-

phism ρA At an intermediate step one uses that the part of the morphism (δA)lowast which

is made up of Ui and Uı ribbons their duals and the basis morphisms λ(iı)0 andϒ(iı)0can be replaced by 1

dim Uimiddot dUi

3 Cardy Algebras

In this section we start by investigating the properties of Frobenius algebras which sat-isfy the so-called modular invariance condition We then give two definitions of a Cardyalgebra and prove their equivalence Finally in Sect 33 we study the properties of thesealgebras and state our main results

We fix a modular tensor category C Recall that C2plusmn is an abbreviation for C+ Cminus

31 Modular invariance In C2plusmn we define the object K and the morphism ω K rarr Kas

K =oplus

i jisinIUi times U j ω =

sumi jisinI

dim Ui dim U j

Dim C idUi timesU j (31)

They have the property (see eg [BK Cor 3111])

ω

K

(Uk times Ul )or

(Uk times Ul )or

Ui times U j

Ui times U j

= δik δ jlDim C

dim Ui dim U jbUi timesU j dUi timesU j (32)

892 L Kong I Runkel

Definition 31

(i) Let A B be objects of C2plusmn A morphism f A otimes B rarr B is called S-invariant iff

A

B

W

W

f =

A

B

W

W

f

K

ω(33)

holds for all for W isin C2plusmn(ii) A C2plusmn-algebra (Aclmcl ηcl) is called modular invariant iff θAcl = idAcl and mcl

is S-invariant

Lemma 32 The morphism f A otimes B rarr B is S-invariant if and only if

A

B

Ui times U j

Ui times U j

f = Dim Cdim Ui dim U j

sumα

A

B

B

f

Ui times U j

Ui times U j

α

α

(34)

holds for all i j isin I

Proof Condition (33) holds for all W iff it holds for all W = Ui times U j i j isin I so itis enough to show that the right-hand side of (33) with W = Ui times U j is equal to theright-hand side of (34) Recall the notation for basis morphisms in (236) Starting from(33) write

idBor otimes idUi timesU j =⎛⎝sum

klα

(b(ktimeslα)

B

)or (

bB(ktimeslα)

)or⎞⎠ otimes idUi timesU j (35)

and then apply (32) The graphical representation of the resulting morphism can bedeformed to give (34) Remark 33 As shown in [K3 Sect 61] the modular invariance condition of a C2plusmn-alge-bra exactly coincides with the modular invariance condition for torus 1-point correlationfunctions of a genus-01 closed CFT In particular the condition θAcl = idAcl is equiv-alent to invariance under the modular transformation T τ rarr τ + 1 and the condition(34) with f = mcl is equivalent to invariance under S τ rarr minus 1

τ Combining the

modular invariance condition with the genus zero properties of a genus-01 closed CFTresults in a modular invariant commutative symmetric Frobenius algebra in C2plusmn

Cardy Algebras and Sewing Constraints I 893

Let Acl be a modular invariant C2plusmn-algebra Evaluating (34) for f = mcl composingit with ηcl otimes idUi timesU j and taking the trace implies the following identity

Zi j = 1

DimCAclUi timesU j where

(36)

Zi j = dimC HomC2plusmn(UitimesU j Acl)

Decomposing Acl into simple objects this gives

Zi j =sum

klisinISik Zkl Sminus1

l j where Si j = si jradic

DimC (37)

which in CFT terms is of course nothing but the invariance of the torus partition functionunder the modular S-transformation

The following theorem gives a simple criterion for modular invariance

Theorem 34 Let Acl be a haploid commutative symmetric Frobenius algebra in C2plusmn

(i) If Acl is modular invariant then dim Acl = Dim C(ii) If dim Acl = Dim C then Acl is special and modular invariant

Proof Part (i) Since Acl is haploid for i = j = 0 Equation (36) reduces to 1 =dim AclDimCPart (ii) By the same reasoning as in the proof of (iii)rArr(i) in Lemma 211 one showsthat Acl is special Thus mcl cl = ζclidAcl for some ζcl = 0

By [KO Thm 45] the category (C2plusmn)locAcl

of local Acl-modules is again a mod-

ular tensor category and Dim (C2plusmn)locAcl

= (Dim C dim Acl)2 (see [Fr Prop 321 amp

Rem 323] for the same statement in the notation used here) Thus by assumption wehave Dim(C2plusmn)loc

Acl= 1 It then follows from [ENO1 Thm 23] that up to isomorphism

(C2plusmn)locAcl

has a unique simple object (namely the tensor unit) In other words every simplelocal Acl-module is isomorphic to Acl (seen as a left-module over itself)

We have the following isomorphisms between morphism spaces [FS Prop 47 amp 411]

HomAcl(Acl otimes (Ui times U j ) Acl) sim= HomC2plusmn(Ui times U j Acl)

HomAcl(Acl Acl otimes (Ui times U j )) sim= HomC2plusmn(AclUi times U j )

Using these to transport the bases (236) from the right to the left we obtain bases bα(i j)αof HomAcl(Acl otimes (Ui times U j ) Acl) and b(i j)

β β of HomAcl(Acl Acl otimes (Ui times U j )) Thesecan be expressed graphically as

894 L Kong I Runkel

bα(i j) =

Acl

Acl

Acl

Ui times U j

α

b(i j)β = DimC

dim Ui dim U j

1

ζcl

Acl

Ui times U jAcl

Acl

β

(38)

where the nonzero factor in b(i j)β is included for convenience Notice that bα(i j) b(i j)

β isa left Acl-module map Since Acl is simple as a left module over itself we have

bα(i j) b(i j)β = λαβ idAcl (39)

for some λαβ isin C By computing tr(bα(i j) b(i j)β ) it is easy to verify that λαβ = δαβ

We will now prove the following identity

1

ζcl

Acl

Acl

Ui times U j

Ui times U j

Acl

=sumα

DimCdim Ui dim U j

1

ζcl

Acl

Ui times U j

Ui times U j

Acl

Acl

Acl

α

α

(310)

One checks that the left-hand side of this equation is an idempotent which we denote byζminus1

cl PlAcl(Ui timesU j ) cf [Fr Sect 31] By [Fr Prop 41] the image Im(ζminus1

cl PlAcl(Ui timesU j ))

is a local Acl-module and hence isomorphic to AoplusNcl for some N isin Zge0

All left-module morphisms from Acl to Aclotimes(Ui timesU j ) are linear combinations of the

b(i j)β Furthermore one verifies that ζminus1

cl PlAcl(Ui timesU j )b(i j)

β = b(i j)β Therefore the b(i j)

β

describe precisely the image of the idempotent ie ζminus1cl Pl

Acl(Ui timesU j ) = sum

α b(i j)α bα(i j)

which is nothing but (310) Composing (310) with ζcl middot εcl otimes idUi timesU j from the left (iefrom the top) produces (34)

In addition since Acl is commutative symmetric Frobenius it satisfies θAcl = idAcl

[Fr Prop 225] Altogether this shows that Acl is modular invariant Remark 35 As we were writing this paper we heard that the results in Theorem 34were obtained independently by Kitaev and Muumlger [Ki]

Remark 36 Setting i = j = 0 in (36) gives the identity dim Acl = Z00 Dim C [KRProp 23] Combining this with Theorem 34 (ii) one may wonder if a general modularinvariant commutative symmetric Frobenius algebra Acl in C2plusmn is isomorphic to a directsum of simple such algebras However this is not so For example one can take the com-mutative symmetric Frobenius algebra Acl = C[x]x2 in the category of vector spacesequipped with the non-degenerate trace ε(ax + b) = a In this case the modular invari-ance condition holds automatically but Acl is clearly not a direct sum of two algebras

Cardy Algebras and Sewing Constraints I 895

For a general modular tensor category C the algebra C[x]x2 R(1) understood as analgebra in C2plusmn via the braided monoidal isomorphism Vect f (C) C2plusmn rarr C2plusmn providesanother counter-example

32 Two definitions Define a morphism PlA A rarr A for a Frobenius algebra A in C

or C2plusmn as follows [Fr Sect 24]

PlA =

A

A

A

m

(311)

If A is also commutative and obeys m A A = ζA idA we have PlA = ζA idA In partic-

ular this holds if A is commutative and special Using the fact that the Frobenius algebraR(1) is commutative and normalised-special one can check that Pl

R(A) R(A) rarr R(A)takes the following form

PlR(A) =

oplusiisinI

A

A

Uori

Uori

A

m

times

Ui

Ui

(312)

With these ingredients we can now give the first definition of a Cardy C|C2plusmn-algebrawhich was introduced in [K3 Def 514] cf Remark 315 below

Definition 37 (CardyC|C2plusmn-algebra I) A CardyC|C2plusmn-algebra is a triple (Aop|Acl ιcl-op)where (Aclmcl ηclcl εcl) is a modular invariant commutative symmetric Frobe-nius C2plusmn-algebra (Aopmop ηopop εop) is a symmetric Frobenius C-algebra andιcl-op Acl rarr R(Aop) an algebra homomorphism such that the following conditionsare satisfied

(i) Centre condition

Acl R(Aop)

R(Aop)

R(Aop)

ιcl-op

m R(Aop)

=

Acl

ιcl-op

R(Aop)

R(Aop)

R(Aop)

m R(Aop)

(313)

896 L Kong I Runkel

(ii) Cardy condition

ιcl-op ιlowastcl-op =

R(Aop)

R(Aop)

R(Aop)

(314)

Remark 38

(i) The name ldquoCardy C|C2plusmn-algebrardquo in Definition 37 was chosen because many ofthe important ingredients were first studied by Cardy the modular invariance ofthe closed theory [C1] the consistency of the annulus amplitude [C2] and thebulk-boundary OPE [CL] On the other hand the boundary-boundary OPE andthe OPE analogue of the centre condition were first considered in [Lw]

(ii) One can easily see that in the special case that C is the category Vect f (C) of finite-dimensional C-vector spaces a Cardy C|C2plusmn-algebra gives exactly the algebraicformulation of two-dimensional open-closed topological field theory over C (cfRemark 614 in [K3]) see [Lz Sect 48] [Mo Thm 11] [AN Thm 45] [LPCor 43] [MS Sect 22] When passing to a general modular tensor category Cthere are two important differences to the two-dimensional topological field the-ory Firstly the algebras Acl and Aop now live in different categories which inparticular affects the formulation of the centre condition and the Cardy conditionSecondly the modular invariance condition has to be imposed on Acl In the caseC = Vect f (C) modular invariance holds automatically

Definition 39 A homomorphism of Cardy C|C2plusmn-algebras (A(1)op |A(1)cl ι(1)cl-op) rarr (A(2)op

|A(2)cl ι(2)cl-op) is a pair ( fop fcl) of Frobenius algebra homomorphisms fop A(1)op rarr A(2)op

and fcl A(1)cl rarr A(2)cl such that the diagram

A(1)cl

fcl

ι(1)cl-op

A(2)cl

ι(2)cl-op

R(A(1)op )

R( fop) R(A(2)op )

(315)

commutes

Remark 310 Since a homomorphism of Frobenius algebras is invertible (cf Lemma218 (iv)) a homomorphism of Cardy algebras is always an isomorphism

For a homomorphism ( fop fcl) of Cardy C|C2plusmn-algebras using the commutativity of(315) and the fact that fcl and fop are both algebra and coalgebra homomorphisms itis easy to show that (315) commutes iff

R(A(1)op )R( fop)

(ι(1)cl-op)

lowast

R(A(2)op )

(ι(2)cl-op)

lowast

A(1)cl

fcl A(2)cl

(316)

Cardy Algebras and Sewing Constraints I 897

commutesLet (Aop|Acl ιcl-op) be a Cardy C|C2plusmn-algebra Define the morphism

ιcl-op = χminus1(ιcl-op) T (Acl) minusrarr Aop (317)

Decompose Acl as Acl = oplusNn=1Cl

n times Crn such that Cl

1 times Cr1 = ηcl(1 times 1) We use ι(n)cl-op

to denote the restriction of ιcl-op to Cln times Cr

n and ι(n)cl-op to denote the restriction of ιcl-op

to Cln otimes Cr

n We introduce the following graphical notation

ι(n)cl-op =

Cln Cr

n

Aop

(318)

By (243) ιcl-op can be expressed in terms of ιcl-op as follows

ιcl-op =Noplus

n=1

oplusiisinI

sumα

Cln

Crn

Aop

α

Uori

times α

Crn

Ui

(319)

Lemma 311 The centre condition (313) is equivalent to the following condition in C

Cln Cr

n

Aop

Aop

Aop

=

Cln Cr

n

Aop

Aop

Aop

for n = 1 N (320)

Proof First insert (319) and the definition (217) of m R(Aop) into (313) Then apply thecommutativity of R(1) to the left hand side of (313) The equivalence between (313)and (320) follows immediately Remark 312 The centre condition (320) is very natural from the open-closed confor-mal field theory point of view Correlators on the upper half plane are expressed in termsof conformal blocks on the full complex plane The objects Cl

n and Crn are associated to

the field insertion at a point z in the upper half plane and at the complex conjugate pointz in the lower half plane respectively The object Aop corresponds to a field inserted ata point r on the real axis The centre condition (320) simply says that the correlationfunctions in the disjoint domains |z| gt r gt 0 and r gt |z| gt 0 are analytic continuationsof each other see [K2 Prop 118]

898 L Kong I Runkel

Recall that we define ιlowastcl-op Aop rarr T (Acl) as in (230) We introduce the graphicalnotation

ιlowastcl-op =Noplus

n=1

Cln Cr

n

Aop

=Noplus

n=1

Cln Cr

n

Aop

Aop T (minus1Acl) (321)

where the second equality follows from (215) (221) and (259)

Lemma 313 The Cardy condition (314) is equivalent to the following identity in C

Noplusn=1

DimCdim Ui

sumα

Cln

Crn

Crn

α

α

Aop

Aop Uori

Uori

=Aop

Aop

Aop

Uori

Uori

op

mop for all i isin I (322)

Proof By (244) (312) and (319) it is easy to see that (322) is equivalent to thefollowing identity

ιcl-op χ(ιlowastcl-op) = PlR(Aop)

(323)

Therefore it is enough to show that

χ(ιlowastcl-op) = ιlowastcl-op (324)

We have

χminus1(ιlowastcl-op)(1)= T (ιlowastcl-op) δ (2)= T (ιcl-op)

lowast δlowastlowast (3)= (ρ T (ιcl-op)

)lowast (4)= (ιcl-op

)lowast (325)

In step (1) we use the expression (253) for χminus1 step (2) follows from Lemma 218 (v)and Lemma 219 Step (3) is Lemma 218 (i) and Lemma 228 and finally step (4)amounts to substituting (253) and (317) Acting with χ on both sides of the aboveequality produces (324)

Combining Lemmas 311 and 313 and Proposition 216 we obtain the followingequivalent definition of Cardy C|C2plusmn-algebra (recall the graphical notation (318) forιcl-op and (321) for ιlowastcl-op)

Cardy Algebras and Sewing Constraints I 899

Definition 314 (CardyC|C2plusmn-algebra II) A CardyC-algebra is a triple (Aop|Acl ιcl-op)where Acl is a commutative symmetric Frobenius C2plusmn-algebra satisfying property (34)with f = mcl Aop is a symmetric Frobenius C-algebra and ιcl-op T (Acl) rarr Aop isan algebra homomorphism satisfying the conditions (320) and (322)

Remark 315 Up to a choice of normalisation Definition 314 is the same as the orig-inal one in [K3 Def 613] The difference between the two definitions is the factorDim C dim Ui on the left hand side of (322) which in [K3 Def 613] is given byradic

Dim C dim Ui The two definitions are related by rescaling the coproduct cl andcounit εcl of Acl by 1

radicDim C and

radicDim C respectively We chose the convention in

(322) to remove all dimension factors from the expression (314) for the Cardy condition

33 Uniqueness and existence theorems In this subsection we investigate the structureof Cardy algebras We start with the following proposition which when combined withthe results of part II provides an alternative proof of [Fj Prop 422]

Proposition 316 Let (Aop|Acl ιcl-op) be a Cardy C|C2plusmn-algebra If Acl is simple anddim Aop = 0 then Aop is simple and special

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis therefore haploid Restricting the Cardy condition (322) to the case Ui = 1 andcomposing both sides with εop from the left we see that εop kills all terms associated toU j times 1 isin Acl in the sum except for a single 1 times 1 term Thus we obtain the followingidentity

β εop = dAop (mop otimes idAorop) (idAop otimes bAop) (326)

where β isin C Composing with ηop from the right in turn implies that βεop ηop =dim Aop which is nonzero by assumption Thus also β = 0 and εop is a nonzero multipleof the morphism on the right hand side of (326) By [FRS Lem 311] Aop is special

Since Aop is a special Frobenius algebra Aop is semi-simple as an Aop-bimodule(apply [FS Prop 524] to Aop tensored with its opposite algebra) Suppose Aop is not

simple so that we can write Aop = A(1)op oplus A(2)op for nonzero Aop-bimodules A(1)op and A(2)op We denote the canonical embeddings and projections associated to this decompositionas ι12 and π12 We have the identities

mop (ι1 otimes ι2) = 0 εop ηop =2sum

i=1

εop ιi πi ηop (327)

The first identity follows since π1 mop (ι1 otimes ι2) = 0 (as mop gives the left action of

Aop on Aop and hence it preserves A(2)op ) and similarly π2 mop (ι1 otimes ι2) = 0 Thesecond identity is just the completeness of ι12 π12

Since εop ηop = 0 without losing generality we can assume εop ι1 π1 ηop = 0Using that π2 is a bimodule map we compute

π2 [LHS of (322)]Ui =1 ι1 π1 ηop = π2 [RHS of (322)]Ui =1 ι1 π1 ηop

= PlA(2)op

π2 ι1 π1 ηop = 0 (328)

900 L Kong I Runkel

On the other hand using that Acl is haploid that ιcl-op is an algebra map and thatι lowastcl-op is a coalgebra map one can check that the left-hand side of (328) is equal toλ(εop ι1 π1 ηop) π2 ηop for some λ = 0 This implies that π2 ηop = 0 Thusηop = sum

i ιi πi ηop = ι1 π1 ηop Hence we have

0 = π2 ι2 = π2 mop (ηop otimes ι2) = π2 mop ((ι1 π1 ηop)otimes ι2) (329)

However the right-hand side is zero by (327) This is a contradiction and hence Aopmust be simple

To formulate the next theorem we need the notion of the full centre of an algebra [FjDef 49] Recall that an algebra A in a braided tensor category has a left centre and aright centre [VZO] both of which are sub-algebras of A Of these two we will onlyneed the left centre The following definition is [Fr Def 231] which in our setting isequivalent to that of [VZO]

Definition 317 Let A be a symmetric special Frobenius algebra such that m A A =ζA idA

(i) The left centre Cl(A) of A is the image of the idempotent ζminus1A Pl

A(ii) The full centre Z(A) is Cl(R(A))

That ζminus1A Pl

A is an idempotent follows from [FRS Lem 52] when keeping track ofthe factors ζA ([FRS] assumes normalised-special ie ζA = 1) Note that Cl(A) is againan object of C while Z(A) is an object of C2plusmn Let el

A Cl(A) rarr A be the embeddingof Cl(A) into A The left centre is in fact the maximal subobject of A such that

m A cAA (elA otimes idA) = m A (330)

see [Fr Lem 232] This observation explains the name left centre and also makes theconnection to [O Def 15]

The full centre is by definition the image of the idempotent ζminus1A Pl

R(A) R(A) rarrR(A) Since C2plusmn is abelian the idempotent splits and we obtain the embedding andrestriction morphisms

e Z(A) rarr R(A) and r R(A) Z(A) (331)

which obey r e = idZ(A) and e r = ζminus1A Pl

R(A) It follows from Proposition 225 and

[Fr Prop 237] that Z(A) is a commutative symmetric Frobenius algebra in C2plusmn withstructure morphisms5

m Z(A) = r m R(A) (e otimes e) ηZ(A) = r ηR(A)

Z(A) = ζA middot (r otimes r) R(A) e εZ(A) = ζminus1A middot εR(A) e

(332)

Moreover if A is simple then Z(A) is simple and if A is simple and dim A = 0 thenZ(A) is simple and special The normalisation of the counit is such that

εZ(A) ηZ(A) = ζminus2A dim A DimC (333)

5 The normalisation of product and unit is the standard one The factors in the coproduct and counit haveto be included in order for (A|Z(A) e) to be a Cardy algebra see Theorem 318 below The normalisation ofthe counit enters the Cardy condition (314) through the definition of ( middot )lowast

Cardy Algebras and Sewing Constraints I 901

Theorem 318 Let A be a special symmetric Frobenius C-algebra Then (A|Z(A) e)is a Cardy C|C2plusmn-algebra

The proof of this theorem makes use the following two lemmas

Lemma 319 e Z(A) rarr R(A) is an algebra map and elowast = ζA middot r

Proof It follows from [Fr Lem 229] (or by direct calculation using in particularm R(A) R(A) = ζAidR(A)) that

m R(A) (e otimes e) = ζminus1A middot Pl

R(A) m R(A) (e otimes e) (334)

Substituting e r = ζminus1A Pl

R(A) shows that e is compatible with multiplication For theunit one finds

e ηZ(A) = e r ηR(A) = ζminus1A Pl

R(A) ηR(A) = ηR(A) (335)

Thus e is an algebra map For the second statement one computes

elowast (1)= ζA

R(A)

r

R(A)

R(A)

Z(A)

R(A)

Z(A)

r

e

e

(2)=

R(A)

r

R(A)

R(A)

R(A)

Z(A)

(3)=

R(A)

r

R(A)

Z(A)

(4)= ζA middot r (336)

where in (1) the definitions (230) and (332) have been substituted step (2) is e r = ζminus1

A PlR(A) step (3) uses that R(A) is symmetric Frobenius and step (4) is again

e r = ζminus1A Pl

R(A) Lemma 320 Let A be a symmetric Frobenius algebra in C The morphism

PlR(A) m R(A)

(Pl

R(A) otimes PlR(A)

) R(A)otimes R(A) minusrarr R(A) (337)

is S-invariant

The proof of this lemma is a slightly lengthy explicit calculation and has been deferredto Appendix A2

902 L Kong I Runkel

Proof of Theorem 318 That e is an algebra map was proved in Lemma 319 The centrecondition (313) holds by property (330) of the left centre The Cardy condition (314)also is an immediate consequence of Lemma 319

ιcl-op ιlowastcl-op = e (ζA r) = PlR(A) (338)

The full centre Z(A) is a commutative symmetric Frobenius algebra It remains to provemodular invariance That θZ(A) = idZ(A) is implied by commutativity and symmetryof Z(A) [Fr Prop 225] The S-invariance condition (33) follows from Lemma 320In (337) substitute Pl

R(A) = ζA e r and then put the resulting morphism into (33)Compose the resulting equation with e otimes idW from the right (ie from the bottom) andsubstitute the definition (332) of m Z(A) This results in the statement that m Z(A) isS-invariant

The following theorem is analogous to [LR Prop 29] and [Fj Thm 426] whichroughly speaking answer the question under which circumstances the restriction of atwo-dimensional conformal field theory to the boundary already determines the entireconformal field theory The first work is set in Minkowski space and uses operator alge-bras and subfactors while the second work is set in Euclidean space and uses modulartensor categories

Theorem 321 Let (A|Acl ιcl-op) be a Cardy C|C2plusmn-algebra such that dim A = 0 andAcl is simple Then A is special and (A|Acl ιcl-op) sim= (A|Z(A) e) as Cardy algebras

Proof By Proposition 316 A is simple and special Since Acl is simple the algebramap ιcl-op Acl rarr R(A) is either zero or a monomorphism But ιcl-op ηcl = ηR(A)and so ιcl-op = 0 Thus ιcl-op is monic By Lemma 218 (ii) ζminus1

A ιlowastcl-op is epi The Cardycondition (314) implies

ιcl-op ζminus1A ιlowastcl-op = ζminus1

A PlR(A) = e r (339)

Composing this with e r from the left yields e r ιcl-op ζminus1A ιlowastcl-op = e r =

ιcl-op ζminus1A ιlowastcl-op Since ζminus1

A ιlowastcl-op is epi we have

e r ιcl-op = ιcl-op (340)

Actually (340) also follows from (313) and specialness of R(A) We will prove that

( fop fcl) (A|Acl ιcl-op) minusrarr (A|Z(A) e) where fop = idA fcl = r ιcl-op

(341)

is an isomorphism of Cardy algebrasfcl is an algebra map Compatibility with the units follows since ιcl-op is an algebramap

fcl ηcl = r ιcl-op ηcl = r ηR(A) = ηZ(A) (342)

Compatibility with the multiplication also follows since ιcl-op is an algebra map

m Z(A) ( fcl otimes fcl) = r m R(A) (e otimes e) (r otimes r) (ιcl-op otimes ιcl-op)

= r m R(A) (ιcl-op otimes ιcl-op) = r ιcl-op mcl = fcl mcl (343)

where in the second step we used (340)

Cardy Algebras and Sewing Constraints I 903

fcl is an isomorphism As above since fcl is an algebra map and since Acl is simple fcl

has to be monic By Lemma 319 rlowast = ζminus1A e Thus f lowast

cl = ιlowastcl-op rlowast = ζminus1A ιlowastcl-op e and

fcl f lowastcl = r ιcl-op ζminus1

A ιlowastcl-op e = r e r e = idZ(A) (344)

and so fcl is also epi and hence isofcl is a coalgebra map Since fcl is an algebra map so is f minus1

cl By (344) f minus1cl = f lowast

cland by Lemma 218 (iii) this implies that fcl is a also coalgebra mapThe diagram (315) commutes Commutativity of (315) is equivalent to e fcl = ιcl-opwhich holds by (340)

Let A be a special symmetric Frobenius algebra So far we have seen that (A|Z(A) e)is a Cardy algebra and that all Cardy algebras with Aop = A and simple Acl are of thisform It is now natural to ask if every simple Acl does occur as part of a Cardy algebraThe following theorem provides an affirmative answer Recall that for an A-left moduleM the object Mor otimesA M is an algebra (see eg [KR Lem 42])

Theorem 322 If Acl is a simple modular invariant commutative symmetric FrobeniusC2plusmn-algebra then there exist a simple special symmetric Frobenius C-algebra A and amorphism ιcl-op Acl rarr R(A) such that

(i) Acl sim= Z(A) as Frobenius algebras(ii) (A|Acl ιcl-op) is a Cardy C|C2plusmn-algebra

(iii) T (Acl) sim= oplusκisinJ Morκ otimesA Mκ as algebras where MκκisinJ is a set of representa-

tives of the isomorphism classes of simple A-left modules

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis haploid It then follows from Theorem 34 that Acl is special By Proposition 224T (Acl) is a special symmetric Frobenius algebra in C Thus T (Acl) = oplusi Ai where theAi are simple symmetric Frobenius algebras We will show that at least one of the Aiis special Since T (Acl) is special we have mT (Acl) T (Acl) = ζ idT (Acl) for someζ isin C

times Restricting this to the summand Ai shows mi i = ζ idAi FurthermoreεT (Acl) ηT (Acl) = ξ id1 for some ξ isin C

times But εT (Acl) ηT (Acl) = sumi εi ηi and so

at least one of the εi ηi has to be nonzero Therefore at least one of the Ai is speciallet A equiv Ai be this summand We denote the embedding A rarr T (Acl) by e0 and therestriction T (Acl) A by r0 Notice that r0 is an algebra homomorphism Define

ιcl-op = χ (r0) Acl minusrarr R(A) (345)

By Proposition 216 ιcl-op is an algebra homomorphism Next we verify the centre con-dition (313) or rather its equivalent form (320) By substituting the definitions onecan convince oneself that the commutativity mcl cAclAcl = mcl of Acl in C2plusmn implies thecondition mT (Acl) = mT (Acl) in C see [K2 Prop 36] Here T (Acl)otimes T (Acl) rarrT (Acl)otimes T (Acl) is given by

=oplusmn

(idCl

notimes cCl

m Crnotimes idCr

m

)

(cCl

m Clnotimes cminus1

Crn C

rm

)

(idCl

motimes cminus1

Cln C

rm

otimes idCrn

)

(346)

and we decomposed Acl as Acl = oplusnCln times Cr

n As a consequence we obtain the identity

r0 mT (Acl) (idAcl otimes e0) = r0 mT (Acl) (idAcl otimes e0) (347)

904 L Kong I Runkel

Using that r0 is an algebra map and that by definition ιcl-op = r0 we obtain (320)In order to show that (A|Acl ιcl-op) is a Cardy algebra it remains to show that the

Cardy condition (314) is satisfied We will demonstrate this via a detour by first provingthat Acl sim= Z(A) as Frobenius algebras

Recall the notations e and r given in (331) Using the centre condition (320) onecan check that Pl

R(A) ιcl-op = m R(A) R(A) ιcl-op By specialness of A we have

m A A = ζA idA and so together with e r = ζminus1A Pl

R(A) we get

e r ιcl-op = ιcl-op (348)

Next consider the morphism

fcl = r ιcl-op Acl minusrarr Z(A) (349)

By the same derivation as in (342) and (343) one sees that fcl is an algebra map Inparticular fcl ηcl = ηZ(A) = 0 and so fcl = 0 Since Acl is simple fcl has to be amonomorphism

By the same argument as used in the proof of Theorem 34 (ii) up to isomorphismAcl is the unique simple local Acl-(left-)module The algebra monomorphism fcl turnsZ(A) into an Acl-module Since Z(A) is commutative it is local as an Acl-module andso Z(A) sim= AoplusN

cl for some N ge 1 By construction A is a simple special symmetricFrobenius algebra Proposition 225 and Corollary 227 show that R(A) inherits all theseproperties and thus Z(A) is simple (see the comment below Eq (332)) By Theorem318 Z(A) is modular invariant and then by Theorem 34 (i) dim Z(A) = Dim C Thisimplies that N = 1 in Z(A) sim= AoplusN

cl and so fcl is in fact an isomorphismSince Acl and Z(A) are both haploid we have εZ(A) fcl = ξ εcl for some ξ isin C

timesThe counit uniquely determines the Frobenius structure on Acl and Z(A) (see eg [FRSLemma 37]) so that fcl is a coalgebra isomorphism iff ξ = 1 To compute ξ we composethe above identity with ηcl from the right Defining ζcl via εcl ηcl = ζminus1

cl Dim C middot id1 andusing (333) gives ξ = dim A ζclζ

2A By rescaling the comultiplication and the counit

of A and consequently changing ζA we can always achieve ξ = 1 This proves part (i)of the theorem

Equation (348) implies that ιcl-op = e fcl Since fcl is an isomorphism of Frobeniusalgebras by Lemmas 218 and 319 we have

ιcl-op ιlowastcl-op = e fcl f lowastcl elowast = ζA e r = Pl

R(A) (350)

Thus (A|Acl ιcl-op) is a Cardy algebra This proves part (ii) of the theoremPart (iii) can be seen as follows By [KR Prop 43] T Z(A) sim= oplusκisinJ Mor

κ otimesA Mκ asalgebras Together with the observation that T ( fcl) T (Acl) rarr T Z(A) is an isomor-phism of algebras this proves part (iii) Remark 323 Part (i) of Theorem 322 was announced by Muumlger [Muuml2] We provide anindependent proof in the setting of Cardy algebras

The above theorem together with Lemma 211 and Theorem 34 shows that a sim-ple commutative symmetric Frobenius C2plusmn-algebra Acl with dim Acl = Dim C is alwayspart of a Cardy algebra (Aop|Acl ιcl-op) for some simple special symmetric Frobeniusalgebra Aop in C However the above proof also illustrates that Aop is not unique Thisraises the question how two Cardy algebras with a given Acl can differ This question isanswered by [KR Thm 11] which in the present framework can be restated as follows

Cardy Algebras and Sewing Constraints I 905

Theorem 324 If (A(i)op |A(i)cl ι(i)cl-op) i = 1 2 are two Cardy C|C2plusmn-algebras such that

A(i)cl is simple and dim A(i)op = 0 for i = 1 2 then A(1)clsim= A(2)cl as algebras if and only

if A(1)op and A(2)op are Morita equivalent

Proof Theorem 11 in [KR] is stated for A(i)op being non-degenerate algebras and A(i)cl =Z(A(i)op ) for i = 1 2 By Proposition 316 A(i)op are simple and special for i = 1 2

Then by [KR Lem 21] A(i)op are non-degenerate algebras By Theorem 321 we have

A(i)clsim= Z(A(i)op ) as Frobenius algebras Finally by [KR Thm 11] Z(A(1)op ) sim= Z(A(2)op )

as algebras iff A(1)op and A(2)op are Morita equivalent Let Cmax(C2plusmn) be the set of equivalence classes [B] of simple modular invariant

commutative symmetric Frobenius algebras B in C2plusmn Two such algebras B and B prime areequivalent if B and B prime are isomorphic as algebras (but not necessarily as Frobenius alge-bras) Let Msimp(C) be the set of Morita classes of simple special symmetric Frobeniusalgebras in C Define the map z Msimp(C) rarr Cmax(C2plusmn) by z A rarr [Z(A)] whereA denotes the Morita class of A From Theorem 322 (i) and [KR Thm 11] we learn

Corollary 325 The map z Msimp(C) rarr Cmax(C2plusmn) is a bijection

A Appendix

A1 Proof of Lemma 27 We will show that if (F ψ F2 ψ

F0 ) is a colax tensor functor

from C1 to C2 then (G φG2 φ

G0 ) is a lax tensor functor from C2 to C1 Applying this

result to the opposed categories then gives the converse statementWe need to show that φG

0 and φG2 make the diagrams (21) and (22) commute We

first prove the commutativity of (21) Consider the following diagram

G(A)otimes (G(B)otimes G(C))

GψF2 δ

idG(A)otimesφG2 G(A)otimes G(B otimes C)

GψF2 δ

G(FG(A)otimes F(G(B)otimes G(C)))

G(F(idG(A))otimesF(φG2 ))

G(idFG(A)otimesψ F2 )

G(FG(A)otimes FG(B otimes C))

G(ρAotimesρBotimesC )

G(FG(A)otimes (FG(B)otimes FG(C)))

G(ρAotimes(ρBotimesρC )) G(A otimes (B otimes C))

(A1)

The top subdiagram is commutative because of the naturality of Gψ F2 δ The commu-

tativity of the bottom subdiagram follows from the following identities

(ρB otimes ρC ) ψ F2 = (ρB otimes ρC ) ψ F

2 ρF Fδ

= ρBotimesC FG(ρB otimes ρC ) FG(ψ F2 ) Fδ

= ρBotimesC F(φG2 ) (A2)

as a map F(G(B) otimes G(C)) rarr B otimes C The commutativity of (A1) implies that thecomposition of maps in the left column in (21) can be replaced by

G(ρA otimes (ρB otimes ρC )) G((idFG(A) otimes ψ F

2 ) ψ F2

) δ (A3)

906 L Kong I Runkel

Similarly we can show that the composition of maps in the right column in (21) can bereplaced by

G((ρA otimes ρB)otimes ρC ) G((ψ F

2 otimes idFG(C)) ψ F2

) δ (A4)

Using the commutativity of (23) it is easy to see that (21) with the left and right col-umns of (21) replaced by (A3) and (A4) respectively is commutative Hence (21) iscommutative

Now we prove the commutativity of the first diagram in (22)

φG2 (φG

0 otimes idG(A))

(1)= G(ρ12 otimes ρA) Gψ F2 δ

[(Gψ F

0 δ11)otimes idG(A)

]

(2)= G(ρ12 otimes ρA) Gψ F2 G F(Gψ0 otimes idG(A)) G F(δ11 otimes idG(A)) δ

(3)= G(ρ12 otimes ρA) G(FG(ψ F0 )otimes idFG(A)) Gψ F

2 G F(δ11 otimes idG(A)) δ(4)= G(id12 otimes ρA) G

([ψ F

0 ρF(11) (Fδ)11] otimes idFG(A)

) Gψ F

2 δ(5)= G(id12 otimes ρA) G(ψ F

0 otimes idFG(A)) Gψ F2 δ

(6)= G(id12 otimes ρA) G(lminus1FG(A)) G F(lG(A)) δ

(7)= G(lminus1A ) GρA δG lG(A)

(8)= G(lminus1A ) lG(A) (A5)

where in step (1) we substituted the definition of φG0 φ

G2 given in (210) in step (2)

we used the naturality of δ in step (3) we used the naturality of Gψ F2 in step (4) we

switched the position between Gρ12 and G FG(ψ F0 ) and the position between Gψ2 and

G F(δ12 otimes idG(A)) using the naturality of ρ and FψG2 respectively in step (5) we applied

the second identity in (28) in step (6) we used (24) in step (7) we used the naturalityof lminus1 and δ in step (8) we used the first identity in (28)

The proof of the commutativity of the second diagram in (22) is similar Thus wehave shown that G is a lax tensor functor

A2 Proof of Lemma 320 To prepare the proof recall that for a given object B isin Cthe modular group P SL(2Z) acts on the space oplusi HomC(B otimes Ui Ui ) see eg [BKSect 31] and [K3 Eq (455)] We will only need the action of S and Sminus1 Let f isinoplusi HomC(B otimes Ui Ui ) Then

S oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A6)

Cardy Algebras and Sewing Constraints I 907

Sminus1 oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A7)

By Lemma 32 to establish that (337) is S-invariant it is enough to prove the identity(34) when f is given by (337) Using (A6) and (A7) we can see that Eq (34) simplysays that oplusi j [RHS of (34)] is invariant under the action of S times S Consider the elementg of oplus jkisinIHomC2plusmn(R(A)otimes (Uor

j times Uk)Uorj times Uk) given by

g =oplusjkisinI

sumα

R(A)

R(A)

R(A)

R(A)

Uorj times Uk

Uorj times Uk

α

α

(A8)

By the above arguments proving S-invariance of (337) is equivalent to proving invari-ance of g under the action of S times S

For i isin I we denote by gi the component of g in(oplus jisinIHomC(A otimes Uor

i otimes Uorj U

orj )

)otimes (opluskisinIHomC(Ui otimes UkUk))

We view the second Hom-space in the above tensor product as a Hom-space in C+ insteadof Cminus It is enough to show that gi is invariant under the action of S times Sminus1 Note thatthe action of Sminus1 in C+ is equivalent to that of S in Cminus

The morphism gi can be canonically identified with a bilinear pairing

( middot middot )i ⎛⎝oplus

jisinIHomC(Uor

j A otimes Uori otimes Uor

j )

⎞⎠

times(oplus

kisinIHomC(UkUi otimes Uk)

)minusrarr C (A9)

as follows For h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) we set

(h1 h2)i = (dim U j dim Uk)minus1 trUor

j timesUk[gi (h1 times h2)] (A10)

908 L Kong I Runkel

When substituting the explicit form of the product m R(A) of R(A) = (A times 1) otimes R(1)after a short calculation one finds

(h1 h2)i =sumα

1

dim U j

A

A

A

A

A A

A α

α

h1

h2

U jUk

Uorj

Ui

Uk (A11)

Here the top morphism PlR(A) has been simplified with the help of the identity

PR(A) m R(A) (PR(A) otimes PR(A))

=(oplus

iisinI((m A A)otimes idUor

i)times idUi

) m R(A) (PR(A) otimes PR(A)) (A12)

which can be checked by direct calculation along the same lines as in the proof of [FrLem 310]

The action of the modular transformation S on oplusiisinI(B otimes Ui Ui ) for B isin C nat-urally induces an action on oplusiisinI(Ui B otimes Ui ) [K3 Prop 514] which we denote bySlowast In the present case we get an action of Slowast on oplus jisinIHomC(Uor

j A otimes Uori otimes Uor

j ) and

opluskisinIHomC(UkUi otimes Uk) Then to show gi is invariant under the action of S times Sminus1

amounts to showing that

(h1 h2)i = ((Sminus1)lowast h1 Slowast h2)i (A13)

Cardy Algebras and Sewing Constraints I 909

for all h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) We have

((Sminus1)lowast h1 Slowast h2)i =sum

mnα

dim Un

DimC

A

A

A

A

A A

A

Uk

α

α

h1

h2

Ui

U j

UmUn

(A14)

Now drag the upper vertex indexed by α in the above graph along its Uorm -leg until it

meets the lower vertex also indexed by α then sum over α and m This gives

((Sminus1)lowast h1 Slowast h2)i =sum

n

dim Un

DimC

A

A

A

A

A A

A

Uk

Un

h1

h2

Ui

U j

(A15)

910 L Kong I Runkel

If we just look at the neighbourhood of the Un-loop in the above graph we see thefollowing subgraphs

sumn

dim Un

DimC

A Aor

Uorj U j

Uk Uork

Un=

sumα

1

dim U j

A Aor Uk Uork

Uorj U j

αα

(A16)

where we have applied [BK Cor 3111] Substituting this subgraph back to the originalgraph in (A15) we obtain

((Sminus1)lowast h1 Slowast h2)i =sumα

1

dim U j

A

A

A

A

A A

A

A

A

Uorj

α

α

h1

h2m A

Ui

U jUk

Uk

(A17)

The graph in (A17) is equal to that in (A11) In order to see this we first drag theldquobubblerdquo (m A A) along A lines and through the m A vertex (because m A A is abimodule map) until it reaches the lower-left leg of the upper vertex indexed by α Thendrag the m A vertex along the (red) dotted line in the above graph Finally we applythe associativity of A (A12) and [Fr Lem 311] Then we see that the graph in (A17)exactly matches with the one in (A11)

Acknowledgements We would like to thank the organisers of the Oberwolfach Arbeitsgemeinschaft ldquoAlge-braic structures in conformal field theoriesrdquo (April 2007) where this work was started for an inspiring meetingWe would further like to thank the Hausdorff Institute for Mathematics in Bonn and the organisers of the stim-ulating meeting ldquoGeometry and Physicsrdquo (May 2008) We are indebted to Alexei Davydov Jens FjelstadJuumlrgen Fuchs Yi-Zhi Huang Alexei Kitaev Urs Schreiber Christoph Schweigert Stephan Stolz and PeterTeichner for helpful discussions andor comments on a draft of this paper The research of IR was partiallysupported by the EPSRC First Grant EPE0050471 the PPARC rolling grant PPC5071451 and the MarieCurie network lsquoSuperstring Theoryrsquo (MRTN-CT-2004-512194)

Cardy Algebras and Sewing Constraints I 911

References

[AN] Alexeevski A Natanzon SM Noncommutative two-dimensional topological field theories andhurwitz numbers for real algebraic curves Sel Math New Ser 12 307ndash377 (2006)

[Bi] Bichon J Cosovereign hopf algebras J Pure Appl Alg 157 121ndash133 (2001)[BK] Bakalov B Kirillov AA Lectures on Tensor Categories and Modular Functors Providence

RI Amer Math Soc 2001[C1] Cardy JL Operator content of two-dimensional conformal invariant theories Nucl Phys

B 270 186ndash204 (1986)[C2] Cardy JL Boundary conditions fusion rules and the verlinde formula Nucl Phys B 324

581ndash596 (1989)[CL] Cardy JL Lewellen DC Bulk and boundary operators in conformal field theory Phys Lett

B 259 274ndash278 (1991)[DMZ] Dong C-Y Mason G Zhu Y-C Discrete series of the Virasoro algebra and the moonshine

module In Algebraic Groups and Their Generalizations Quantum and infinite-dimensional Meth-ods Proc Symp Pure Math 56 Part 2 Providence RI Amer MathSoc 1994 pp 295ndash316

[DP] Day B Pastro C Note on frobenius monoidal functors New York J Math 14 733ndash742 (2008)[ENO1] Etingof PI Nikshych D Ostrik V On fusion categories Ann Math 162 581ndash642 (2005)[ENO2] Etingof PI Nikshych D Ostrik V An analogue of radfordrsquos s4 formula for finite tensor

categories Int Math Research Notices 54 2915ndash2933 (2004)[Fe] Felder G Froumlhlich J Fuchs J Schweigert C Correlation functions and boundary conditions in

rational conformal field theory and three-dimensional topology Comp Math 131 189ndash237 (2002)[FHL] Frenkel IB Huang Y-Z Lepowsky J On axiomatic approaches to vertex operator algebras

and modules Mem Amer Math Soc 104 (1993)[Fj] Fjelstad J Fuchs J Runkel I Schweigert C Uniqueness of openclosed rational cft with

given algebra of open states Adv Theor Math Phys 12 1283ndash1375 (2008)[FK] Froumlhlich J King C The chern-simons theory and knot polynomials Commun Math

Phys 126 167ndash199 (1989)[Fr] Froumlhlich J Fuchs J Runkel I Schweigert C Correspondences of ribbon categories Adv

Math 199 192ndash329 (2006)[FRS] Fuchs J Runkel I Schweigert C Tft construction of rcft correlators I partition functions Nucl

Phys B 646 353ndash497 (2002)[FS] Fuchs J Schweigert C Category theory for conformal boundary conditions Fields Inst Com-

mun 39 25ndash71 (2003)[H1] Huang Y-Z Two-dimensional conformal geometry and vertex operator algebras Progress in

Mathematics Vol 148 Boston Birkhaumluser 1997[H2] Huang Y-Z Rigidity and modularity of vertex tensor categories Comm Contemp Math 10

871ndash911 (2008)[HK1] Huang Y-Z Kong L Open-string vertex algebra category and operads Commun Math

Phys 250 433ndash471 (2004)[HK2] Huang Y-Z Kong L Full field algebras Commun Math Phys 272 345ndash396 (2007)[HK3] Huang Y-Z Kong L Modular invariance for conformal full field algebras httparxivorgabs

math0609570v2[mathQA] 2006[HL] Huang Y-Z Lepowsky J Tensor products of modules for a vertex operator algebra and vertex

tensor categories In Lie Theory and Geometry in honor of Bertram Kostant ed R BrylinskiJ-L Brylinski V Guillemin V Kac Boston Birkhaumluser 1994 pp 349ndash383

[JS] Joyal A Street R Braided tensor categories Adv Math 102 20ndash78 (1993)[K1] Kong L Full field algebras operads and tensor categories Adv Math 213 271ndash340 (2007)[K2] Kong L Open-closed field algebras Commun Math Phys 280 207ndash261 (2008)[K3] Kong L Cardy condition for open-closed field algebras Commun Math Phys 283 25ndash92 (2008)[Ki] Kitaev A Private communication[KO] Kirillov AA Ostrik V On q-analog of mckay correspondence and ade classification of sl(2)

conformal field theories Adv Math 171 183ndash227 (2002)[KR] Kong L Runkel I Morita classes of algebras in modular tensor categories Adv Math 219

1548ndash1576 (2008)[Ld] Lauda AD Frobenius algebras and ambidextrous adjunctions Theo Appl Cat 16 84ndash

122 (2006)[Li1] Li H-S Regular representations of vertex operator algebras Commun Contemp Math 4

639ndash683 (2002)[Li2] Li H-S Regular representations and huang-lepowsky tensor functors for vertex operator

algebras J Alge 255 423ndash462 (2002)

Cardy Algebras and Sewing Constraints I 887

Thus χ is natural Let (γ RA )

lowast HomC2plusmn(R(A)M) rarr HomC2plusmn(R(A)M) denote the

pull-back of γ RA The isomorphism χ is equal to the composition of (γ R

A )lowast and the

following sequence of natural isomorphisms

HomC(A T (M)) = oplusn HomC(AMln otimes Mr

n)

sim= oplusni HomC(AMln otimes Ui )otimes HomC(Ui Mr

n)

sim= oplusni HomC(A otimes Uori Ml

n)otimes HomC(Ui Mrn) = HomC2plusmn(R(A)M) (246)

We have proved that both χ and χ are natural isomorphisms There are four natural transformations associated to χ and χ namely

idC2plusmnδminusrarr RT

ρminusrarr idC2plusmn and idCδminusrarr T R

ρminusrarr idC (247)

defined by for A isin C M isin C2plusmn

δM = χ(idT (M)) ρA = χminus1(idR(A))

ρM = χ (idT (M)) δA = χminus1(idR(A))(248)

They can be expressed graphically as follows with M = oplusNn=1 Ml

n times Mrn

δM =oplusni

sumα

Mln

Mln Mr

n Uori

αtimes

Ui

Mrn

α

ρA =oplusiisinI

A

A

Uori Ui

(249)

ρM =oplusni

sumα

Mln

Mln

Mrn Uor

i

α times

Ui

Mrn

α DimCdim Ui

δA =oplusiisinI

A

A Uori Ui

dim UiDimC

Note that

ρM δM = Dim C middot idM and ρA δA = idA (250)

Lemma 222 The functors T and R as maps on the sets of morphisms have left inversesand thus are injective

Proof Let f A rarr B be a morphism in C We define a map Q R HomC2plusmn(R(A)

R(B)) rarr HomC(A B) by f prime rarr ρB T ( f prime) δA Then we have

Q R R( f ) = ρB T R( f ) δA = ρB δB f = f (251)

888 L Kong I Runkel

where we used naturality of δ and (250) in the second and third equalities respectivelySo Q R is a left inverse of R on morphisms Thus R is injective on morphisms Similarlylet g M rarr N be a morphism in C2plusmn We define a map QT HomC(T (M) T (N )) rarrHomC2plusmn(M N ) by gprime rarr (Dim C)minus1 middot ρN R(gprime) δM Then we have

QT T (g) = (Dim C)minus1 middot ρN RT (g) δM = (Dim C)minus1 middot ρN δN g = g (252)

So QT is a left inverse of T on morphisms Thus T is injective on morphisms Using (29) and (250) one can express the two inverse maps χminus1 χminus1 as follows

for f isin HomC2plusmn(M R(A)) and g isin HomC2plusmn(R(A)M)

χminus1( f ) = ρ T ( f ) χminus1(g) = T (g) δ (253)

By Proposition 25 and Lemma 27 R is both a lax and colax tensor functor In particularφR

0 1 times 1 rarr R(1) is given by

φR0 = χ (ψT

0 ) = R(ψT0 ) δ1 times 1 = id1times1 (254)

and φR2 R(A)otimes R(B) rarr R(A otimes B) by φR

2 = R(ρA otimes ρB) R(ψT2 ) δ which can

be expressed graphically as

φR2 =

oplusi jkisinI

sumα

A

A

Uori B

B

Uorj

Uork

α

times

Ui U j

α

Uk

(255)

Similarly ψ R0 R(1) rarr 1 times 1 is given by

ψ R0 = ρ1 R(φT

0 ) = Dim C id1times1 (256)

and ψ R2 R(A otimes B) rarr R(A) otimes R(B) by ψ R

2 = ρ R(φT2 ) R(δA otimes δB) which in

graphical notation reads

ψ R2 =

oplusi jkisinI

sumα

A

A Uori

B

B Uorj

Uork

α

times

Ui U j

α

Uk

dim Ui dim U j

dim Uk Dim C

(257)

If C has more than one simple object then R does not take the tensor unit of C to thetensor unit of C2plusmn and so is clearly not a tensor functor However we will show that R isstill a Frobenius functor This will imply that if A is a Frobenius algebra in C then

R(A) = (R(A)m R(A) ηR(A)R(A) εR(A)) (258)

Cardy Algebras and Sewing Constraints I 889

is a Frobenius algebra in C2plusmn where the structure morphisms were given in (217) and(218) In the case A = 1 it was proved in [Muuml1 Prop 41] (see also [Fr Lem 619]and [K1 Thm 52]) that (258) is a commutative simple symmetric normalised-specialFrobenius algebra in C2plusmn In fact given a Frobenius algebra A in C it is straightforwardto verify that the structure morphisms in (258) are precisely those of (A times 1)otimes R(1)cf Sect 22

Proposition 223 (R φR2 φ

R0 ψ

R2 ψ

R0 ) is a Frobenius functor

Proof Using the explicit graphical expression of φR2 φ

R0 ψ

R2 ψ

R0 it is easy to see that

the commutativity of the diagrams (25) and (26) are equivalent to the statement thatR(1) with structure morphisms as in (258) is a Frobenius algebra in C2plusmn The latterstatement is true by [Muuml1 Prop 41]

From Lemma 220 and Proposition 223 we see that T and R take Frobenius alge-bras to Frobenius algebras The following two propositions show how the properties ofFrobenius algebras are transported

Proposition 224 Let A be a Frobenius algebra in C2plusmn Then T (A) is a Frobenius algebrain C and

(i) A is symmetric iff T (A) is symmetric(ii) A is (normalised-)special iff T (A) is (normalised-)special

Proof For part (i) write A as a direct sum oplusNn=1 Al

n times Arn Then the maps IT (A) I prime

T (A) T (Aor) rarr T (A)or defined in (220) are given by

IT (A) = I primeT (A) = oplusN

n=1c(Aln)

or(Arn)

or (259)

Therefore by (221) T (A) = primeT (A) is equivalent to T (A) = T (prime

A) Since byLemma 222 T is injective on morphisms this proves part (i) Part (ii) can be checkedin the same way for example the condition mT (A) T (A) = ζ idT (A) is easily checkedto be equivalent to T (m A A) = ζ T (idA) Proposition 225 Let A be a Frobenius algebra A in C Then R(A) is a Frobeniusalgebra in C2plusmn and

(i) A is symmetric iff R(A) is symmetric(ii) A is (normalised-)special iff R(A) is (normalised-)special

Proof Recall that the structure morphisms of the Frobenius algebra R(A) are equal tothose of (A times 1) otimes R(1) Using this equality part (i) and (ii) follow because R(1) issymmetric and normalised-special For example

m R(A) R(A) = [(m A A)times id1] otimes idR(1) = R(m A A) (260)

so that m R(A) R(A) = ζ idR(A) is equivalent to R(m A A) = ζ R(idA) which byLemma 222 is equivalent to m A A = ζ idA

The functor R has one additional property not shared by T namely R takes absolutelysimple algebras to absolutely simple algebras We will see explicitly in Sect 33 thatthis is not true for T

890 L Kong I Runkel

Lemma 226 For a C-algebra A the map

R HomA|A(A A) rarr HomR(A)|R(A)(R(A) R(A)) (261)

given by f rarr R( f ) is well-defined and an isomorphism

Proof Since R is a lax tensor functor R(A) is naturally a R(A)-bimodule It is easy tosee that R in (261) is a well-defined map R(A) is also naturally a R(1)-bimodule whichcan be identified with the induced R(1)-bimodule structure on (A times 1)otimes R(1) wherethe left R(1) action on (Atimes1)otimes R(1) is given by (idAtimes1 otimesm R(1))(cminus1

Atimes1R(1)otimes idR(1))We have the following natural isomorphisms

HomR(1)|R(1)(R(A) R(A))sim=minusrarr HomC2plusmn(A times 1 R(A))

χminus1

minusminusrarr HomC(A A) (262)

which by (253) are given by for f isin HomR(1)|R(1)(R(A) R(A))

f rarr f prime = f (idAtimes1 otimes ηR(1)) rarr f primeprime = ρ T ( f prime) (263)

and its inverse is given by for g isin HomC(A A)

g rarr gprime = R(g) δ rarr gprimeprime = (idAtimes1 otimes m R(1)) (gprime otimes idR(1)) (264)

where gprimeprime is indeed a R(1)-bimodule map due to the commutativity of R(1) It is easy tocheck that gprimeprime = R(g) in (264) Therefore R gives an isomorphism from HomC(A A)to HomR(1)|R(1)(R(A) R(A)) Moreover one verifies that R(g) is an R(A)-bimodulemap iff g is an A-bimodule map In other words R g rarr R(g) gives an isomorphism

HomA|A(A A)sim=minusrarr HomR(A)|R(A)(R(A) R(A))

Corollary 227 Let A be a C-algebra

(i) A is absolutely simple iff R(A) is absolutely simple(ii) Let A be in addition Frobenius Then A is simple and special iff R(A) is simple and

special

Proof Part (i) immediately follows from Lemma 226 The statement of part (ii) withoutthe qualifier lsquosimplersquo is proved in Proposition 225 But as in the proof of (iii)rArr(i) inLemma 211 a special Frobenius algebra in a semi-simple category has a semi-simplecategory of bimodules and for a semi-simple C-linear category simple and absolutelysimple are equivalent Part (ii) then follows from part (i)

The following lemma will be needed in Sect 32 below to discuss the properties ofCardy algebras

Lemma 228 Let A be a Frobenius algebra in C Then (δA)lowast = ρA

Proof Recall from (250) that δA is a morphism A rarr T R(A) Since T and R are bothFrobenius functors T R(A) is a Frobenius algebra in C Substituting the definitions aftera short calculation one finds

Cardy Algebras and Sewing Constraints I 891

εT R(A) mT R(A) = Dim(C)oplusiisinI

A Uori Ui A U or

ı Uı

(265)

Substituting this in the definition of (δA)lowast gives again after a short calculation the mor-

phism ρA At an intermediate step one uses that the part of the morphism (δA)lowast which

is made up of Ui and Uı ribbons their duals and the basis morphisms λ(iı)0 andϒ(iı)0can be replaced by 1

dim Uimiddot dUi

3 Cardy Algebras

In this section we start by investigating the properties of Frobenius algebras which sat-isfy the so-called modular invariance condition We then give two definitions of a Cardyalgebra and prove their equivalence Finally in Sect 33 we study the properties of thesealgebras and state our main results

We fix a modular tensor category C Recall that C2plusmn is an abbreviation for C+ Cminus

31 Modular invariance In C2plusmn we define the object K and the morphism ω K rarr Kas

K =oplus

i jisinIUi times U j ω =

sumi jisinI

dim Ui dim U j

Dim C idUi timesU j (31)

They have the property (see eg [BK Cor 3111])

ω

K

(Uk times Ul )or

(Uk times Ul )or

Ui times U j

Ui times U j

= δik δ jlDim C

dim Ui dim U jbUi timesU j dUi timesU j (32)

892 L Kong I Runkel

Definition 31

(i) Let A B be objects of C2plusmn A morphism f A otimes B rarr B is called S-invariant iff

A

B

W

W

f =

A

B

W

W

f

K

ω(33)

holds for all for W isin C2plusmn(ii) A C2plusmn-algebra (Aclmcl ηcl) is called modular invariant iff θAcl = idAcl and mcl

is S-invariant

Lemma 32 The morphism f A otimes B rarr B is S-invariant if and only if

A

B

Ui times U j

Ui times U j

f = Dim Cdim Ui dim U j

sumα

A

B

B

f

Ui times U j

Ui times U j

α

α

(34)

holds for all i j isin I

Proof Condition (33) holds for all W iff it holds for all W = Ui times U j i j isin I so itis enough to show that the right-hand side of (33) with W = Ui times U j is equal to theright-hand side of (34) Recall the notation for basis morphisms in (236) Starting from(33) write

idBor otimes idUi timesU j =⎛⎝sum

klα

(b(ktimeslα)

B

)or (

bB(ktimeslα)

)or⎞⎠ otimes idUi timesU j (35)

and then apply (32) The graphical representation of the resulting morphism can bedeformed to give (34) Remark 33 As shown in [K3 Sect 61] the modular invariance condition of a C2plusmn-alge-bra exactly coincides with the modular invariance condition for torus 1-point correlationfunctions of a genus-01 closed CFT In particular the condition θAcl = idAcl is equiv-alent to invariance under the modular transformation T τ rarr τ + 1 and the condition(34) with f = mcl is equivalent to invariance under S τ rarr minus 1

τ Combining the

modular invariance condition with the genus zero properties of a genus-01 closed CFTresults in a modular invariant commutative symmetric Frobenius algebra in C2plusmn

Cardy Algebras and Sewing Constraints I 893

Let Acl be a modular invariant C2plusmn-algebra Evaluating (34) for f = mcl composingit with ηcl otimes idUi timesU j and taking the trace implies the following identity

Zi j = 1

DimCAclUi timesU j where

(36)

Zi j = dimC HomC2plusmn(UitimesU j Acl)

Decomposing Acl into simple objects this gives

Zi j =sum

klisinISik Zkl Sminus1

l j where Si j = si jradic

DimC (37)

which in CFT terms is of course nothing but the invariance of the torus partition functionunder the modular S-transformation

The following theorem gives a simple criterion for modular invariance

Theorem 34 Let Acl be a haploid commutative symmetric Frobenius algebra in C2plusmn

(i) If Acl is modular invariant then dim Acl = Dim C(ii) If dim Acl = Dim C then Acl is special and modular invariant

Proof Part (i) Since Acl is haploid for i = j = 0 Equation (36) reduces to 1 =dim AclDimCPart (ii) By the same reasoning as in the proof of (iii)rArr(i) in Lemma 211 one showsthat Acl is special Thus mcl cl = ζclidAcl for some ζcl = 0

By [KO Thm 45] the category (C2plusmn)locAcl

of local Acl-modules is again a mod-

ular tensor category and Dim (C2plusmn)locAcl

= (Dim C dim Acl)2 (see [Fr Prop 321 amp

Rem 323] for the same statement in the notation used here) Thus by assumption wehave Dim(C2plusmn)loc

Acl= 1 It then follows from [ENO1 Thm 23] that up to isomorphism

(C2plusmn)locAcl

has a unique simple object (namely the tensor unit) In other words every simplelocal Acl-module is isomorphic to Acl (seen as a left-module over itself)

We have the following isomorphisms between morphism spaces [FS Prop 47 amp 411]

HomAcl(Acl otimes (Ui times U j ) Acl) sim= HomC2plusmn(Ui times U j Acl)

HomAcl(Acl Acl otimes (Ui times U j )) sim= HomC2plusmn(AclUi times U j )

Using these to transport the bases (236) from the right to the left we obtain bases bα(i j)αof HomAcl(Acl otimes (Ui times U j ) Acl) and b(i j)

β β of HomAcl(Acl Acl otimes (Ui times U j )) Thesecan be expressed graphically as

894 L Kong I Runkel

bα(i j) =

Acl

Acl

Acl

Ui times U j

α

b(i j)β = DimC

dim Ui dim U j

1

ζcl

Acl

Ui times U jAcl

Acl

β

(38)

where the nonzero factor in b(i j)β is included for convenience Notice that bα(i j) b(i j)

β isa left Acl-module map Since Acl is simple as a left module over itself we have

bα(i j) b(i j)β = λαβ idAcl (39)

for some λαβ isin C By computing tr(bα(i j) b(i j)β ) it is easy to verify that λαβ = δαβ

We will now prove the following identity

1

ζcl

Acl

Acl

Ui times U j

Ui times U j

Acl

=sumα

DimCdim Ui dim U j

1

ζcl

Acl

Ui times U j

Ui times U j

Acl

Acl

Acl

α

α

(310)

One checks that the left-hand side of this equation is an idempotent which we denote byζminus1

cl PlAcl(Ui timesU j ) cf [Fr Sect 31] By [Fr Prop 41] the image Im(ζminus1

cl PlAcl(Ui timesU j ))

is a local Acl-module and hence isomorphic to AoplusNcl for some N isin Zge0

All left-module morphisms from Acl to Aclotimes(Ui timesU j ) are linear combinations of the

b(i j)β Furthermore one verifies that ζminus1

cl PlAcl(Ui timesU j )b(i j)

β = b(i j)β Therefore the b(i j)

β

describe precisely the image of the idempotent ie ζminus1cl Pl

Acl(Ui timesU j ) = sum

α b(i j)α bα(i j)

which is nothing but (310) Composing (310) with ζcl middot εcl otimes idUi timesU j from the left (iefrom the top) produces (34)

In addition since Acl is commutative symmetric Frobenius it satisfies θAcl = idAcl

[Fr Prop 225] Altogether this shows that Acl is modular invariant Remark 35 As we were writing this paper we heard that the results in Theorem 34were obtained independently by Kitaev and Muumlger [Ki]

Remark 36 Setting i = j = 0 in (36) gives the identity dim Acl = Z00 Dim C [KRProp 23] Combining this with Theorem 34 (ii) one may wonder if a general modularinvariant commutative symmetric Frobenius algebra Acl in C2plusmn is isomorphic to a directsum of simple such algebras However this is not so For example one can take the com-mutative symmetric Frobenius algebra Acl = C[x]x2 in the category of vector spacesequipped with the non-degenerate trace ε(ax + b) = a In this case the modular invari-ance condition holds automatically but Acl is clearly not a direct sum of two algebras

Cardy Algebras and Sewing Constraints I 895

For a general modular tensor category C the algebra C[x]x2 R(1) understood as analgebra in C2plusmn via the braided monoidal isomorphism Vect f (C) C2plusmn rarr C2plusmn providesanother counter-example

32 Two definitions Define a morphism PlA A rarr A for a Frobenius algebra A in C

or C2plusmn as follows [Fr Sect 24]

PlA =

A

A

A

m

(311)

If A is also commutative and obeys m A A = ζA idA we have PlA = ζA idA In partic-

ular this holds if A is commutative and special Using the fact that the Frobenius algebraR(1) is commutative and normalised-special one can check that Pl

R(A) R(A) rarr R(A)takes the following form

PlR(A) =

oplusiisinI

A

A

Uori

Uori

A

m

times

Ui

Ui

(312)

With these ingredients we can now give the first definition of a Cardy C|C2plusmn-algebrawhich was introduced in [K3 Def 514] cf Remark 315 below

Definition 37 (CardyC|C2plusmn-algebra I) A CardyC|C2plusmn-algebra is a triple (Aop|Acl ιcl-op)where (Aclmcl ηclcl εcl) is a modular invariant commutative symmetric Frobe-nius C2plusmn-algebra (Aopmop ηopop εop) is a symmetric Frobenius C-algebra andιcl-op Acl rarr R(Aop) an algebra homomorphism such that the following conditionsare satisfied

(i) Centre condition

Acl R(Aop)

R(Aop)

R(Aop)

ιcl-op

m R(Aop)

=

Acl

ιcl-op

R(Aop)

R(Aop)

R(Aop)

m R(Aop)

(313)

896 L Kong I Runkel

(ii) Cardy condition

ιcl-op ιlowastcl-op =

R(Aop)

R(Aop)

R(Aop)

(314)

Remark 38

(i) The name ldquoCardy C|C2plusmn-algebrardquo in Definition 37 was chosen because many ofthe important ingredients were first studied by Cardy the modular invariance ofthe closed theory [C1] the consistency of the annulus amplitude [C2] and thebulk-boundary OPE [CL] On the other hand the boundary-boundary OPE andthe OPE analogue of the centre condition were first considered in [Lw]

(ii) One can easily see that in the special case that C is the category Vect f (C) of finite-dimensional C-vector spaces a Cardy C|C2plusmn-algebra gives exactly the algebraicformulation of two-dimensional open-closed topological field theory over C (cfRemark 614 in [K3]) see [Lz Sect 48] [Mo Thm 11] [AN Thm 45] [LPCor 43] [MS Sect 22] When passing to a general modular tensor category Cthere are two important differences to the two-dimensional topological field the-ory Firstly the algebras Acl and Aop now live in different categories which inparticular affects the formulation of the centre condition and the Cardy conditionSecondly the modular invariance condition has to be imposed on Acl In the caseC = Vect f (C) modular invariance holds automatically

Definition 39 A homomorphism of Cardy C|C2plusmn-algebras (A(1)op |A(1)cl ι(1)cl-op) rarr (A(2)op

|A(2)cl ι(2)cl-op) is a pair ( fop fcl) of Frobenius algebra homomorphisms fop A(1)op rarr A(2)op

and fcl A(1)cl rarr A(2)cl such that the diagram

A(1)cl

fcl

ι(1)cl-op

A(2)cl

ι(2)cl-op

R(A(1)op )

R( fop) R(A(2)op )

(315)

commutes

Remark 310 Since a homomorphism of Frobenius algebras is invertible (cf Lemma218 (iv)) a homomorphism of Cardy algebras is always an isomorphism

For a homomorphism ( fop fcl) of Cardy C|C2plusmn-algebras using the commutativity of(315) and the fact that fcl and fop are both algebra and coalgebra homomorphisms itis easy to show that (315) commutes iff

R(A(1)op )R( fop)

(ι(1)cl-op)

lowast

R(A(2)op )

(ι(2)cl-op)

lowast

A(1)cl

fcl A(2)cl

(316)

Cardy Algebras and Sewing Constraints I 897

commutesLet (Aop|Acl ιcl-op) be a Cardy C|C2plusmn-algebra Define the morphism

ιcl-op = χminus1(ιcl-op) T (Acl) minusrarr Aop (317)

Decompose Acl as Acl = oplusNn=1Cl

n times Crn such that Cl

1 times Cr1 = ηcl(1 times 1) We use ι(n)cl-op

to denote the restriction of ιcl-op to Cln times Cr

n and ι(n)cl-op to denote the restriction of ιcl-op

to Cln otimes Cr

n We introduce the following graphical notation

ι(n)cl-op =

Cln Cr

n

Aop

(318)

By (243) ιcl-op can be expressed in terms of ιcl-op as follows

ιcl-op =Noplus

n=1

oplusiisinI

sumα

Cln

Crn

Aop

α

Uori

times α

Crn

Ui

(319)

Lemma 311 The centre condition (313) is equivalent to the following condition in C

Cln Cr

n

Aop

Aop

Aop

=

Cln Cr

n

Aop

Aop

Aop

for n = 1 N (320)

Proof First insert (319) and the definition (217) of m R(Aop) into (313) Then apply thecommutativity of R(1) to the left hand side of (313) The equivalence between (313)and (320) follows immediately Remark 312 The centre condition (320) is very natural from the open-closed confor-mal field theory point of view Correlators on the upper half plane are expressed in termsof conformal blocks on the full complex plane The objects Cl

n and Crn are associated to

the field insertion at a point z in the upper half plane and at the complex conjugate pointz in the lower half plane respectively The object Aop corresponds to a field inserted ata point r on the real axis The centre condition (320) simply says that the correlationfunctions in the disjoint domains |z| gt r gt 0 and r gt |z| gt 0 are analytic continuationsof each other see [K2 Prop 118]

898 L Kong I Runkel

Recall that we define ιlowastcl-op Aop rarr T (Acl) as in (230) We introduce the graphicalnotation

ιlowastcl-op =Noplus

n=1

Cln Cr

n

Aop

=Noplus

n=1

Cln Cr

n

Aop

Aop T (minus1Acl) (321)

where the second equality follows from (215) (221) and (259)

Lemma 313 The Cardy condition (314) is equivalent to the following identity in C

Noplusn=1

DimCdim Ui

sumα

Cln

Crn

Crn

α

α

Aop

Aop Uori

Uori

=Aop

Aop

Aop

Uori

Uori

op

mop for all i isin I (322)

Proof By (244) (312) and (319) it is easy to see that (322) is equivalent to thefollowing identity

ιcl-op χ(ιlowastcl-op) = PlR(Aop)

(323)

Therefore it is enough to show that

χ(ιlowastcl-op) = ιlowastcl-op (324)

We have

χminus1(ιlowastcl-op)(1)= T (ιlowastcl-op) δ (2)= T (ιcl-op)

lowast δlowastlowast (3)= (ρ T (ιcl-op)

)lowast (4)= (ιcl-op

)lowast (325)

In step (1) we use the expression (253) for χminus1 step (2) follows from Lemma 218 (v)and Lemma 219 Step (3) is Lemma 218 (i) and Lemma 228 and finally step (4)amounts to substituting (253) and (317) Acting with χ on both sides of the aboveequality produces (324)

Combining Lemmas 311 and 313 and Proposition 216 we obtain the followingequivalent definition of Cardy C|C2plusmn-algebra (recall the graphical notation (318) forιcl-op and (321) for ιlowastcl-op)

Cardy Algebras and Sewing Constraints I 899

Definition 314 (CardyC|C2plusmn-algebra II) A CardyC-algebra is a triple (Aop|Acl ιcl-op)where Acl is a commutative symmetric Frobenius C2plusmn-algebra satisfying property (34)with f = mcl Aop is a symmetric Frobenius C-algebra and ιcl-op T (Acl) rarr Aop isan algebra homomorphism satisfying the conditions (320) and (322)

Remark 315 Up to a choice of normalisation Definition 314 is the same as the orig-inal one in [K3 Def 613] The difference between the two definitions is the factorDim C dim Ui on the left hand side of (322) which in [K3 Def 613] is given byradic

Dim C dim Ui The two definitions are related by rescaling the coproduct cl andcounit εcl of Acl by 1

radicDim C and

radicDim C respectively We chose the convention in

(322) to remove all dimension factors from the expression (314) for the Cardy condition

33 Uniqueness and existence theorems In this subsection we investigate the structureof Cardy algebras We start with the following proposition which when combined withthe results of part II provides an alternative proof of [Fj Prop 422]

Proposition 316 Let (Aop|Acl ιcl-op) be a Cardy C|C2plusmn-algebra If Acl is simple anddim Aop = 0 then Aop is simple and special

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis therefore haploid Restricting the Cardy condition (322) to the case Ui = 1 andcomposing both sides with εop from the left we see that εop kills all terms associated toU j times 1 isin Acl in the sum except for a single 1 times 1 term Thus we obtain the followingidentity

β εop = dAop (mop otimes idAorop) (idAop otimes bAop) (326)

where β isin C Composing with ηop from the right in turn implies that βεop ηop =dim Aop which is nonzero by assumption Thus also β = 0 and εop is a nonzero multipleof the morphism on the right hand side of (326) By [FRS Lem 311] Aop is special

Since Aop is a special Frobenius algebra Aop is semi-simple as an Aop-bimodule(apply [FS Prop 524] to Aop tensored with its opposite algebra) Suppose Aop is not

simple so that we can write Aop = A(1)op oplus A(2)op for nonzero Aop-bimodules A(1)op and A(2)op We denote the canonical embeddings and projections associated to this decompositionas ι12 and π12 We have the identities

mop (ι1 otimes ι2) = 0 εop ηop =2sum

i=1

εop ιi πi ηop (327)

The first identity follows since π1 mop (ι1 otimes ι2) = 0 (as mop gives the left action of

Aop on Aop and hence it preserves A(2)op ) and similarly π2 mop (ι1 otimes ι2) = 0 Thesecond identity is just the completeness of ι12 π12

Since εop ηop = 0 without losing generality we can assume εop ι1 π1 ηop = 0Using that π2 is a bimodule map we compute

π2 [LHS of (322)]Ui =1 ι1 π1 ηop = π2 [RHS of (322)]Ui =1 ι1 π1 ηop

= PlA(2)op

π2 ι1 π1 ηop = 0 (328)

900 L Kong I Runkel

On the other hand using that Acl is haploid that ιcl-op is an algebra map and thatι lowastcl-op is a coalgebra map one can check that the left-hand side of (328) is equal toλ(εop ι1 π1 ηop) π2 ηop for some λ = 0 This implies that π2 ηop = 0 Thusηop = sum

i ιi πi ηop = ι1 π1 ηop Hence we have

0 = π2 ι2 = π2 mop (ηop otimes ι2) = π2 mop ((ι1 π1 ηop)otimes ι2) (329)

However the right-hand side is zero by (327) This is a contradiction and hence Aopmust be simple

To formulate the next theorem we need the notion of the full centre of an algebra [FjDef 49] Recall that an algebra A in a braided tensor category has a left centre and aright centre [VZO] both of which are sub-algebras of A Of these two we will onlyneed the left centre The following definition is [Fr Def 231] which in our setting isequivalent to that of [VZO]

Definition 317 Let A be a symmetric special Frobenius algebra such that m A A =ζA idA

(i) The left centre Cl(A) of A is the image of the idempotent ζminus1A Pl

A(ii) The full centre Z(A) is Cl(R(A))

That ζminus1A Pl

A is an idempotent follows from [FRS Lem 52] when keeping track ofthe factors ζA ([FRS] assumes normalised-special ie ζA = 1) Note that Cl(A) is againan object of C while Z(A) is an object of C2plusmn Let el

A Cl(A) rarr A be the embeddingof Cl(A) into A The left centre is in fact the maximal subobject of A such that

m A cAA (elA otimes idA) = m A (330)

see [Fr Lem 232] This observation explains the name left centre and also makes theconnection to [O Def 15]

The full centre is by definition the image of the idempotent ζminus1A Pl

R(A) R(A) rarrR(A) Since C2plusmn is abelian the idempotent splits and we obtain the embedding andrestriction morphisms

e Z(A) rarr R(A) and r R(A) Z(A) (331)

which obey r e = idZ(A) and e r = ζminus1A Pl

R(A) It follows from Proposition 225 and

[Fr Prop 237] that Z(A) is a commutative symmetric Frobenius algebra in C2plusmn withstructure morphisms5

m Z(A) = r m R(A) (e otimes e) ηZ(A) = r ηR(A)

Z(A) = ζA middot (r otimes r) R(A) e εZ(A) = ζminus1A middot εR(A) e

(332)

Moreover if A is simple then Z(A) is simple and if A is simple and dim A = 0 thenZ(A) is simple and special The normalisation of the counit is such that

εZ(A) ηZ(A) = ζminus2A dim A DimC (333)

5 The normalisation of product and unit is the standard one The factors in the coproduct and counit haveto be included in order for (A|Z(A) e) to be a Cardy algebra see Theorem 318 below The normalisation ofthe counit enters the Cardy condition (314) through the definition of ( middot )lowast

Cardy Algebras and Sewing Constraints I 901

Theorem 318 Let A be a special symmetric Frobenius C-algebra Then (A|Z(A) e)is a Cardy C|C2plusmn-algebra

The proof of this theorem makes use the following two lemmas

Lemma 319 e Z(A) rarr R(A) is an algebra map and elowast = ζA middot r

Proof It follows from [Fr Lem 229] (or by direct calculation using in particularm R(A) R(A) = ζAidR(A)) that

m R(A) (e otimes e) = ζminus1A middot Pl

R(A) m R(A) (e otimes e) (334)

Substituting e r = ζminus1A Pl

R(A) shows that e is compatible with multiplication For theunit one finds

e ηZ(A) = e r ηR(A) = ζminus1A Pl

R(A) ηR(A) = ηR(A) (335)

Thus e is an algebra map For the second statement one computes

elowast (1)= ζA

R(A)

r

R(A)

R(A)

Z(A)

R(A)

Z(A)

r

e

e

(2)=

R(A)

r

R(A)

R(A)

R(A)

Z(A)

(3)=

R(A)

r

R(A)

Z(A)

(4)= ζA middot r (336)

where in (1) the definitions (230) and (332) have been substituted step (2) is e r = ζminus1

A PlR(A) step (3) uses that R(A) is symmetric Frobenius and step (4) is again

e r = ζminus1A Pl

R(A) Lemma 320 Let A be a symmetric Frobenius algebra in C The morphism

PlR(A) m R(A)

(Pl

R(A) otimes PlR(A)

) R(A)otimes R(A) minusrarr R(A) (337)

is S-invariant

The proof of this lemma is a slightly lengthy explicit calculation and has been deferredto Appendix A2

902 L Kong I Runkel

Proof of Theorem 318 That e is an algebra map was proved in Lemma 319 The centrecondition (313) holds by property (330) of the left centre The Cardy condition (314)also is an immediate consequence of Lemma 319

ιcl-op ιlowastcl-op = e (ζA r) = PlR(A) (338)

The full centre Z(A) is a commutative symmetric Frobenius algebra It remains to provemodular invariance That θZ(A) = idZ(A) is implied by commutativity and symmetryof Z(A) [Fr Prop 225] The S-invariance condition (33) follows from Lemma 320In (337) substitute Pl

R(A) = ζA e r and then put the resulting morphism into (33)Compose the resulting equation with e otimes idW from the right (ie from the bottom) andsubstitute the definition (332) of m Z(A) This results in the statement that m Z(A) isS-invariant

The following theorem is analogous to [LR Prop 29] and [Fj Thm 426] whichroughly speaking answer the question under which circumstances the restriction of atwo-dimensional conformal field theory to the boundary already determines the entireconformal field theory The first work is set in Minkowski space and uses operator alge-bras and subfactors while the second work is set in Euclidean space and uses modulartensor categories

Theorem 321 Let (A|Acl ιcl-op) be a Cardy C|C2plusmn-algebra such that dim A = 0 andAcl is simple Then A is special and (A|Acl ιcl-op) sim= (A|Z(A) e) as Cardy algebras

Proof By Proposition 316 A is simple and special Since Acl is simple the algebramap ιcl-op Acl rarr R(A) is either zero or a monomorphism But ιcl-op ηcl = ηR(A)and so ιcl-op = 0 Thus ιcl-op is monic By Lemma 218 (ii) ζminus1

A ιlowastcl-op is epi The Cardycondition (314) implies

ιcl-op ζminus1A ιlowastcl-op = ζminus1

A PlR(A) = e r (339)

Composing this with e r from the left yields e r ιcl-op ζminus1A ιlowastcl-op = e r =

ιcl-op ζminus1A ιlowastcl-op Since ζminus1

A ιlowastcl-op is epi we have

e r ιcl-op = ιcl-op (340)

Actually (340) also follows from (313) and specialness of R(A) We will prove that

( fop fcl) (A|Acl ιcl-op) minusrarr (A|Z(A) e) where fop = idA fcl = r ιcl-op

(341)

is an isomorphism of Cardy algebrasfcl is an algebra map Compatibility with the units follows since ιcl-op is an algebramap

fcl ηcl = r ιcl-op ηcl = r ηR(A) = ηZ(A) (342)

Compatibility with the multiplication also follows since ιcl-op is an algebra map

m Z(A) ( fcl otimes fcl) = r m R(A) (e otimes e) (r otimes r) (ιcl-op otimes ιcl-op)

= r m R(A) (ιcl-op otimes ιcl-op) = r ιcl-op mcl = fcl mcl (343)

where in the second step we used (340)

Cardy Algebras and Sewing Constraints I 903

fcl is an isomorphism As above since fcl is an algebra map and since Acl is simple fcl

has to be monic By Lemma 319 rlowast = ζminus1A e Thus f lowast

cl = ιlowastcl-op rlowast = ζminus1A ιlowastcl-op e and

fcl f lowastcl = r ιcl-op ζminus1

A ιlowastcl-op e = r e r e = idZ(A) (344)

and so fcl is also epi and hence isofcl is a coalgebra map Since fcl is an algebra map so is f minus1

cl By (344) f minus1cl = f lowast

cland by Lemma 218 (iii) this implies that fcl is a also coalgebra mapThe diagram (315) commutes Commutativity of (315) is equivalent to e fcl = ιcl-opwhich holds by (340)

Let A be a special symmetric Frobenius algebra So far we have seen that (A|Z(A) e)is a Cardy algebra and that all Cardy algebras with Aop = A and simple Acl are of thisform It is now natural to ask if every simple Acl does occur as part of a Cardy algebraThe following theorem provides an affirmative answer Recall that for an A-left moduleM the object Mor otimesA M is an algebra (see eg [KR Lem 42])

Theorem 322 If Acl is a simple modular invariant commutative symmetric FrobeniusC2plusmn-algebra then there exist a simple special symmetric Frobenius C-algebra A and amorphism ιcl-op Acl rarr R(A) such that

(i) Acl sim= Z(A) as Frobenius algebras(ii) (A|Acl ιcl-op) is a Cardy C|C2plusmn-algebra

(iii) T (Acl) sim= oplusκisinJ Morκ otimesA Mκ as algebras where MκκisinJ is a set of representa-

tives of the isomorphism classes of simple A-left modules

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis haploid It then follows from Theorem 34 that Acl is special By Proposition 224T (Acl) is a special symmetric Frobenius algebra in C Thus T (Acl) = oplusi Ai where theAi are simple symmetric Frobenius algebras We will show that at least one of the Aiis special Since T (Acl) is special we have mT (Acl) T (Acl) = ζ idT (Acl) for someζ isin C

times Restricting this to the summand Ai shows mi i = ζ idAi FurthermoreεT (Acl) ηT (Acl) = ξ id1 for some ξ isin C

times But εT (Acl) ηT (Acl) = sumi εi ηi and so

at least one of the εi ηi has to be nonzero Therefore at least one of the Ai is speciallet A equiv Ai be this summand We denote the embedding A rarr T (Acl) by e0 and therestriction T (Acl) A by r0 Notice that r0 is an algebra homomorphism Define

ιcl-op = χ (r0) Acl minusrarr R(A) (345)

By Proposition 216 ιcl-op is an algebra homomorphism Next we verify the centre con-dition (313) or rather its equivalent form (320) By substituting the definitions onecan convince oneself that the commutativity mcl cAclAcl = mcl of Acl in C2plusmn implies thecondition mT (Acl) = mT (Acl) in C see [K2 Prop 36] Here T (Acl)otimes T (Acl) rarrT (Acl)otimes T (Acl) is given by

=oplusmn

(idCl

notimes cCl

m Crnotimes idCr

m

)

(cCl

m Clnotimes cminus1

Crn C

rm

)

(idCl

motimes cminus1

Cln C

rm

otimes idCrn

)

(346)

and we decomposed Acl as Acl = oplusnCln times Cr

n As a consequence we obtain the identity

r0 mT (Acl) (idAcl otimes e0) = r0 mT (Acl) (idAcl otimes e0) (347)

904 L Kong I Runkel

Using that r0 is an algebra map and that by definition ιcl-op = r0 we obtain (320)In order to show that (A|Acl ιcl-op) is a Cardy algebra it remains to show that the

Cardy condition (314) is satisfied We will demonstrate this via a detour by first provingthat Acl sim= Z(A) as Frobenius algebras

Recall the notations e and r given in (331) Using the centre condition (320) onecan check that Pl

R(A) ιcl-op = m R(A) R(A) ιcl-op By specialness of A we have

m A A = ζA idA and so together with e r = ζminus1A Pl

R(A) we get

e r ιcl-op = ιcl-op (348)

Next consider the morphism

fcl = r ιcl-op Acl minusrarr Z(A) (349)

By the same derivation as in (342) and (343) one sees that fcl is an algebra map Inparticular fcl ηcl = ηZ(A) = 0 and so fcl = 0 Since Acl is simple fcl has to be amonomorphism

By the same argument as used in the proof of Theorem 34 (ii) up to isomorphismAcl is the unique simple local Acl-(left-)module The algebra monomorphism fcl turnsZ(A) into an Acl-module Since Z(A) is commutative it is local as an Acl-module andso Z(A) sim= AoplusN

cl for some N ge 1 By construction A is a simple special symmetricFrobenius algebra Proposition 225 and Corollary 227 show that R(A) inherits all theseproperties and thus Z(A) is simple (see the comment below Eq (332)) By Theorem318 Z(A) is modular invariant and then by Theorem 34 (i) dim Z(A) = Dim C Thisimplies that N = 1 in Z(A) sim= AoplusN

cl and so fcl is in fact an isomorphismSince Acl and Z(A) are both haploid we have εZ(A) fcl = ξ εcl for some ξ isin C

timesThe counit uniquely determines the Frobenius structure on Acl and Z(A) (see eg [FRSLemma 37]) so that fcl is a coalgebra isomorphism iff ξ = 1 To compute ξ we composethe above identity with ηcl from the right Defining ζcl via εcl ηcl = ζminus1

cl Dim C middot id1 andusing (333) gives ξ = dim A ζclζ

2A By rescaling the comultiplication and the counit

of A and consequently changing ζA we can always achieve ξ = 1 This proves part (i)of the theorem

Equation (348) implies that ιcl-op = e fcl Since fcl is an isomorphism of Frobeniusalgebras by Lemmas 218 and 319 we have

ιcl-op ιlowastcl-op = e fcl f lowastcl elowast = ζA e r = Pl

R(A) (350)

Thus (A|Acl ιcl-op) is a Cardy algebra This proves part (ii) of the theoremPart (iii) can be seen as follows By [KR Prop 43] T Z(A) sim= oplusκisinJ Mor

κ otimesA Mκ asalgebras Together with the observation that T ( fcl) T (Acl) rarr T Z(A) is an isomor-phism of algebras this proves part (iii) Remark 323 Part (i) of Theorem 322 was announced by Muumlger [Muuml2] We provide anindependent proof in the setting of Cardy algebras

The above theorem together with Lemma 211 and Theorem 34 shows that a sim-ple commutative symmetric Frobenius C2plusmn-algebra Acl with dim Acl = Dim C is alwayspart of a Cardy algebra (Aop|Acl ιcl-op) for some simple special symmetric Frobeniusalgebra Aop in C However the above proof also illustrates that Aop is not unique Thisraises the question how two Cardy algebras with a given Acl can differ This question isanswered by [KR Thm 11] which in the present framework can be restated as follows

Cardy Algebras and Sewing Constraints I 905

Theorem 324 If (A(i)op |A(i)cl ι(i)cl-op) i = 1 2 are two Cardy C|C2plusmn-algebras such that

A(i)cl is simple and dim A(i)op = 0 for i = 1 2 then A(1)clsim= A(2)cl as algebras if and only

if A(1)op and A(2)op are Morita equivalent

Proof Theorem 11 in [KR] is stated for A(i)op being non-degenerate algebras and A(i)cl =Z(A(i)op ) for i = 1 2 By Proposition 316 A(i)op are simple and special for i = 1 2

Then by [KR Lem 21] A(i)op are non-degenerate algebras By Theorem 321 we have

A(i)clsim= Z(A(i)op ) as Frobenius algebras Finally by [KR Thm 11] Z(A(1)op ) sim= Z(A(2)op )

as algebras iff A(1)op and A(2)op are Morita equivalent Let Cmax(C2plusmn) be the set of equivalence classes [B] of simple modular invariant

commutative symmetric Frobenius algebras B in C2plusmn Two such algebras B and B prime areequivalent if B and B prime are isomorphic as algebras (but not necessarily as Frobenius alge-bras) Let Msimp(C) be the set of Morita classes of simple special symmetric Frobeniusalgebras in C Define the map z Msimp(C) rarr Cmax(C2plusmn) by z A rarr [Z(A)] whereA denotes the Morita class of A From Theorem 322 (i) and [KR Thm 11] we learn

Corollary 325 The map z Msimp(C) rarr Cmax(C2plusmn) is a bijection

A Appendix

A1 Proof of Lemma 27 We will show that if (F ψ F2 ψ

F0 ) is a colax tensor functor

from C1 to C2 then (G φG2 φ

G0 ) is a lax tensor functor from C2 to C1 Applying this

result to the opposed categories then gives the converse statementWe need to show that φG

0 and φG2 make the diagrams (21) and (22) commute We

first prove the commutativity of (21) Consider the following diagram

G(A)otimes (G(B)otimes G(C))

GψF2 δ

idG(A)otimesφG2 G(A)otimes G(B otimes C)

GψF2 δ

G(FG(A)otimes F(G(B)otimes G(C)))

G(F(idG(A))otimesF(φG2 ))

G(idFG(A)otimesψ F2 )

G(FG(A)otimes FG(B otimes C))

G(ρAotimesρBotimesC )

G(FG(A)otimes (FG(B)otimes FG(C)))

G(ρAotimes(ρBotimesρC )) G(A otimes (B otimes C))

(A1)

The top subdiagram is commutative because of the naturality of Gψ F2 δ The commu-

tativity of the bottom subdiagram follows from the following identities

(ρB otimes ρC ) ψ F2 = (ρB otimes ρC ) ψ F

2 ρF Fδ

= ρBotimesC FG(ρB otimes ρC ) FG(ψ F2 ) Fδ

= ρBotimesC F(φG2 ) (A2)

as a map F(G(B) otimes G(C)) rarr B otimes C The commutativity of (A1) implies that thecomposition of maps in the left column in (21) can be replaced by

G(ρA otimes (ρB otimes ρC )) G((idFG(A) otimes ψ F

2 ) ψ F2

) δ (A3)

906 L Kong I Runkel

Similarly we can show that the composition of maps in the right column in (21) can bereplaced by

G((ρA otimes ρB)otimes ρC ) G((ψ F

2 otimes idFG(C)) ψ F2

) δ (A4)

Using the commutativity of (23) it is easy to see that (21) with the left and right col-umns of (21) replaced by (A3) and (A4) respectively is commutative Hence (21) iscommutative

Now we prove the commutativity of the first diagram in (22)

φG2 (φG

0 otimes idG(A))

(1)= G(ρ12 otimes ρA) Gψ F2 δ

[(Gψ F

0 δ11)otimes idG(A)

]

(2)= G(ρ12 otimes ρA) Gψ F2 G F(Gψ0 otimes idG(A)) G F(δ11 otimes idG(A)) δ

(3)= G(ρ12 otimes ρA) G(FG(ψ F0 )otimes idFG(A)) Gψ F

2 G F(δ11 otimes idG(A)) δ(4)= G(id12 otimes ρA) G

([ψ F

0 ρF(11) (Fδ)11] otimes idFG(A)

) Gψ F

2 δ(5)= G(id12 otimes ρA) G(ψ F

0 otimes idFG(A)) Gψ F2 δ

(6)= G(id12 otimes ρA) G(lminus1FG(A)) G F(lG(A)) δ

(7)= G(lminus1A ) GρA δG lG(A)

(8)= G(lminus1A ) lG(A) (A5)

where in step (1) we substituted the definition of φG0 φ

G2 given in (210) in step (2)

we used the naturality of δ in step (3) we used the naturality of Gψ F2 in step (4) we

switched the position between Gρ12 and G FG(ψ F0 ) and the position between Gψ2 and

G F(δ12 otimes idG(A)) using the naturality of ρ and FψG2 respectively in step (5) we applied

the second identity in (28) in step (6) we used (24) in step (7) we used the naturalityof lminus1 and δ in step (8) we used the first identity in (28)

The proof of the commutativity of the second diagram in (22) is similar Thus wehave shown that G is a lax tensor functor

A2 Proof of Lemma 320 To prepare the proof recall that for a given object B isin Cthe modular group P SL(2Z) acts on the space oplusi HomC(B otimes Ui Ui ) see eg [BKSect 31] and [K3 Eq (455)] We will only need the action of S and Sminus1 Let f isinoplusi HomC(B otimes Ui Ui ) Then

S oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A6)

Cardy Algebras and Sewing Constraints I 907

Sminus1 oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A7)

By Lemma 32 to establish that (337) is S-invariant it is enough to prove the identity(34) when f is given by (337) Using (A6) and (A7) we can see that Eq (34) simplysays that oplusi j [RHS of (34)] is invariant under the action of S times S Consider the elementg of oplus jkisinIHomC2plusmn(R(A)otimes (Uor

j times Uk)Uorj times Uk) given by

g =oplusjkisinI

sumα

R(A)

R(A)

R(A)

R(A)

Uorj times Uk

Uorj times Uk

α

α

(A8)

By the above arguments proving S-invariance of (337) is equivalent to proving invari-ance of g under the action of S times S

For i isin I we denote by gi the component of g in(oplus jisinIHomC(A otimes Uor

i otimes Uorj U

orj )

)otimes (opluskisinIHomC(Ui otimes UkUk))

We view the second Hom-space in the above tensor product as a Hom-space in C+ insteadof Cminus It is enough to show that gi is invariant under the action of S times Sminus1 Note thatthe action of Sminus1 in C+ is equivalent to that of S in Cminus

The morphism gi can be canonically identified with a bilinear pairing

( middot middot )i ⎛⎝oplus

jisinIHomC(Uor

j A otimes Uori otimes Uor

j )

⎞⎠

times(oplus

kisinIHomC(UkUi otimes Uk)

)minusrarr C (A9)

as follows For h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) we set

(h1 h2)i = (dim U j dim Uk)minus1 trUor

j timesUk[gi (h1 times h2)] (A10)

908 L Kong I Runkel

When substituting the explicit form of the product m R(A) of R(A) = (A times 1) otimes R(1)after a short calculation one finds

(h1 h2)i =sumα

1

dim U j

A

A

A

A

A A

A α

α

h1

h2

U jUk

Uorj

Ui

Uk (A11)

Here the top morphism PlR(A) has been simplified with the help of the identity

PR(A) m R(A) (PR(A) otimes PR(A))

=(oplus

iisinI((m A A)otimes idUor

i)times idUi

) m R(A) (PR(A) otimes PR(A)) (A12)

which can be checked by direct calculation along the same lines as in the proof of [FrLem 310]

The action of the modular transformation S on oplusiisinI(B otimes Ui Ui ) for B isin C nat-urally induces an action on oplusiisinI(Ui B otimes Ui ) [K3 Prop 514] which we denote bySlowast In the present case we get an action of Slowast on oplus jisinIHomC(Uor

j A otimes Uori otimes Uor

j ) and

opluskisinIHomC(UkUi otimes Uk) Then to show gi is invariant under the action of S times Sminus1

amounts to showing that

(h1 h2)i = ((Sminus1)lowast h1 Slowast h2)i (A13)

Cardy Algebras and Sewing Constraints I 909

for all h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) We have

((Sminus1)lowast h1 Slowast h2)i =sum

mnα

dim Un

DimC

A

A

A

A

A A

A

Uk

α

α

h1

h2

Ui

U j

UmUn

(A14)

Now drag the upper vertex indexed by α in the above graph along its Uorm -leg until it

meets the lower vertex also indexed by α then sum over α and m This gives

((Sminus1)lowast h1 Slowast h2)i =sum

n

dim Un

DimC

A

A

A

A

A A

A

Uk

Un

h1

h2

Ui

U j

(A15)

910 L Kong I Runkel

If we just look at the neighbourhood of the Un-loop in the above graph we see thefollowing subgraphs

sumn

dim Un

DimC

A Aor

Uorj U j

Uk Uork

Un=

sumα

1

dim U j

A Aor Uk Uork

Uorj U j

αα

(A16)

where we have applied [BK Cor 3111] Substituting this subgraph back to the originalgraph in (A15) we obtain

((Sminus1)lowast h1 Slowast h2)i =sumα

1

dim U j

A

A

A

A

A A

A

A

A

Uorj

α

α

h1

h2m A

Ui

U jUk

Uk

(A17)

The graph in (A17) is equal to that in (A11) In order to see this we first drag theldquobubblerdquo (m A A) along A lines and through the m A vertex (because m A A is abimodule map) until it reaches the lower-left leg of the upper vertex indexed by α Thendrag the m A vertex along the (red) dotted line in the above graph Finally we applythe associativity of A (A12) and [Fr Lem 311] Then we see that the graph in (A17)exactly matches with the one in (A11)

Acknowledgements We would like to thank the organisers of the Oberwolfach Arbeitsgemeinschaft ldquoAlge-braic structures in conformal field theoriesrdquo (April 2007) where this work was started for an inspiring meetingWe would further like to thank the Hausdorff Institute for Mathematics in Bonn and the organisers of the stim-ulating meeting ldquoGeometry and Physicsrdquo (May 2008) We are indebted to Alexei Davydov Jens FjelstadJuumlrgen Fuchs Yi-Zhi Huang Alexei Kitaev Urs Schreiber Christoph Schweigert Stephan Stolz and PeterTeichner for helpful discussions andor comments on a draft of this paper The research of IR was partiallysupported by the EPSRC First Grant EPE0050471 the PPARC rolling grant PPC5071451 and the MarieCurie network lsquoSuperstring Theoryrsquo (MRTN-CT-2004-512194)

Cardy Algebras and Sewing Constraints I 911

References

[AN] Alexeevski A Natanzon SM Noncommutative two-dimensional topological field theories andhurwitz numbers for real algebraic curves Sel Math New Ser 12 307ndash377 (2006)

[Bi] Bichon J Cosovereign hopf algebras J Pure Appl Alg 157 121ndash133 (2001)[BK] Bakalov B Kirillov AA Lectures on Tensor Categories and Modular Functors Providence

RI Amer Math Soc 2001[C1] Cardy JL Operator content of two-dimensional conformal invariant theories Nucl Phys

B 270 186ndash204 (1986)[C2] Cardy JL Boundary conditions fusion rules and the verlinde formula Nucl Phys B 324

581ndash596 (1989)[CL] Cardy JL Lewellen DC Bulk and boundary operators in conformal field theory Phys Lett

B 259 274ndash278 (1991)[DMZ] Dong C-Y Mason G Zhu Y-C Discrete series of the Virasoro algebra and the moonshine

module In Algebraic Groups and Their Generalizations Quantum and infinite-dimensional Meth-ods Proc Symp Pure Math 56 Part 2 Providence RI Amer MathSoc 1994 pp 295ndash316

[DP] Day B Pastro C Note on frobenius monoidal functors New York J Math 14 733ndash742 (2008)[ENO1] Etingof PI Nikshych D Ostrik V On fusion categories Ann Math 162 581ndash642 (2005)[ENO2] Etingof PI Nikshych D Ostrik V An analogue of radfordrsquos s4 formula for finite tensor

categories Int Math Research Notices 54 2915ndash2933 (2004)[Fe] Felder G Froumlhlich J Fuchs J Schweigert C Correlation functions and boundary conditions in

rational conformal field theory and three-dimensional topology Comp Math 131 189ndash237 (2002)[FHL] Frenkel IB Huang Y-Z Lepowsky J On axiomatic approaches to vertex operator algebras

and modules Mem Amer Math Soc 104 (1993)[Fj] Fjelstad J Fuchs J Runkel I Schweigert C Uniqueness of openclosed rational cft with

given algebra of open states Adv Theor Math Phys 12 1283ndash1375 (2008)[FK] Froumlhlich J King C The chern-simons theory and knot polynomials Commun Math

Phys 126 167ndash199 (1989)[Fr] Froumlhlich J Fuchs J Runkel I Schweigert C Correspondences of ribbon categories Adv

Math 199 192ndash329 (2006)[FRS] Fuchs J Runkel I Schweigert C Tft construction of rcft correlators I partition functions Nucl

Phys B 646 353ndash497 (2002)[FS] Fuchs J Schweigert C Category theory for conformal boundary conditions Fields Inst Com-

mun 39 25ndash71 (2003)[H1] Huang Y-Z Two-dimensional conformal geometry and vertex operator algebras Progress in

Mathematics Vol 148 Boston Birkhaumluser 1997[H2] Huang Y-Z Rigidity and modularity of vertex tensor categories Comm Contemp Math 10

871ndash911 (2008)[HK1] Huang Y-Z Kong L Open-string vertex algebra category and operads Commun Math

Phys 250 433ndash471 (2004)[HK2] Huang Y-Z Kong L Full field algebras Commun Math Phys 272 345ndash396 (2007)[HK3] Huang Y-Z Kong L Modular invariance for conformal full field algebras httparxivorgabs

math0609570v2[mathQA] 2006[HL] Huang Y-Z Lepowsky J Tensor products of modules for a vertex operator algebra and vertex

tensor categories In Lie Theory and Geometry in honor of Bertram Kostant ed R BrylinskiJ-L Brylinski V Guillemin V Kac Boston Birkhaumluser 1994 pp 349ndash383

[JS] Joyal A Street R Braided tensor categories Adv Math 102 20ndash78 (1993)[K1] Kong L Full field algebras operads and tensor categories Adv Math 213 271ndash340 (2007)[K2] Kong L Open-closed field algebras Commun Math Phys 280 207ndash261 (2008)[K3] Kong L Cardy condition for open-closed field algebras Commun Math Phys 283 25ndash92 (2008)[Ki] Kitaev A Private communication[KO] Kirillov AA Ostrik V On q-analog of mckay correspondence and ade classification of sl(2)

conformal field theories Adv Math 171 183ndash227 (2002)[KR] Kong L Runkel I Morita classes of algebras in modular tensor categories Adv Math 219

1548ndash1576 (2008)[Ld] Lauda AD Frobenius algebras and ambidextrous adjunctions Theo Appl Cat 16 84ndash

122 (2006)[Li1] Li H-S Regular representations of vertex operator algebras Commun Contemp Math 4

639ndash683 (2002)[Li2] Li H-S Regular representations and huang-lepowsky tensor functors for vertex operator

algebras J Alge 255 423ndash462 (2002)

888 L Kong I Runkel

where we used naturality of δ and (250) in the second and third equalities respectivelySo Q R is a left inverse of R on morphisms Thus R is injective on morphisms Similarlylet g M rarr N be a morphism in C2plusmn We define a map QT HomC(T (M) T (N )) rarrHomC2plusmn(M N ) by gprime rarr (Dim C)minus1 middot ρN R(gprime) δM Then we have

QT T (g) = (Dim C)minus1 middot ρN RT (g) δM = (Dim C)minus1 middot ρN δN g = g (252)

So QT is a left inverse of T on morphisms Thus T is injective on morphisms Using (29) and (250) one can express the two inverse maps χminus1 χminus1 as follows

for f isin HomC2plusmn(M R(A)) and g isin HomC2plusmn(R(A)M)

χminus1( f ) = ρ T ( f ) χminus1(g) = T (g) δ (253)

By Proposition 25 and Lemma 27 R is both a lax and colax tensor functor In particularφR

0 1 times 1 rarr R(1) is given by

φR0 = χ (ψT

0 ) = R(ψT0 ) δ1 times 1 = id1times1 (254)

and φR2 R(A)otimes R(B) rarr R(A otimes B) by φR

2 = R(ρA otimes ρB) R(ψT2 ) δ which can

be expressed graphically as

φR2 =

oplusi jkisinI

sumα

A

A

Uori B

B

Uorj

Uork

α

times

Ui U j

α

Uk

(255)

Similarly ψ R0 R(1) rarr 1 times 1 is given by

ψ R0 = ρ1 R(φT

0 ) = Dim C id1times1 (256)

and ψ R2 R(A otimes B) rarr R(A) otimes R(B) by ψ R

2 = ρ R(φT2 ) R(δA otimes δB) which in

graphical notation reads

ψ R2 =

oplusi jkisinI

sumα

A

A Uori

B

B Uorj

Uork

α

times

Ui U j

α

Uk

dim Ui dim U j

dim Uk Dim C

(257)

If C has more than one simple object then R does not take the tensor unit of C to thetensor unit of C2plusmn and so is clearly not a tensor functor However we will show that R isstill a Frobenius functor This will imply that if A is a Frobenius algebra in C then

R(A) = (R(A)m R(A) ηR(A)R(A) εR(A)) (258)

Cardy Algebras and Sewing Constraints I 889

is a Frobenius algebra in C2plusmn where the structure morphisms were given in (217) and(218) In the case A = 1 it was proved in [Muuml1 Prop 41] (see also [Fr Lem 619]and [K1 Thm 52]) that (258) is a commutative simple symmetric normalised-specialFrobenius algebra in C2plusmn In fact given a Frobenius algebra A in C it is straightforwardto verify that the structure morphisms in (258) are precisely those of (A times 1)otimes R(1)cf Sect 22

Proposition 223 (R φR2 φ

R0 ψ

R2 ψ

R0 ) is a Frobenius functor

Proof Using the explicit graphical expression of φR2 φ

R0 ψ

R2 ψ

R0 it is easy to see that

the commutativity of the diagrams (25) and (26) are equivalent to the statement thatR(1) with structure morphisms as in (258) is a Frobenius algebra in C2plusmn The latterstatement is true by [Muuml1 Prop 41]

From Lemma 220 and Proposition 223 we see that T and R take Frobenius alge-bras to Frobenius algebras The following two propositions show how the properties ofFrobenius algebras are transported

Proposition 224 Let A be a Frobenius algebra in C2plusmn Then T (A) is a Frobenius algebrain C and

(i) A is symmetric iff T (A) is symmetric(ii) A is (normalised-)special iff T (A) is (normalised-)special

Proof For part (i) write A as a direct sum oplusNn=1 Al

n times Arn Then the maps IT (A) I prime

T (A) T (Aor) rarr T (A)or defined in (220) are given by

IT (A) = I primeT (A) = oplusN

n=1c(Aln)

or(Arn)

or (259)

Therefore by (221) T (A) = primeT (A) is equivalent to T (A) = T (prime

A) Since byLemma 222 T is injective on morphisms this proves part (i) Part (ii) can be checkedin the same way for example the condition mT (A) T (A) = ζ idT (A) is easily checkedto be equivalent to T (m A A) = ζ T (idA) Proposition 225 Let A be a Frobenius algebra A in C Then R(A) is a Frobeniusalgebra in C2plusmn and

(i) A is symmetric iff R(A) is symmetric(ii) A is (normalised-)special iff R(A) is (normalised-)special

Proof Recall that the structure morphisms of the Frobenius algebra R(A) are equal tothose of (A times 1) otimes R(1) Using this equality part (i) and (ii) follow because R(1) issymmetric and normalised-special For example

m R(A) R(A) = [(m A A)times id1] otimes idR(1) = R(m A A) (260)

so that m R(A) R(A) = ζ idR(A) is equivalent to R(m A A) = ζ R(idA) which byLemma 222 is equivalent to m A A = ζ idA

The functor R has one additional property not shared by T namely R takes absolutelysimple algebras to absolutely simple algebras We will see explicitly in Sect 33 thatthis is not true for T

890 L Kong I Runkel

Lemma 226 For a C-algebra A the map

R HomA|A(A A) rarr HomR(A)|R(A)(R(A) R(A)) (261)

given by f rarr R( f ) is well-defined and an isomorphism

Proof Since R is a lax tensor functor R(A) is naturally a R(A)-bimodule It is easy tosee that R in (261) is a well-defined map R(A) is also naturally a R(1)-bimodule whichcan be identified with the induced R(1)-bimodule structure on (A times 1)otimes R(1) wherethe left R(1) action on (Atimes1)otimes R(1) is given by (idAtimes1 otimesm R(1))(cminus1

Atimes1R(1)otimes idR(1))We have the following natural isomorphisms

HomR(1)|R(1)(R(A) R(A))sim=minusrarr HomC2plusmn(A times 1 R(A))

χminus1

minusminusrarr HomC(A A) (262)

which by (253) are given by for f isin HomR(1)|R(1)(R(A) R(A))

f rarr f prime = f (idAtimes1 otimes ηR(1)) rarr f primeprime = ρ T ( f prime) (263)

and its inverse is given by for g isin HomC(A A)

g rarr gprime = R(g) δ rarr gprimeprime = (idAtimes1 otimes m R(1)) (gprime otimes idR(1)) (264)

where gprimeprime is indeed a R(1)-bimodule map due to the commutativity of R(1) It is easy tocheck that gprimeprime = R(g) in (264) Therefore R gives an isomorphism from HomC(A A)to HomR(1)|R(1)(R(A) R(A)) Moreover one verifies that R(g) is an R(A)-bimodulemap iff g is an A-bimodule map In other words R g rarr R(g) gives an isomorphism

HomA|A(A A)sim=minusrarr HomR(A)|R(A)(R(A) R(A))

Corollary 227 Let A be a C-algebra

(i) A is absolutely simple iff R(A) is absolutely simple(ii) Let A be in addition Frobenius Then A is simple and special iff R(A) is simple and

special

Proof Part (i) immediately follows from Lemma 226 The statement of part (ii) withoutthe qualifier lsquosimplersquo is proved in Proposition 225 But as in the proof of (iii)rArr(i) inLemma 211 a special Frobenius algebra in a semi-simple category has a semi-simplecategory of bimodules and for a semi-simple C-linear category simple and absolutelysimple are equivalent Part (ii) then follows from part (i)

The following lemma will be needed in Sect 32 below to discuss the properties ofCardy algebras

Lemma 228 Let A be a Frobenius algebra in C Then (δA)lowast = ρA

Proof Recall from (250) that δA is a morphism A rarr T R(A) Since T and R are bothFrobenius functors T R(A) is a Frobenius algebra in C Substituting the definitions aftera short calculation one finds

Cardy Algebras and Sewing Constraints I 891

εT R(A) mT R(A) = Dim(C)oplusiisinI

A Uori Ui A U or

ı Uı

(265)

Substituting this in the definition of (δA)lowast gives again after a short calculation the mor-

phism ρA At an intermediate step one uses that the part of the morphism (δA)lowast which

is made up of Ui and Uı ribbons their duals and the basis morphisms λ(iı)0 andϒ(iı)0can be replaced by 1

dim Uimiddot dUi

3 Cardy Algebras

In this section we start by investigating the properties of Frobenius algebras which sat-isfy the so-called modular invariance condition We then give two definitions of a Cardyalgebra and prove their equivalence Finally in Sect 33 we study the properties of thesealgebras and state our main results

We fix a modular tensor category C Recall that C2plusmn is an abbreviation for C+ Cminus

31 Modular invariance In C2plusmn we define the object K and the morphism ω K rarr Kas

K =oplus

i jisinIUi times U j ω =

sumi jisinI

dim Ui dim U j

Dim C idUi timesU j (31)

They have the property (see eg [BK Cor 3111])

ω

K

(Uk times Ul )or

(Uk times Ul )or

Ui times U j

Ui times U j

= δik δ jlDim C

dim Ui dim U jbUi timesU j dUi timesU j (32)

892 L Kong I Runkel

Definition 31

(i) Let A B be objects of C2plusmn A morphism f A otimes B rarr B is called S-invariant iff

A

B

W

W

f =

A

B

W

W

f

K

ω(33)

holds for all for W isin C2plusmn(ii) A C2plusmn-algebra (Aclmcl ηcl) is called modular invariant iff θAcl = idAcl and mcl

is S-invariant

Lemma 32 The morphism f A otimes B rarr B is S-invariant if and only if

A

B

Ui times U j

Ui times U j

f = Dim Cdim Ui dim U j

sumα

A

B

B

f

Ui times U j

Ui times U j

α

α

(34)

holds for all i j isin I

Proof Condition (33) holds for all W iff it holds for all W = Ui times U j i j isin I so itis enough to show that the right-hand side of (33) with W = Ui times U j is equal to theright-hand side of (34) Recall the notation for basis morphisms in (236) Starting from(33) write

idBor otimes idUi timesU j =⎛⎝sum

klα

(b(ktimeslα)

B

)or (

bB(ktimeslα)

)or⎞⎠ otimes idUi timesU j (35)

and then apply (32) The graphical representation of the resulting morphism can bedeformed to give (34) Remark 33 As shown in [K3 Sect 61] the modular invariance condition of a C2plusmn-alge-bra exactly coincides with the modular invariance condition for torus 1-point correlationfunctions of a genus-01 closed CFT In particular the condition θAcl = idAcl is equiv-alent to invariance under the modular transformation T τ rarr τ + 1 and the condition(34) with f = mcl is equivalent to invariance under S τ rarr minus 1

τ Combining the

modular invariance condition with the genus zero properties of a genus-01 closed CFTresults in a modular invariant commutative symmetric Frobenius algebra in C2plusmn

Cardy Algebras and Sewing Constraints I 893

Let Acl be a modular invariant C2plusmn-algebra Evaluating (34) for f = mcl composingit with ηcl otimes idUi timesU j and taking the trace implies the following identity

Zi j = 1

DimCAclUi timesU j where

(36)

Zi j = dimC HomC2plusmn(UitimesU j Acl)

Decomposing Acl into simple objects this gives

Zi j =sum

klisinISik Zkl Sminus1

l j where Si j = si jradic

DimC (37)

which in CFT terms is of course nothing but the invariance of the torus partition functionunder the modular S-transformation

The following theorem gives a simple criterion for modular invariance

Theorem 34 Let Acl be a haploid commutative symmetric Frobenius algebra in C2plusmn

(i) If Acl is modular invariant then dim Acl = Dim C(ii) If dim Acl = Dim C then Acl is special and modular invariant

Proof Part (i) Since Acl is haploid for i = j = 0 Equation (36) reduces to 1 =dim AclDimCPart (ii) By the same reasoning as in the proof of (iii)rArr(i) in Lemma 211 one showsthat Acl is special Thus mcl cl = ζclidAcl for some ζcl = 0

By [KO Thm 45] the category (C2plusmn)locAcl

of local Acl-modules is again a mod-

ular tensor category and Dim (C2plusmn)locAcl

= (Dim C dim Acl)2 (see [Fr Prop 321 amp

Rem 323] for the same statement in the notation used here) Thus by assumption wehave Dim(C2plusmn)loc

Acl= 1 It then follows from [ENO1 Thm 23] that up to isomorphism

(C2plusmn)locAcl

has a unique simple object (namely the tensor unit) In other words every simplelocal Acl-module is isomorphic to Acl (seen as a left-module over itself)

We have the following isomorphisms between morphism spaces [FS Prop 47 amp 411]

HomAcl(Acl otimes (Ui times U j ) Acl) sim= HomC2plusmn(Ui times U j Acl)

HomAcl(Acl Acl otimes (Ui times U j )) sim= HomC2plusmn(AclUi times U j )

Using these to transport the bases (236) from the right to the left we obtain bases bα(i j)αof HomAcl(Acl otimes (Ui times U j ) Acl) and b(i j)

β β of HomAcl(Acl Acl otimes (Ui times U j )) Thesecan be expressed graphically as

894 L Kong I Runkel

bα(i j) =

Acl

Acl

Acl

Ui times U j

α

b(i j)β = DimC

dim Ui dim U j

1

ζcl

Acl

Ui times U jAcl

Acl

β

(38)

where the nonzero factor in b(i j)β is included for convenience Notice that bα(i j) b(i j)

β isa left Acl-module map Since Acl is simple as a left module over itself we have

bα(i j) b(i j)β = λαβ idAcl (39)

for some λαβ isin C By computing tr(bα(i j) b(i j)β ) it is easy to verify that λαβ = δαβ

We will now prove the following identity

1

ζcl

Acl

Acl

Ui times U j

Ui times U j

Acl

=sumα

DimCdim Ui dim U j

1

ζcl

Acl

Ui times U j

Ui times U j

Acl

Acl

Acl

α

α

(310)

One checks that the left-hand side of this equation is an idempotent which we denote byζminus1

cl PlAcl(Ui timesU j ) cf [Fr Sect 31] By [Fr Prop 41] the image Im(ζminus1

cl PlAcl(Ui timesU j ))

is a local Acl-module and hence isomorphic to AoplusNcl for some N isin Zge0

All left-module morphisms from Acl to Aclotimes(Ui timesU j ) are linear combinations of the

b(i j)β Furthermore one verifies that ζminus1

cl PlAcl(Ui timesU j )b(i j)

β = b(i j)β Therefore the b(i j)

β

describe precisely the image of the idempotent ie ζminus1cl Pl

Acl(Ui timesU j ) = sum

α b(i j)α bα(i j)

which is nothing but (310) Composing (310) with ζcl middot εcl otimes idUi timesU j from the left (iefrom the top) produces (34)

In addition since Acl is commutative symmetric Frobenius it satisfies θAcl = idAcl

[Fr Prop 225] Altogether this shows that Acl is modular invariant Remark 35 As we were writing this paper we heard that the results in Theorem 34were obtained independently by Kitaev and Muumlger [Ki]

Remark 36 Setting i = j = 0 in (36) gives the identity dim Acl = Z00 Dim C [KRProp 23] Combining this with Theorem 34 (ii) one may wonder if a general modularinvariant commutative symmetric Frobenius algebra Acl in C2plusmn is isomorphic to a directsum of simple such algebras However this is not so For example one can take the com-mutative symmetric Frobenius algebra Acl = C[x]x2 in the category of vector spacesequipped with the non-degenerate trace ε(ax + b) = a In this case the modular invari-ance condition holds automatically but Acl is clearly not a direct sum of two algebras

Cardy Algebras and Sewing Constraints I 895

For a general modular tensor category C the algebra C[x]x2 R(1) understood as analgebra in C2plusmn via the braided monoidal isomorphism Vect f (C) C2plusmn rarr C2plusmn providesanother counter-example

32 Two definitions Define a morphism PlA A rarr A for a Frobenius algebra A in C

or C2plusmn as follows [Fr Sect 24]

PlA =

A

A

A

m

(311)

If A is also commutative and obeys m A A = ζA idA we have PlA = ζA idA In partic-

ular this holds if A is commutative and special Using the fact that the Frobenius algebraR(1) is commutative and normalised-special one can check that Pl

R(A) R(A) rarr R(A)takes the following form

PlR(A) =

oplusiisinI

A

A

Uori

Uori

A

m

times

Ui

Ui

(312)

With these ingredients we can now give the first definition of a Cardy C|C2plusmn-algebrawhich was introduced in [K3 Def 514] cf Remark 315 below

Definition 37 (CardyC|C2plusmn-algebra I) A CardyC|C2plusmn-algebra is a triple (Aop|Acl ιcl-op)where (Aclmcl ηclcl εcl) is a modular invariant commutative symmetric Frobe-nius C2plusmn-algebra (Aopmop ηopop εop) is a symmetric Frobenius C-algebra andιcl-op Acl rarr R(Aop) an algebra homomorphism such that the following conditionsare satisfied

(i) Centre condition

Acl R(Aop)

R(Aop)

R(Aop)

ιcl-op

m R(Aop)

=

Acl

ιcl-op

R(Aop)

R(Aop)

R(Aop)

m R(Aop)

(313)

896 L Kong I Runkel

(ii) Cardy condition

ιcl-op ιlowastcl-op =

R(Aop)

R(Aop)

R(Aop)

(314)

Remark 38

(i) The name ldquoCardy C|C2plusmn-algebrardquo in Definition 37 was chosen because many ofthe important ingredients were first studied by Cardy the modular invariance ofthe closed theory [C1] the consistency of the annulus amplitude [C2] and thebulk-boundary OPE [CL] On the other hand the boundary-boundary OPE andthe OPE analogue of the centre condition were first considered in [Lw]

(ii) One can easily see that in the special case that C is the category Vect f (C) of finite-dimensional C-vector spaces a Cardy C|C2plusmn-algebra gives exactly the algebraicformulation of two-dimensional open-closed topological field theory over C (cfRemark 614 in [K3]) see [Lz Sect 48] [Mo Thm 11] [AN Thm 45] [LPCor 43] [MS Sect 22] When passing to a general modular tensor category Cthere are two important differences to the two-dimensional topological field the-ory Firstly the algebras Acl and Aop now live in different categories which inparticular affects the formulation of the centre condition and the Cardy conditionSecondly the modular invariance condition has to be imposed on Acl In the caseC = Vect f (C) modular invariance holds automatically

Definition 39 A homomorphism of Cardy C|C2plusmn-algebras (A(1)op |A(1)cl ι(1)cl-op) rarr (A(2)op

|A(2)cl ι(2)cl-op) is a pair ( fop fcl) of Frobenius algebra homomorphisms fop A(1)op rarr A(2)op

and fcl A(1)cl rarr A(2)cl such that the diagram

A(1)cl

fcl

ι(1)cl-op

A(2)cl

ι(2)cl-op

R(A(1)op )

R( fop) R(A(2)op )

(315)

commutes

Remark 310 Since a homomorphism of Frobenius algebras is invertible (cf Lemma218 (iv)) a homomorphism of Cardy algebras is always an isomorphism

For a homomorphism ( fop fcl) of Cardy C|C2plusmn-algebras using the commutativity of(315) and the fact that fcl and fop are both algebra and coalgebra homomorphisms itis easy to show that (315) commutes iff

R(A(1)op )R( fop)

(ι(1)cl-op)

lowast

R(A(2)op )

(ι(2)cl-op)

lowast

A(1)cl

fcl A(2)cl

(316)

Cardy Algebras and Sewing Constraints I 897

commutesLet (Aop|Acl ιcl-op) be a Cardy C|C2plusmn-algebra Define the morphism

ιcl-op = χminus1(ιcl-op) T (Acl) minusrarr Aop (317)

Decompose Acl as Acl = oplusNn=1Cl

n times Crn such that Cl

1 times Cr1 = ηcl(1 times 1) We use ι(n)cl-op

to denote the restriction of ιcl-op to Cln times Cr

n and ι(n)cl-op to denote the restriction of ιcl-op

to Cln otimes Cr

n We introduce the following graphical notation

ι(n)cl-op =

Cln Cr

n

Aop

(318)

By (243) ιcl-op can be expressed in terms of ιcl-op as follows

ιcl-op =Noplus

n=1

oplusiisinI

sumα

Cln

Crn

Aop

α

Uori

times α

Crn

Ui

(319)

Lemma 311 The centre condition (313) is equivalent to the following condition in C

Cln Cr

n

Aop

Aop

Aop

=

Cln Cr

n

Aop

Aop

Aop

for n = 1 N (320)

Proof First insert (319) and the definition (217) of m R(Aop) into (313) Then apply thecommutativity of R(1) to the left hand side of (313) The equivalence between (313)and (320) follows immediately Remark 312 The centre condition (320) is very natural from the open-closed confor-mal field theory point of view Correlators on the upper half plane are expressed in termsof conformal blocks on the full complex plane The objects Cl

n and Crn are associated to

the field insertion at a point z in the upper half plane and at the complex conjugate pointz in the lower half plane respectively The object Aop corresponds to a field inserted ata point r on the real axis The centre condition (320) simply says that the correlationfunctions in the disjoint domains |z| gt r gt 0 and r gt |z| gt 0 are analytic continuationsof each other see [K2 Prop 118]

898 L Kong I Runkel

Recall that we define ιlowastcl-op Aop rarr T (Acl) as in (230) We introduce the graphicalnotation

ιlowastcl-op =Noplus

n=1

Cln Cr

n

Aop

=Noplus

n=1

Cln Cr

n

Aop

Aop T (minus1Acl) (321)

where the second equality follows from (215) (221) and (259)

Lemma 313 The Cardy condition (314) is equivalent to the following identity in C

Noplusn=1

DimCdim Ui

sumα

Cln

Crn

Crn

α

α

Aop

Aop Uori

Uori

=Aop

Aop

Aop

Uori

Uori

op

mop for all i isin I (322)

Proof By (244) (312) and (319) it is easy to see that (322) is equivalent to thefollowing identity

ιcl-op χ(ιlowastcl-op) = PlR(Aop)

(323)

Therefore it is enough to show that

χ(ιlowastcl-op) = ιlowastcl-op (324)

We have

χminus1(ιlowastcl-op)(1)= T (ιlowastcl-op) δ (2)= T (ιcl-op)

lowast δlowastlowast (3)= (ρ T (ιcl-op)

)lowast (4)= (ιcl-op

)lowast (325)

In step (1) we use the expression (253) for χminus1 step (2) follows from Lemma 218 (v)and Lemma 219 Step (3) is Lemma 218 (i) and Lemma 228 and finally step (4)amounts to substituting (253) and (317) Acting with χ on both sides of the aboveequality produces (324)

Combining Lemmas 311 and 313 and Proposition 216 we obtain the followingequivalent definition of Cardy C|C2plusmn-algebra (recall the graphical notation (318) forιcl-op and (321) for ιlowastcl-op)

Cardy Algebras and Sewing Constraints I 899

Definition 314 (CardyC|C2plusmn-algebra II) A CardyC-algebra is a triple (Aop|Acl ιcl-op)where Acl is a commutative symmetric Frobenius C2plusmn-algebra satisfying property (34)with f = mcl Aop is a symmetric Frobenius C-algebra and ιcl-op T (Acl) rarr Aop isan algebra homomorphism satisfying the conditions (320) and (322)

Remark 315 Up to a choice of normalisation Definition 314 is the same as the orig-inal one in [K3 Def 613] The difference between the two definitions is the factorDim C dim Ui on the left hand side of (322) which in [K3 Def 613] is given byradic

Dim C dim Ui The two definitions are related by rescaling the coproduct cl andcounit εcl of Acl by 1

radicDim C and

radicDim C respectively We chose the convention in

(322) to remove all dimension factors from the expression (314) for the Cardy condition

33 Uniqueness and existence theorems In this subsection we investigate the structureof Cardy algebras We start with the following proposition which when combined withthe results of part II provides an alternative proof of [Fj Prop 422]

Proposition 316 Let (Aop|Acl ιcl-op) be a Cardy C|C2plusmn-algebra If Acl is simple anddim Aop = 0 then Aop is simple and special

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis therefore haploid Restricting the Cardy condition (322) to the case Ui = 1 andcomposing both sides with εop from the left we see that εop kills all terms associated toU j times 1 isin Acl in the sum except for a single 1 times 1 term Thus we obtain the followingidentity

β εop = dAop (mop otimes idAorop) (idAop otimes bAop) (326)

where β isin C Composing with ηop from the right in turn implies that βεop ηop =dim Aop which is nonzero by assumption Thus also β = 0 and εop is a nonzero multipleof the morphism on the right hand side of (326) By [FRS Lem 311] Aop is special

Since Aop is a special Frobenius algebra Aop is semi-simple as an Aop-bimodule(apply [FS Prop 524] to Aop tensored with its opposite algebra) Suppose Aop is not

simple so that we can write Aop = A(1)op oplus A(2)op for nonzero Aop-bimodules A(1)op and A(2)op We denote the canonical embeddings and projections associated to this decompositionas ι12 and π12 We have the identities

mop (ι1 otimes ι2) = 0 εop ηop =2sum

i=1

εop ιi πi ηop (327)

The first identity follows since π1 mop (ι1 otimes ι2) = 0 (as mop gives the left action of

Aop on Aop and hence it preserves A(2)op ) and similarly π2 mop (ι1 otimes ι2) = 0 Thesecond identity is just the completeness of ι12 π12

Since εop ηop = 0 without losing generality we can assume εop ι1 π1 ηop = 0Using that π2 is a bimodule map we compute

π2 [LHS of (322)]Ui =1 ι1 π1 ηop = π2 [RHS of (322)]Ui =1 ι1 π1 ηop

= PlA(2)op

π2 ι1 π1 ηop = 0 (328)

900 L Kong I Runkel

On the other hand using that Acl is haploid that ιcl-op is an algebra map and thatι lowastcl-op is a coalgebra map one can check that the left-hand side of (328) is equal toλ(εop ι1 π1 ηop) π2 ηop for some λ = 0 This implies that π2 ηop = 0 Thusηop = sum

i ιi πi ηop = ι1 π1 ηop Hence we have

0 = π2 ι2 = π2 mop (ηop otimes ι2) = π2 mop ((ι1 π1 ηop)otimes ι2) (329)

However the right-hand side is zero by (327) This is a contradiction and hence Aopmust be simple

To formulate the next theorem we need the notion of the full centre of an algebra [FjDef 49] Recall that an algebra A in a braided tensor category has a left centre and aright centre [VZO] both of which are sub-algebras of A Of these two we will onlyneed the left centre The following definition is [Fr Def 231] which in our setting isequivalent to that of [VZO]

Definition 317 Let A be a symmetric special Frobenius algebra such that m A A =ζA idA

(i) The left centre Cl(A) of A is the image of the idempotent ζminus1A Pl

A(ii) The full centre Z(A) is Cl(R(A))

That ζminus1A Pl

A is an idempotent follows from [FRS Lem 52] when keeping track ofthe factors ζA ([FRS] assumes normalised-special ie ζA = 1) Note that Cl(A) is againan object of C while Z(A) is an object of C2plusmn Let el

A Cl(A) rarr A be the embeddingof Cl(A) into A The left centre is in fact the maximal subobject of A such that

m A cAA (elA otimes idA) = m A (330)

see [Fr Lem 232] This observation explains the name left centre and also makes theconnection to [O Def 15]

The full centre is by definition the image of the idempotent ζminus1A Pl

R(A) R(A) rarrR(A) Since C2plusmn is abelian the idempotent splits and we obtain the embedding andrestriction morphisms

e Z(A) rarr R(A) and r R(A) Z(A) (331)

which obey r e = idZ(A) and e r = ζminus1A Pl

R(A) It follows from Proposition 225 and

[Fr Prop 237] that Z(A) is a commutative symmetric Frobenius algebra in C2plusmn withstructure morphisms5

m Z(A) = r m R(A) (e otimes e) ηZ(A) = r ηR(A)

Z(A) = ζA middot (r otimes r) R(A) e εZ(A) = ζminus1A middot εR(A) e

(332)

Moreover if A is simple then Z(A) is simple and if A is simple and dim A = 0 thenZ(A) is simple and special The normalisation of the counit is such that

εZ(A) ηZ(A) = ζminus2A dim A DimC (333)

5 The normalisation of product and unit is the standard one The factors in the coproduct and counit haveto be included in order for (A|Z(A) e) to be a Cardy algebra see Theorem 318 below The normalisation ofthe counit enters the Cardy condition (314) through the definition of ( middot )lowast

Cardy Algebras and Sewing Constraints I 901

Theorem 318 Let A be a special symmetric Frobenius C-algebra Then (A|Z(A) e)is a Cardy C|C2plusmn-algebra

The proof of this theorem makes use the following two lemmas

Lemma 319 e Z(A) rarr R(A) is an algebra map and elowast = ζA middot r

Proof It follows from [Fr Lem 229] (or by direct calculation using in particularm R(A) R(A) = ζAidR(A)) that

m R(A) (e otimes e) = ζminus1A middot Pl

R(A) m R(A) (e otimes e) (334)

Substituting e r = ζminus1A Pl

R(A) shows that e is compatible with multiplication For theunit one finds

e ηZ(A) = e r ηR(A) = ζminus1A Pl

R(A) ηR(A) = ηR(A) (335)

Thus e is an algebra map For the second statement one computes

elowast (1)= ζA

R(A)

r

R(A)

R(A)

Z(A)

R(A)

Z(A)

r

e

e

(2)=

R(A)

r

R(A)

R(A)

R(A)

Z(A)

(3)=

R(A)

r

R(A)

Z(A)

(4)= ζA middot r (336)

where in (1) the definitions (230) and (332) have been substituted step (2) is e r = ζminus1

A PlR(A) step (3) uses that R(A) is symmetric Frobenius and step (4) is again

e r = ζminus1A Pl

R(A) Lemma 320 Let A be a symmetric Frobenius algebra in C The morphism

PlR(A) m R(A)

(Pl

R(A) otimes PlR(A)

) R(A)otimes R(A) minusrarr R(A) (337)

is S-invariant

The proof of this lemma is a slightly lengthy explicit calculation and has been deferredto Appendix A2

902 L Kong I Runkel

Proof of Theorem 318 That e is an algebra map was proved in Lemma 319 The centrecondition (313) holds by property (330) of the left centre The Cardy condition (314)also is an immediate consequence of Lemma 319

ιcl-op ιlowastcl-op = e (ζA r) = PlR(A) (338)

The full centre Z(A) is a commutative symmetric Frobenius algebra It remains to provemodular invariance That θZ(A) = idZ(A) is implied by commutativity and symmetryof Z(A) [Fr Prop 225] The S-invariance condition (33) follows from Lemma 320In (337) substitute Pl

R(A) = ζA e r and then put the resulting morphism into (33)Compose the resulting equation with e otimes idW from the right (ie from the bottom) andsubstitute the definition (332) of m Z(A) This results in the statement that m Z(A) isS-invariant

The following theorem is analogous to [LR Prop 29] and [Fj Thm 426] whichroughly speaking answer the question under which circumstances the restriction of atwo-dimensional conformal field theory to the boundary already determines the entireconformal field theory The first work is set in Minkowski space and uses operator alge-bras and subfactors while the second work is set in Euclidean space and uses modulartensor categories

Theorem 321 Let (A|Acl ιcl-op) be a Cardy C|C2plusmn-algebra such that dim A = 0 andAcl is simple Then A is special and (A|Acl ιcl-op) sim= (A|Z(A) e) as Cardy algebras

Proof By Proposition 316 A is simple and special Since Acl is simple the algebramap ιcl-op Acl rarr R(A) is either zero or a monomorphism But ιcl-op ηcl = ηR(A)and so ιcl-op = 0 Thus ιcl-op is monic By Lemma 218 (ii) ζminus1

A ιlowastcl-op is epi The Cardycondition (314) implies

ιcl-op ζminus1A ιlowastcl-op = ζminus1

A PlR(A) = e r (339)

Composing this with e r from the left yields e r ιcl-op ζminus1A ιlowastcl-op = e r =

ιcl-op ζminus1A ιlowastcl-op Since ζminus1

A ιlowastcl-op is epi we have

e r ιcl-op = ιcl-op (340)

Actually (340) also follows from (313) and specialness of R(A) We will prove that

( fop fcl) (A|Acl ιcl-op) minusrarr (A|Z(A) e) where fop = idA fcl = r ιcl-op

(341)

is an isomorphism of Cardy algebrasfcl is an algebra map Compatibility with the units follows since ιcl-op is an algebramap

fcl ηcl = r ιcl-op ηcl = r ηR(A) = ηZ(A) (342)

Compatibility with the multiplication also follows since ιcl-op is an algebra map

m Z(A) ( fcl otimes fcl) = r m R(A) (e otimes e) (r otimes r) (ιcl-op otimes ιcl-op)

= r m R(A) (ιcl-op otimes ιcl-op) = r ιcl-op mcl = fcl mcl (343)

where in the second step we used (340)

Cardy Algebras and Sewing Constraints I 903

fcl is an isomorphism As above since fcl is an algebra map and since Acl is simple fcl

has to be monic By Lemma 319 rlowast = ζminus1A e Thus f lowast

cl = ιlowastcl-op rlowast = ζminus1A ιlowastcl-op e and

fcl f lowastcl = r ιcl-op ζminus1

A ιlowastcl-op e = r e r e = idZ(A) (344)

and so fcl is also epi and hence isofcl is a coalgebra map Since fcl is an algebra map so is f minus1

cl By (344) f minus1cl = f lowast

cland by Lemma 218 (iii) this implies that fcl is a also coalgebra mapThe diagram (315) commutes Commutativity of (315) is equivalent to e fcl = ιcl-opwhich holds by (340)

Let A be a special symmetric Frobenius algebra So far we have seen that (A|Z(A) e)is a Cardy algebra and that all Cardy algebras with Aop = A and simple Acl are of thisform It is now natural to ask if every simple Acl does occur as part of a Cardy algebraThe following theorem provides an affirmative answer Recall that for an A-left moduleM the object Mor otimesA M is an algebra (see eg [KR Lem 42])

Theorem 322 If Acl is a simple modular invariant commutative symmetric FrobeniusC2plusmn-algebra then there exist a simple special symmetric Frobenius C-algebra A and amorphism ιcl-op Acl rarr R(A) such that

(i) Acl sim= Z(A) as Frobenius algebras(ii) (A|Acl ιcl-op) is a Cardy C|C2plusmn-algebra

(iii) T (Acl) sim= oplusκisinJ Morκ otimesA Mκ as algebras where MκκisinJ is a set of representa-

tives of the isomorphism classes of simple A-left modules

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis haploid It then follows from Theorem 34 that Acl is special By Proposition 224T (Acl) is a special symmetric Frobenius algebra in C Thus T (Acl) = oplusi Ai where theAi are simple symmetric Frobenius algebras We will show that at least one of the Aiis special Since T (Acl) is special we have mT (Acl) T (Acl) = ζ idT (Acl) for someζ isin C

times Restricting this to the summand Ai shows mi i = ζ idAi FurthermoreεT (Acl) ηT (Acl) = ξ id1 for some ξ isin C

times But εT (Acl) ηT (Acl) = sumi εi ηi and so

at least one of the εi ηi has to be nonzero Therefore at least one of the Ai is speciallet A equiv Ai be this summand We denote the embedding A rarr T (Acl) by e0 and therestriction T (Acl) A by r0 Notice that r0 is an algebra homomorphism Define

ιcl-op = χ (r0) Acl minusrarr R(A) (345)

By Proposition 216 ιcl-op is an algebra homomorphism Next we verify the centre con-dition (313) or rather its equivalent form (320) By substituting the definitions onecan convince oneself that the commutativity mcl cAclAcl = mcl of Acl in C2plusmn implies thecondition mT (Acl) = mT (Acl) in C see [K2 Prop 36] Here T (Acl)otimes T (Acl) rarrT (Acl)otimes T (Acl) is given by

=oplusmn

(idCl

notimes cCl

m Crnotimes idCr

m

)

(cCl

m Clnotimes cminus1

Crn C

rm

)

(idCl

motimes cminus1

Cln C

rm

otimes idCrn

)

(346)

and we decomposed Acl as Acl = oplusnCln times Cr

n As a consequence we obtain the identity

r0 mT (Acl) (idAcl otimes e0) = r0 mT (Acl) (idAcl otimes e0) (347)

904 L Kong I Runkel

Using that r0 is an algebra map and that by definition ιcl-op = r0 we obtain (320)In order to show that (A|Acl ιcl-op) is a Cardy algebra it remains to show that the

Cardy condition (314) is satisfied We will demonstrate this via a detour by first provingthat Acl sim= Z(A) as Frobenius algebras

Recall the notations e and r given in (331) Using the centre condition (320) onecan check that Pl

R(A) ιcl-op = m R(A) R(A) ιcl-op By specialness of A we have

m A A = ζA idA and so together with e r = ζminus1A Pl

R(A) we get

e r ιcl-op = ιcl-op (348)

Next consider the morphism

fcl = r ιcl-op Acl minusrarr Z(A) (349)

By the same derivation as in (342) and (343) one sees that fcl is an algebra map Inparticular fcl ηcl = ηZ(A) = 0 and so fcl = 0 Since Acl is simple fcl has to be amonomorphism

By the same argument as used in the proof of Theorem 34 (ii) up to isomorphismAcl is the unique simple local Acl-(left-)module The algebra monomorphism fcl turnsZ(A) into an Acl-module Since Z(A) is commutative it is local as an Acl-module andso Z(A) sim= AoplusN

cl for some N ge 1 By construction A is a simple special symmetricFrobenius algebra Proposition 225 and Corollary 227 show that R(A) inherits all theseproperties and thus Z(A) is simple (see the comment below Eq (332)) By Theorem318 Z(A) is modular invariant and then by Theorem 34 (i) dim Z(A) = Dim C Thisimplies that N = 1 in Z(A) sim= AoplusN

cl and so fcl is in fact an isomorphismSince Acl and Z(A) are both haploid we have εZ(A) fcl = ξ εcl for some ξ isin C

timesThe counit uniquely determines the Frobenius structure on Acl and Z(A) (see eg [FRSLemma 37]) so that fcl is a coalgebra isomorphism iff ξ = 1 To compute ξ we composethe above identity with ηcl from the right Defining ζcl via εcl ηcl = ζminus1

cl Dim C middot id1 andusing (333) gives ξ = dim A ζclζ

2A By rescaling the comultiplication and the counit

of A and consequently changing ζA we can always achieve ξ = 1 This proves part (i)of the theorem

Equation (348) implies that ιcl-op = e fcl Since fcl is an isomorphism of Frobeniusalgebras by Lemmas 218 and 319 we have

ιcl-op ιlowastcl-op = e fcl f lowastcl elowast = ζA e r = Pl

R(A) (350)

Thus (A|Acl ιcl-op) is a Cardy algebra This proves part (ii) of the theoremPart (iii) can be seen as follows By [KR Prop 43] T Z(A) sim= oplusκisinJ Mor

κ otimesA Mκ asalgebras Together with the observation that T ( fcl) T (Acl) rarr T Z(A) is an isomor-phism of algebras this proves part (iii) Remark 323 Part (i) of Theorem 322 was announced by Muumlger [Muuml2] We provide anindependent proof in the setting of Cardy algebras

The above theorem together with Lemma 211 and Theorem 34 shows that a sim-ple commutative symmetric Frobenius C2plusmn-algebra Acl with dim Acl = Dim C is alwayspart of a Cardy algebra (Aop|Acl ιcl-op) for some simple special symmetric Frobeniusalgebra Aop in C However the above proof also illustrates that Aop is not unique Thisraises the question how two Cardy algebras with a given Acl can differ This question isanswered by [KR Thm 11] which in the present framework can be restated as follows

Cardy Algebras and Sewing Constraints I 905

Theorem 324 If (A(i)op |A(i)cl ι(i)cl-op) i = 1 2 are two Cardy C|C2plusmn-algebras such that

A(i)cl is simple and dim A(i)op = 0 for i = 1 2 then A(1)clsim= A(2)cl as algebras if and only

if A(1)op and A(2)op are Morita equivalent

Proof Theorem 11 in [KR] is stated for A(i)op being non-degenerate algebras and A(i)cl =Z(A(i)op ) for i = 1 2 By Proposition 316 A(i)op are simple and special for i = 1 2

Then by [KR Lem 21] A(i)op are non-degenerate algebras By Theorem 321 we have

A(i)clsim= Z(A(i)op ) as Frobenius algebras Finally by [KR Thm 11] Z(A(1)op ) sim= Z(A(2)op )

as algebras iff A(1)op and A(2)op are Morita equivalent Let Cmax(C2plusmn) be the set of equivalence classes [B] of simple modular invariant

commutative symmetric Frobenius algebras B in C2plusmn Two such algebras B and B prime areequivalent if B and B prime are isomorphic as algebras (but not necessarily as Frobenius alge-bras) Let Msimp(C) be the set of Morita classes of simple special symmetric Frobeniusalgebras in C Define the map z Msimp(C) rarr Cmax(C2plusmn) by z A rarr [Z(A)] whereA denotes the Morita class of A From Theorem 322 (i) and [KR Thm 11] we learn

Corollary 325 The map z Msimp(C) rarr Cmax(C2plusmn) is a bijection

A Appendix

A1 Proof of Lemma 27 We will show that if (F ψ F2 ψ

F0 ) is a colax tensor functor

from C1 to C2 then (G φG2 φ

G0 ) is a lax tensor functor from C2 to C1 Applying this

result to the opposed categories then gives the converse statementWe need to show that φG

0 and φG2 make the diagrams (21) and (22) commute We

first prove the commutativity of (21) Consider the following diagram

G(A)otimes (G(B)otimes G(C))

GψF2 δ

idG(A)otimesφG2 G(A)otimes G(B otimes C)

GψF2 δ

G(FG(A)otimes F(G(B)otimes G(C)))

G(F(idG(A))otimesF(φG2 ))

G(idFG(A)otimesψ F2 )

G(FG(A)otimes FG(B otimes C))

G(ρAotimesρBotimesC )

G(FG(A)otimes (FG(B)otimes FG(C)))

G(ρAotimes(ρBotimesρC )) G(A otimes (B otimes C))

(A1)

The top subdiagram is commutative because of the naturality of Gψ F2 δ The commu-

tativity of the bottom subdiagram follows from the following identities

(ρB otimes ρC ) ψ F2 = (ρB otimes ρC ) ψ F

2 ρF Fδ

= ρBotimesC FG(ρB otimes ρC ) FG(ψ F2 ) Fδ

= ρBotimesC F(φG2 ) (A2)

as a map F(G(B) otimes G(C)) rarr B otimes C The commutativity of (A1) implies that thecomposition of maps in the left column in (21) can be replaced by

G(ρA otimes (ρB otimes ρC )) G((idFG(A) otimes ψ F

2 ) ψ F2

) δ (A3)

906 L Kong I Runkel

Similarly we can show that the composition of maps in the right column in (21) can bereplaced by

G((ρA otimes ρB)otimes ρC ) G((ψ F

2 otimes idFG(C)) ψ F2

) δ (A4)

Using the commutativity of (23) it is easy to see that (21) with the left and right col-umns of (21) replaced by (A3) and (A4) respectively is commutative Hence (21) iscommutative

Now we prove the commutativity of the first diagram in (22)

φG2 (φG

0 otimes idG(A))

(1)= G(ρ12 otimes ρA) Gψ F2 δ

[(Gψ F

0 δ11)otimes idG(A)

]

(2)= G(ρ12 otimes ρA) Gψ F2 G F(Gψ0 otimes idG(A)) G F(δ11 otimes idG(A)) δ

(3)= G(ρ12 otimes ρA) G(FG(ψ F0 )otimes idFG(A)) Gψ F

2 G F(δ11 otimes idG(A)) δ(4)= G(id12 otimes ρA) G

([ψ F

0 ρF(11) (Fδ)11] otimes idFG(A)

) Gψ F

2 δ(5)= G(id12 otimes ρA) G(ψ F

0 otimes idFG(A)) Gψ F2 δ

(6)= G(id12 otimes ρA) G(lminus1FG(A)) G F(lG(A)) δ

(7)= G(lminus1A ) GρA δG lG(A)

(8)= G(lminus1A ) lG(A) (A5)

where in step (1) we substituted the definition of φG0 φ

G2 given in (210) in step (2)

we used the naturality of δ in step (3) we used the naturality of Gψ F2 in step (4) we

switched the position between Gρ12 and G FG(ψ F0 ) and the position between Gψ2 and

G F(δ12 otimes idG(A)) using the naturality of ρ and FψG2 respectively in step (5) we applied

the second identity in (28) in step (6) we used (24) in step (7) we used the naturalityof lminus1 and δ in step (8) we used the first identity in (28)

The proof of the commutativity of the second diagram in (22) is similar Thus wehave shown that G is a lax tensor functor

A2 Proof of Lemma 320 To prepare the proof recall that for a given object B isin Cthe modular group P SL(2Z) acts on the space oplusi HomC(B otimes Ui Ui ) see eg [BKSect 31] and [K3 Eq (455)] We will only need the action of S and Sminus1 Let f isinoplusi HomC(B otimes Ui Ui ) Then

S oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A6)

Cardy Algebras and Sewing Constraints I 907

Sminus1 oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A7)

By Lemma 32 to establish that (337) is S-invariant it is enough to prove the identity(34) when f is given by (337) Using (A6) and (A7) we can see that Eq (34) simplysays that oplusi j [RHS of (34)] is invariant under the action of S times S Consider the elementg of oplus jkisinIHomC2plusmn(R(A)otimes (Uor

j times Uk)Uorj times Uk) given by

g =oplusjkisinI

sumα

R(A)

R(A)

R(A)

R(A)

Uorj times Uk

Uorj times Uk

α

α

(A8)

By the above arguments proving S-invariance of (337) is equivalent to proving invari-ance of g under the action of S times S

For i isin I we denote by gi the component of g in(oplus jisinIHomC(A otimes Uor

i otimes Uorj U

orj )

)otimes (opluskisinIHomC(Ui otimes UkUk))

We view the second Hom-space in the above tensor product as a Hom-space in C+ insteadof Cminus It is enough to show that gi is invariant under the action of S times Sminus1 Note thatthe action of Sminus1 in C+ is equivalent to that of S in Cminus

The morphism gi can be canonically identified with a bilinear pairing

( middot middot )i ⎛⎝oplus

jisinIHomC(Uor

j A otimes Uori otimes Uor

j )

⎞⎠

times(oplus

kisinIHomC(UkUi otimes Uk)

)minusrarr C (A9)

as follows For h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) we set

(h1 h2)i = (dim U j dim Uk)minus1 trUor

j timesUk[gi (h1 times h2)] (A10)

908 L Kong I Runkel

When substituting the explicit form of the product m R(A) of R(A) = (A times 1) otimes R(1)after a short calculation one finds

(h1 h2)i =sumα

1

dim U j

A

A

A

A

A A

A α

α

h1

h2

U jUk

Uorj

Ui

Uk (A11)

Here the top morphism PlR(A) has been simplified with the help of the identity

PR(A) m R(A) (PR(A) otimes PR(A))

=(oplus

iisinI((m A A)otimes idUor

i)times idUi

) m R(A) (PR(A) otimes PR(A)) (A12)

which can be checked by direct calculation along the same lines as in the proof of [FrLem 310]

The action of the modular transformation S on oplusiisinI(B otimes Ui Ui ) for B isin C nat-urally induces an action on oplusiisinI(Ui B otimes Ui ) [K3 Prop 514] which we denote bySlowast In the present case we get an action of Slowast on oplus jisinIHomC(Uor

j A otimes Uori otimes Uor

j ) and

opluskisinIHomC(UkUi otimes Uk) Then to show gi is invariant under the action of S times Sminus1

amounts to showing that

(h1 h2)i = ((Sminus1)lowast h1 Slowast h2)i (A13)

Cardy Algebras and Sewing Constraints I 909

for all h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) We have

((Sminus1)lowast h1 Slowast h2)i =sum

mnα

dim Un

DimC

A

A

A

A

A A

A

Uk

α

α

h1

h2

Ui

U j

UmUn

(A14)

Now drag the upper vertex indexed by α in the above graph along its Uorm -leg until it

meets the lower vertex also indexed by α then sum over α and m This gives

((Sminus1)lowast h1 Slowast h2)i =sum

n

dim Un

DimC

A

A

A

A

A A

A

Uk

Un

h1

h2

Ui

U j

(A15)

910 L Kong I Runkel

If we just look at the neighbourhood of the Un-loop in the above graph we see thefollowing subgraphs

sumn

dim Un

DimC

A Aor

Uorj U j

Uk Uork

Un=

sumα

1

dim U j

A Aor Uk Uork

Uorj U j

αα

(A16)

where we have applied [BK Cor 3111] Substituting this subgraph back to the originalgraph in (A15) we obtain

((Sminus1)lowast h1 Slowast h2)i =sumα

1

dim U j

A

A

A

A

A A

A

A

A

Uorj

α

α

h1

h2m A

Ui

U jUk

Uk

(A17)

The graph in (A17) is equal to that in (A11) In order to see this we first drag theldquobubblerdquo (m A A) along A lines and through the m A vertex (because m A A is abimodule map) until it reaches the lower-left leg of the upper vertex indexed by α Thendrag the m A vertex along the (red) dotted line in the above graph Finally we applythe associativity of A (A12) and [Fr Lem 311] Then we see that the graph in (A17)exactly matches with the one in (A11)

Acknowledgements We would like to thank the organisers of the Oberwolfach Arbeitsgemeinschaft ldquoAlge-braic structures in conformal field theoriesrdquo (April 2007) where this work was started for an inspiring meetingWe would further like to thank the Hausdorff Institute for Mathematics in Bonn and the organisers of the stim-ulating meeting ldquoGeometry and Physicsrdquo (May 2008) We are indebted to Alexei Davydov Jens FjelstadJuumlrgen Fuchs Yi-Zhi Huang Alexei Kitaev Urs Schreiber Christoph Schweigert Stephan Stolz and PeterTeichner for helpful discussions andor comments on a draft of this paper The research of IR was partiallysupported by the EPSRC First Grant EPE0050471 the PPARC rolling grant PPC5071451 and the MarieCurie network lsquoSuperstring Theoryrsquo (MRTN-CT-2004-512194)

Cardy Algebras and Sewing Constraints I 911

References

[AN] Alexeevski A Natanzon SM Noncommutative two-dimensional topological field theories andhurwitz numbers for real algebraic curves Sel Math New Ser 12 307ndash377 (2006)

[Bi] Bichon J Cosovereign hopf algebras J Pure Appl Alg 157 121ndash133 (2001)[BK] Bakalov B Kirillov AA Lectures on Tensor Categories and Modular Functors Providence

RI Amer Math Soc 2001[C1] Cardy JL Operator content of two-dimensional conformal invariant theories Nucl Phys

B 270 186ndash204 (1986)[C2] Cardy JL Boundary conditions fusion rules and the verlinde formula Nucl Phys B 324

581ndash596 (1989)[CL] Cardy JL Lewellen DC Bulk and boundary operators in conformal field theory Phys Lett

B 259 274ndash278 (1991)[DMZ] Dong C-Y Mason G Zhu Y-C Discrete series of the Virasoro algebra and the moonshine

module In Algebraic Groups and Their Generalizations Quantum and infinite-dimensional Meth-ods Proc Symp Pure Math 56 Part 2 Providence RI Amer MathSoc 1994 pp 295ndash316

[DP] Day B Pastro C Note on frobenius monoidal functors New York J Math 14 733ndash742 (2008)[ENO1] Etingof PI Nikshych D Ostrik V On fusion categories Ann Math 162 581ndash642 (2005)[ENO2] Etingof PI Nikshych D Ostrik V An analogue of radfordrsquos s4 formula for finite tensor

categories Int Math Research Notices 54 2915ndash2933 (2004)[Fe] Felder G Froumlhlich J Fuchs J Schweigert C Correlation functions and boundary conditions in

rational conformal field theory and three-dimensional topology Comp Math 131 189ndash237 (2002)[FHL] Frenkel IB Huang Y-Z Lepowsky J On axiomatic approaches to vertex operator algebras

and modules Mem Amer Math Soc 104 (1993)[Fj] Fjelstad J Fuchs J Runkel I Schweigert C Uniqueness of openclosed rational cft with

given algebra of open states Adv Theor Math Phys 12 1283ndash1375 (2008)[FK] Froumlhlich J King C The chern-simons theory and knot polynomials Commun Math

Phys 126 167ndash199 (1989)[Fr] Froumlhlich J Fuchs J Runkel I Schweigert C Correspondences of ribbon categories Adv

Math 199 192ndash329 (2006)[FRS] Fuchs J Runkel I Schweigert C Tft construction of rcft correlators I partition functions Nucl

Phys B 646 353ndash497 (2002)[FS] Fuchs J Schweigert C Category theory for conformal boundary conditions Fields Inst Com-

mun 39 25ndash71 (2003)[H1] Huang Y-Z Two-dimensional conformal geometry and vertex operator algebras Progress in

Mathematics Vol 148 Boston Birkhaumluser 1997[H2] Huang Y-Z Rigidity and modularity of vertex tensor categories Comm Contemp Math 10

871ndash911 (2008)[HK1] Huang Y-Z Kong L Open-string vertex algebra category and operads Commun Math

Phys 250 433ndash471 (2004)[HK2] Huang Y-Z Kong L Full field algebras Commun Math Phys 272 345ndash396 (2007)[HK3] Huang Y-Z Kong L Modular invariance for conformal full field algebras httparxivorgabs

math0609570v2[mathQA] 2006[HL] Huang Y-Z Lepowsky J Tensor products of modules for a vertex operator algebra and vertex

tensor categories In Lie Theory and Geometry in honor of Bertram Kostant ed R BrylinskiJ-L Brylinski V Guillemin V Kac Boston Birkhaumluser 1994 pp 349ndash383

[JS] Joyal A Street R Braided tensor categories Adv Math 102 20ndash78 (1993)[K1] Kong L Full field algebras operads and tensor categories Adv Math 213 271ndash340 (2007)[K2] Kong L Open-closed field algebras Commun Math Phys 280 207ndash261 (2008)[K3] Kong L Cardy condition for open-closed field algebras Commun Math Phys 283 25ndash92 (2008)[Ki] Kitaev A Private communication[KO] Kirillov AA Ostrik V On q-analog of mckay correspondence and ade classification of sl(2)

conformal field theories Adv Math 171 183ndash227 (2002)[KR] Kong L Runkel I Morita classes of algebras in modular tensor categories Adv Math 219

1548ndash1576 (2008)[Ld] Lauda AD Frobenius algebras and ambidextrous adjunctions Theo Appl Cat 16 84ndash

122 (2006)[Li1] Li H-S Regular representations of vertex operator algebras Commun Contemp Math 4

639ndash683 (2002)[Li2] Li H-S Regular representations and huang-lepowsky tensor functors for vertex operator

algebras J Alge 255 423ndash462 (2002)

Cardy Algebras and Sewing Constraints I 889

is a Frobenius algebra in C2plusmn where the structure morphisms were given in (217) and(218) In the case A = 1 it was proved in [Muuml1 Prop 41] (see also [Fr Lem 619]and [K1 Thm 52]) that (258) is a commutative simple symmetric normalised-specialFrobenius algebra in C2plusmn In fact given a Frobenius algebra A in C it is straightforwardto verify that the structure morphisms in (258) are precisely those of (A times 1)otimes R(1)cf Sect 22

Proposition 223 (R φR2 φ

R0 ψ

R2 ψ

R0 ) is a Frobenius functor

Proof Using the explicit graphical expression of φR2 φ

R0 ψ

R2 ψ

R0 it is easy to see that

the commutativity of the diagrams (25) and (26) are equivalent to the statement thatR(1) with structure morphisms as in (258) is a Frobenius algebra in C2plusmn The latterstatement is true by [Muuml1 Prop 41]

From Lemma 220 and Proposition 223 we see that T and R take Frobenius alge-bras to Frobenius algebras The following two propositions show how the properties ofFrobenius algebras are transported

Proposition 224 Let A be a Frobenius algebra in C2plusmn Then T (A) is a Frobenius algebrain C and

(i) A is symmetric iff T (A) is symmetric(ii) A is (normalised-)special iff T (A) is (normalised-)special

Proof For part (i) write A as a direct sum oplusNn=1 Al

n times Arn Then the maps IT (A) I prime

T (A) T (Aor) rarr T (A)or defined in (220) are given by

IT (A) = I primeT (A) = oplusN

n=1c(Aln)

or(Arn)

or (259)

Therefore by (221) T (A) = primeT (A) is equivalent to T (A) = T (prime

A) Since byLemma 222 T is injective on morphisms this proves part (i) Part (ii) can be checkedin the same way for example the condition mT (A) T (A) = ζ idT (A) is easily checkedto be equivalent to T (m A A) = ζ T (idA) Proposition 225 Let A be a Frobenius algebra A in C Then R(A) is a Frobeniusalgebra in C2plusmn and

(i) A is symmetric iff R(A) is symmetric(ii) A is (normalised-)special iff R(A) is (normalised-)special

Proof Recall that the structure morphisms of the Frobenius algebra R(A) are equal tothose of (A times 1) otimes R(1) Using this equality part (i) and (ii) follow because R(1) issymmetric and normalised-special For example

m R(A) R(A) = [(m A A)times id1] otimes idR(1) = R(m A A) (260)

so that m R(A) R(A) = ζ idR(A) is equivalent to R(m A A) = ζ R(idA) which byLemma 222 is equivalent to m A A = ζ idA

The functor R has one additional property not shared by T namely R takes absolutelysimple algebras to absolutely simple algebras We will see explicitly in Sect 33 thatthis is not true for T

890 L Kong I Runkel

Lemma 226 For a C-algebra A the map

R HomA|A(A A) rarr HomR(A)|R(A)(R(A) R(A)) (261)

given by f rarr R( f ) is well-defined and an isomorphism

Proof Since R is a lax tensor functor R(A) is naturally a R(A)-bimodule It is easy tosee that R in (261) is a well-defined map R(A) is also naturally a R(1)-bimodule whichcan be identified with the induced R(1)-bimodule structure on (A times 1)otimes R(1) wherethe left R(1) action on (Atimes1)otimes R(1) is given by (idAtimes1 otimesm R(1))(cminus1

Atimes1R(1)otimes idR(1))We have the following natural isomorphisms

HomR(1)|R(1)(R(A) R(A))sim=minusrarr HomC2plusmn(A times 1 R(A))

χminus1

minusminusrarr HomC(A A) (262)

which by (253) are given by for f isin HomR(1)|R(1)(R(A) R(A))

f rarr f prime = f (idAtimes1 otimes ηR(1)) rarr f primeprime = ρ T ( f prime) (263)

and its inverse is given by for g isin HomC(A A)

g rarr gprime = R(g) δ rarr gprimeprime = (idAtimes1 otimes m R(1)) (gprime otimes idR(1)) (264)

where gprimeprime is indeed a R(1)-bimodule map due to the commutativity of R(1) It is easy tocheck that gprimeprime = R(g) in (264) Therefore R gives an isomorphism from HomC(A A)to HomR(1)|R(1)(R(A) R(A)) Moreover one verifies that R(g) is an R(A)-bimodulemap iff g is an A-bimodule map In other words R g rarr R(g) gives an isomorphism

HomA|A(A A)sim=minusrarr HomR(A)|R(A)(R(A) R(A))

Corollary 227 Let A be a C-algebra

(i) A is absolutely simple iff R(A) is absolutely simple(ii) Let A be in addition Frobenius Then A is simple and special iff R(A) is simple and

special

Proof Part (i) immediately follows from Lemma 226 The statement of part (ii) withoutthe qualifier lsquosimplersquo is proved in Proposition 225 But as in the proof of (iii)rArr(i) inLemma 211 a special Frobenius algebra in a semi-simple category has a semi-simplecategory of bimodules and for a semi-simple C-linear category simple and absolutelysimple are equivalent Part (ii) then follows from part (i)

The following lemma will be needed in Sect 32 below to discuss the properties ofCardy algebras

Lemma 228 Let A be a Frobenius algebra in C Then (δA)lowast = ρA

Proof Recall from (250) that δA is a morphism A rarr T R(A) Since T and R are bothFrobenius functors T R(A) is a Frobenius algebra in C Substituting the definitions aftera short calculation one finds

Cardy Algebras and Sewing Constraints I 891

εT R(A) mT R(A) = Dim(C)oplusiisinI

A Uori Ui A U or

ı Uı

(265)

Substituting this in the definition of (δA)lowast gives again after a short calculation the mor-

phism ρA At an intermediate step one uses that the part of the morphism (δA)lowast which

is made up of Ui and Uı ribbons their duals and the basis morphisms λ(iı)0 andϒ(iı)0can be replaced by 1

dim Uimiddot dUi

3 Cardy Algebras

In this section we start by investigating the properties of Frobenius algebras which sat-isfy the so-called modular invariance condition We then give two definitions of a Cardyalgebra and prove their equivalence Finally in Sect 33 we study the properties of thesealgebras and state our main results

We fix a modular tensor category C Recall that C2plusmn is an abbreviation for C+ Cminus

31 Modular invariance In C2plusmn we define the object K and the morphism ω K rarr Kas

K =oplus

i jisinIUi times U j ω =

sumi jisinI

dim Ui dim U j

Dim C idUi timesU j (31)

They have the property (see eg [BK Cor 3111])

ω

K

(Uk times Ul )or

(Uk times Ul )or

Ui times U j

Ui times U j

= δik δ jlDim C

dim Ui dim U jbUi timesU j dUi timesU j (32)

892 L Kong I Runkel

Definition 31

(i) Let A B be objects of C2plusmn A morphism f A otimes B rarr B is called S-invariant iff

A

B

W

W

f =

A

B

W

W

f

K

ω(33)

holds for all for W isin C2plusmn(ii) A C2plusmn-algebra (Aclmcl ηcl) is called modular invariant iff θAcl = idAcl and mcl

is S-invariant

Lemma 32 The morphism f A otimes B rarr B is S-invariant if and only if

A

B

Ui times U j

Ui times U j

f = Dim Cdim Ui dim U j

sumα

A

B

B

f

Ui times U j

Ui times U j

α

α

(34)

holds for all i j isin I

Proof Condition (33) holds for all W iff it holds for all W = Ui times U j i j isin I so itis enough to show that the right-hand side of (33) with W = Ui times U j is equal to theright-hand side of (34) Recall the notation for basis morphisms in (236) Starting from(33) write

idBor otimes idUi timesU j =⎛⎝sum

klα

(b(ktimeslα)

B

)or (

bB(ktimeslα)

)or⎞⎠ otimes idUi timesU j (35)

and then apply (32) The graphical representation of the resulting morphism can bedeformed to give (34) Remark 33 As shown in [K3 Sect 61] the modular invariance condition of a C2plusmn-alge-bra exactly coincides with the modular invariance condition for torus 1-point correlationfunctions of a genus-01 closed CFT In particular the condition θAcl = idAcl is equiv-alent to invariance under the modular transformation T τ rarr τ + 1 and the condition(34) with f = mcl is equivalent to invariance under S τ rarr minus 1

τ Combining the

modular invariance condition with the genus zero properties of a genus-01 closed CFTresults in a modular invariant commutative symmetric Frobenius algebra in C2plusmn

Cardy Algebras and Sewing Constraints I 893

Let Acl be a modular invariant C2plusmn-algebra Evaluating (34) for f = mcl composingit with ηcl otimes idUi timesU j and taking the trace implies the following identity

Zi j = 1

DimCAclUi timesU j where

(36)

Zi j = dimC HomC2plusmn(UitimesU j Acl)

Decomposing Acl into simple objects this gives

Zi j =sum

klisinISik Zkl Sminus1

l j where Si j = si jradic

DimC (37)

which in CFT terms is of course nothing but the invariance of the torus partition functionunder the modular S-transformation

The following theorem gives a simple criterion for modular invariance

Theorem 34 Let Acl be a haploid commutative symmetric Frobenius algebra in C2plusmn

(i) If Acl is modular invariant then dim Acl = Dim C(ii) If dim Acl = Dim C then Acl is special and modular invariant

Proof Part (i) Since Acl is haploid for i = j = 0 Equation (36) reduces to 1 =dim AclDimCPart (ii) By the same reasoning as in the proof of (iii)rArr(i) in Lemma 211 one showsthat Acl is special Thus mcl cl = ζclidAcl for some ζcl = 0

By [KO Thm 45] the category (C2plusmn)locAcl

of local Acl-modules is again a mod-

ular tensor category and Dim (C2plusmn)locAcl

= (Dim C dim Acl)2 (see [Fr Prop 321 amp

Rem 323] for the same statement in the notation used here) Thus by assumption wehave Dim(C2plusmn)loc

Acl= 1 It then follows from [ENO1 Thm 23] that up to isomorphism

(C2plusmn)locAcl

has a unique simple object (namely the tensor unit) In other words every simplelocal Acl-module is isomorphic to Acl (seen as a left-module over itself)

We have the following isomorphisms between morphism spaces [FS Prop 47 amp 411]

HomAcl(Acl otimes (Ui times U j ) Acl) sim= HomC2plusmn(Ui times U j Acl)

HomAcl(Acl Acl otimes (Ui times U j )) sim= HomC2plusmn(AclUi times U j )

Using these to transport the bases (236) from the right to the left we obtain bases bα(i j)αof HomAcl(Acl otimes (Ui times U j ) Acl) and b(i j)

β β of HomAcl(Acl Acl otimes (Ui times U j )) Thesecan be expressed graphically as

894 L Kong I Runkel

bα(i j) =

Acl

Acl

Acl

Ui times U j

α

b(i j)β = DimC

dim Ui dim U j

1

ζcl

Acl

Ui times U jAcl

Acl

β

(38)

where the nonzero factor in b(i j)β is included for convenience Notice that bα(i j) b(i j)

β isa left Acl-module map Since Acl is simple as a left module over itself we have

bα(i j) b(i j)β = λαβ idAcl (39)

for some λαβ isin C By computing tr(bα(i j) b(i j)β ) it is easy to verify that λαβ = δαβ

We will now prove the following identity

1

ζcl

Acl

Acl

Ui times U j

Ui times U j

Acl

=sumα

DimCdim Ui dim U j

1

ζcl

Acl

Ui times U j

Ui times U j

Acl

Acl

Acl

α

α

(310)

One checks that the left-hand side of this equation is an idempotent which we denote byζminus1

cl PlAcl(Ui timesU j ) cf [Fr Sect 31] By [Fr Prop 41] the image Im(ζminus1

cl PlAcl(Ui timesU j ))

is a local Acl-module and hence isomorphic to AoplusNcl for some N isin Zge0

All left-module morphisms from Acl to Aclotimes(Ui timesU j ) are linear combinations of the

b(i j)β Furthermore one verifies that ζminus1

cl PlAcl(Ui timesU j )b(i j)

β = b(i j)β Therefore the b(i j)

β

describe precisely the image of the idempotent ie ζminus1cl Pl

Acl(Ui timesU j ) = sum

α b(i j)α bα(i j)

which is nothing but (310) Composing (310) with ζcl middot εcl otimes idUi timesU j from the left (iefrom the top) produces (34)

In addition since Acl is commutative symmetric Frobenius it satisfies θAcl = idAcl

[Fr Prop 225] Altogether this shows that Acl is modular invariant Remark 35 As we were writing this paper we heard that the results in Theorem 34were obtained independently by Kitaev and Muumlger [Ki]

Remark 36 Setting i = j = 0 in (36) gives the identity dim Acl = Z00 Dim C [KRProp 23] Combining this with Theorem 34 (ii) one may wonder if a general modularinvariant commutative symmetric Frobenius algebra Acl in C2plusmn is isomorphic to a directsum of simple such algebras However this is not so For example one can take the com-mutative symmetric Frobenius algebra Acl = C[x]x2 in the category of vector spacesequipped with the non-degenerate trace ε(ax + b) = a In this case the modular invari-ance condition holds automatically but Acl is clearly not a direct sum of two algebras

Cardy Algebras and Sewing Constraints I 895

For a general modular tensor category C the algebra C[x]x2 R(1) understood as analgebra in C2plusmn via the braided monoidal isomorphism Vect f (C) C2plusmn rarr C2plusmn providesanother counter-example

32 Two definitions Define a morphism PlA A rarr A for a Frobenius algebra A in C

or C2plusmn as follows [Fr Sect 24]

PlA =

A

A

A

m

(311)

If A is also commutative and obeys m A A = ζA idA we have PlA = ζA idA In partic-

ular this holds if A is commutative and special Using the fact that the Frobenius algebraR(1) is commutative and normalised-special one can check that Pl

R(A) R(A) rarr R(A)takes the following form

PlR(A) =

oplusiisinI

A

A

Uori

Uori

A

m

times

Ui

Ui

(312)

With these ingredients we can now give the first definition of a Cardy C|C2plusmn-algebrawhich was introduced in [K3 Def 514] cf Remark 315 below

Definition 37 (CardyC|C2plusmn-algebra I) A CardyC|C2plusmn-algebra is a triple (Aop|Acl ιcl-op)where (Aclmcl ηclcl εcl) is a modular invariant commutative symmetric Frobe-nius C2plusmn-algebra (Aopmop ηopop εop) is a symmetric Frobenius C-algebra andιcl-op Acl rarr R(Aop) an algebra homomorphism such that the following conditionsare satisfied

(i) Centre condition

Acl R(Aop)

R(Aop)

R(Aop)

ιcl-op

m R(Aop)

=

Acl

ιcl-op

R(Aop)

R(Aop)

R(Aop)

m R(Aop)

(313)

896 L Kong I Runkel

(ii) Cardy condition

ιcl-op ιlowastcl-op =

R(Aop)

R(Aop)

R(Aop)

(314)

Remark 38

(i) The name ldquoCardy C|C2plusmn-algebrardquo in Definition 37 was chosen because many ofthe important ingredients were first studied by Cardy the modular invariance ofthe closed theory [C1] the consistency of the annulus amplitude [C2] and thebulk-boundary OPE [CL] On the other hand the boundary-boundary OPE andthe OPE analogue of the centre condition were first considered in [Lw]

(ii) One can easily see that in the special case that C is the category Vect f (C) of finite-dimensional C-vector spaces a Cardy C|C2plusmn-algebra gives exactly the algebraicformulation of two-dimensional open-closed topological field theory over C (cfRemark 614 in [K3]) see [Lz Sect 48] [Mo Thm 11] [AN Thm 45] [LPCor 43] [MS Sect 22] When passing to a general modular tensor category Cthere are two important differences to the two-dimensional topological field the-ory Firstly the algebras Acl and Aop now live in different categories which inparticular affects the formulation of the centre condition and the Cardy conditionSecondly the modular invariance condition has to be imposed on Acl In the caseC = Vect f (C) modular invariance holds automatically

Definition 39 A homomorphism of Cardy C|C2plusmn-algebras (A(1)op |A(1)cl ι(1)cl-op) rarr (A(2)op

|A(2)cl ι(2)cl-op) is a pair ( fop fcl) of Frobenius algebra homomorphisms fop A(1)op rarr A(2)op

and fcl A(1)cl rarr A(2)cl such that the diagram

A(1)cl

fcl

ι(1)cl-op

A(2)cl

ι(2)cl-op

R(A(1)op )

R( fop) R(A(2)op )

(315)

commutes

Remark 310 Since a homomorphism of Frobenius algebras is invertible (cf Lemma218 (iv)) a homomorphism of Cardy algebras is always an isomorphism

For a homomorphism ( fop fcl) of Cardy C|C2plusmn-algebras using the commutativity of(315) and the fact that fcl and fop are both algebra and coalgebra homomorphisms itis easy to show that (315) commutes iff

R(A(1)op )R( fop)

(ι(1)cl-op)

lowast

R(A(2)op )

(ι(2)cl-op)

lowast

A(1)cl

fcl A(2)cl

(316)

Cardy Algebras and Sewing Constraints I 897

commutesLet (Aop|Acl ιcl-op) be a Cardy C|C2plusmn-algebra Define the morphism

ιcl-op = χminus1(ιcl-op) T (Acl) minusrarr Aop (317)

Decompose Acl as Acl = oplusNn=1Cl

n times Crn such that Cl

1 times Cr1 = ηcl(1 times 1) We use ι(n)cl-op

to denote the restriction of ιcl-op to Cln times Cr

n and ι(n)cl-op to denote the restriction of ιcl-op

to Cln otimes Cr

n We introduce the following graphical notation

ι(n)cl-op =

Cln Cr

n

Aop

(318)

By (243) ιcl-op can be expressed in terms of ιcl-op as follows

ιcl-op =Noplus

n=1

oplusiisinI

sumα

Cln

Crn

Aop

α

Uori

times α

Crn

Ui

(319)

Lemma 311 The centre condition (313) is equivalent to the following condition in C

Cln Cr

n

Aop

Aop

Aop

=

Cln Cr

n

Aop

Aop

Aop

for n = 1 N (320)

Proof First insert (319) and the definition (217) of m R(Aop) into (313) Then apply thecommutativity of R(1) to the left hand side of (313) The equivalence between (313)and (320) follows immediately Remark 312 The centre condition (320) is very natural from the open-closed confor-mal field theory point of view Correlators on the upper half plane are expressed in termsof conformal blocks on the full complex plane The objects Cl

n and Crn are associated to

the field insertion at a point z in the upper half plane and at the complex conjugate pointz in the lower half plane respectively The object Aop corresponds to a field inserted ata point r on the real axis The centre condition (320) simply says that the correlationfunctions in the disjoint domains |z| gt r gt 0 and r gt |z| gt 0 are analytic continuationsof each other see [K2 Prop 118]

898 L Kong I Runkel

Recall that we define ιlowastcl-op Aop rarr T (Acl) as in (230) We introduce the graphicalnotation

ιlowastcl-op =Noplus

n=1

Cln Cr

n

Aop

=Noplus

n=1

Cln Cr

n

Aop

Aop T (minus1Acl) (321)

where the second equality follows from (215) (221) and (259)

Lemma 313 The Cardy condition (314) is equivalent to the following identity in C

Noplusn=1

DimCdim Ui

sumα

Cln

Crn

Crn

α

α

Aop

Aop Uori

Uori

=Aop

Aop

Aop

Uori

Uori

op

mop for all i isin I (322)

Proof By (244) (312) and (319) it is easy to see that (322) is equivalent to thefollowing identity

ιcl-op χ(ιlowastcl-op) = PlR(Aop)

(323)

Therefore it is enough to show that

χ(ιlowastcl-op) = ιlowastcl-op (324)

We have

χminus1(ιlowastcl-op)(1)= T (ιlowastcl-op) δ (2)= T (ιcl-op)

lowast δlowastlowast (3)= (ρ T (ιcl-op)

)lowast (4)= (ιcl-op

)lowast (325)

In step (1) we use the expression (253) for χminus1 step (2) follows from Lemma 218 (v)and Lemma 219 Step (3) is Lemma 218 (i) and Lemma 228 and finally step (4)amounts to substituting (253) and (317) Acting with χ on both sides of the aboveequality produces (324)

Combining Lemmas 311 and 313 and Proposition 216 we obtain the followingequivalent definition of Cardy C|C2plusmn-algebra (recall the graphical notation (318) forιcl-op and (321) for ιlowastcl-op)

Cardy Algebras and Sewing Constraints I 899

Definition 314 (CardyC|C2plusmn-algebra II) A CardyC-algebra is a triple (Aop|Acl ιcl-op)where Acl is a commutative symmetric Frobenius C2plusmn-algebra satisfying property (34)with f = mcl Aop is a symmetric Frobenius C-algebra and ιcl-op T (Acl) rarr Aop isan algebra homomorphism satisfying the conditions (320) and (322)

Remark 315 Up to a choice of normalisation Definition 314 is the same as the orig-inal one in [K3 Def 613] The difference between the two definitions is the factorDim C dim Ui on the left hand side of (322) which in [K3 Def 613] is given byradic

Dim C dim Ui The two definitions are related by rescaling the coproduct cl andcounit εcl of Acl by 1

radicDim C and

radicDim C respectively We chose the convention in

(322) to remove all dimension factors from the expression (314) for the Cardy condition

33 Uniqueness and existence theorems In this subsection we investigate the structureof Cardy algebras We start with the following proposition which when combined withthe results of part II provides an alternative proof of [Fj Prop 422]

Proposition 316 Let (Aop|Acl ιcl-op) be a Cardy C|C2plusmn-algebra If Acl is simple anddim Aop = 0 then Aop is simple and special

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis therefore haploid Restricting the Cardy condition (322) to the case Ui = 1 andcomposing both sides with εop from the left we see that εop kills all terms associated toU j times 1 isin Acl in the sum except for a single 1 times 1 term Thus we obtain the followingidentity

β εop = dAop (mop otimes idAorop) (idAop otimes bAop) (326)

where β isin C Composing with ηop from the right in turn implies that βεop ηop =dim Aop which is nonzero by assumption Thus also β = 0 and εop is a nonzero multipleof the morphism on the right hand side of (326) By [FRS Lem 311] Aop is special

Since Aop is a special Frobenius algebra Aop is semi-simple as an Aop-bimodule(apply [FS Prop 524] to Aop tensored with its opposite algebra) Suppose Aop is not

simple so that we can write Aop = A(1)op oplus A(2)op for nonzero Aop-bimodules A(1)op and A(2)op We denote the canonical embeddings and projections associated to this decompositionas ι12 and π12 We have the identities

mop (ι1 otimes ι2) = 0 εop ηop =2sum

i=1

εop ιi πi ηop (327)

The first identity follows since π1 mop (ι1 otimes ι2) = 0 (as mop gives the left action of

Aop on Aop and hence it preserves A(2)op ) and similarly π2 mop (ι1 otimes ι2) = 0 Thesecond identity is just the completeness of ι12 π12

Since εop ηop = 0 without losing generality we can assume εop ι1 π1 ηop = 0Using that π2 is a bimodule map we compute

π2 [LHS of (322)]Ui =1 ι1 π1 ηop = π2 [RHS of (322)]Ui =1 ι1 π1 ηop

= PlA(2)op

π2 ι1 π1 ηop = 0 (328)

900 L Kong I Runkel

On the other hand using that Acl is haploid that ιcl-op is an algebra map and thatι lowastcl-op is a coalgebra map one can check that the left-hand side of (328) is equal toλ(εop ι1 π1 ηop) π2 ηop for some λ = 0 This implies that π2 ηop = 0 Thusηop = sum

i ιi πi ηop = ι1 π1 ηop Hence we have

0 = π2 ι2 = π2 mop (ηop otimes ι2) = π2 mop ((ι1 π1 ηop)otimes ι2) (329)

However the right-hand side is zero by (327) This is a contradiction and hence Aopmust be simple

To formulate the next theorem we need the notion of the full centre of an algebra [FjDef 49] Recall that an algebra A in a braided tensor category has a left centre and aright centre [VZO] both of which are sub-algebras of A Of these two we will onlyneed the left centre The following definition is [Fr Def 231] which in our setting isequivalent to that of [VZO]

Definition 317 Let A be a symmetric special Frobenius algebra such that m A A =ζA idA

(i) The left centre Cl(A) of A is the image of the idempotent ζminus1A Pl

A(ii) The full centre Z(A) is Cl(R(A))

That ζminus1A Pl

A is an idempotent follows from [FRS Lem 52] when keeping track ofthe factors ζA ([FRS] assumes normalised-special ie ζA = 1) Note that Cl(A) is againan object of C while Z(A) is an object of C2plusmn Let el

A Cl(A) rarr A be the embeddingof Cl(A) into A The left centre is in fact the maximal subobject of A such that

m A cAA (elA otimes idA) = m A (330)

see [Fr Lem 232] This observation explains the name left centre and also makes theconnection to [O Def 15]

The full centre is by definition the image of the idempotent ζminus1A Pl

R(A) R(A) rarrR(A) Since C2plusmn is abelian the idempotent splits and we obtain the embedding andrestriction morphisms

e Z(A) rarr R(A) and r R(A) Z(A) (331)

which obey r e = idZ(A) and e r = ζminus1A Pl

R(A) It follows from Proposition 225 and

[Fr Prop 237] that Z(A) is a commutative symmetric Frobenius algebra in C2plusmn withstructure morphisms5

m Z(A) = r m R(A) (e otimes e) ηZ(A) = r ηR(A)

Z(A) = ζA middot (r otimes r) R(A) e εZ(A) = ζminus1A middot εR(A) e

(332)

Moreover if A is simple then Z(A) is simple and if A is simple and dim A = 0 thenZ(A) is simple and special The normalisation of the counit is such that

εZ(A) ηZ(A) = ζminus2A dim A DimC (333)

5 The normalisation of product and unit is the standard one The factors in the coproduct and counit haveto be included in order for (A|Z(A) e) to be a Cardy algebra see Theorem 318 below The normalisation ofthe counit enters the Cardy condition (314) through the definition of ( middot )lowast

Cardy Algebras and Sewing Constraints I 901

Theorem 318 Let A be a special symmetric Frobenius C-algebra Then (A|Z(A) e)is a Cardy C|C2plusmn-algebra

The proof of this theorem makes use the following two lemmas

Lemma 319 e Z(A) rarr R(A) is an algebra map and elowast = ζA middot r

Proof It follows from [Fr Lem 229] (or by direct calculation using in particularm R(A) R(A) = ζAidR(A)) that

m R(A) (e otimes e) = ζminus1A middot Pl

R(A) m R(A) (e otimes e) (334)

Substituting e r = ζminus1A Pl

R(A) shows that e is compatible with multiplication For theunit one finds

e ηZ(A) = e r ηR(A) = ζminus1A Pl

R(A) ηR(A) = ηR(A) (335)

Thus e is an algebra map For the second statement one computes

elowast (1)= ζA

R(A)

r

R(A)

R(A)

Z(A)

R(A)

Z(A)

r

e

e

(2)=

R(A)

r

R(A)

R(A)

R(A)

Z(A)

(3)=

R(A)

r

R(A)

Z(A)

(4)= ζA middot r (336)

where in (1) the definitions (230) and (332) have been substituted step (2) is e r = ζminus1

A PlR(A) step (3) uses that R(A) is symmetric Frobenius and step (4) is again

e r = ζminus1A Pl

R(A) Lemma 320 Let A be a symmetric Frobenius algebra in C The morphism

PlR(A) m R(A)

(Pl

R(A) otimes PlR(A)

) R(A)otimes R(A) minusrarr R(A) (337)

is S-invariant

The proof of this lemma is a slightly lengthy explicit calculation and has been deferredto Appendix A2

902 L Kong I Runkel

Proof of Theorem 318 That e is an algebra map was proved in Lemma 319 The centrecondition (313) holds by property (330) of the left centre The Cardy condition (314)also is an immediate consequence of Lemma 319

ιcl-op ιlowastcl-op = e (ζA r) = PlR(A) (338)

The full centre Z(A) is a commutative symmetric Frobenius algebra It remains to provemodular invariance That θZ(A) = idZ(A) is implied by commutativity and symmetryof Z(A) [Fr Prop 225] The S-invariance condition (33) follows from Lemma 320In (337) substitute Pl

R(A) = ζA e r and then put the resulting morphism into (33)Compose the resulting equation with e otimes idW from the right (ie from the bottom) andsubstitute the definition (332) of m Z(A) This results in the statement that m Z(A) isS-invariant

The following theorem is analogous to [LR Prop 29] and [Fj Thm 426] whichroughly speaking answer the question under which circumstances the restriction of atwo-dimensional conformal field theory to the boundary already determines the entireconformal field theory The first work is set in Minkowski space and uses operator alge-bras and subfactors while the second work is set in Euclidean space and uses modulartensor categories

Theorem 321 Let (A|Acl ιcl-op) be a Cardy C|C2plusmn-algebra such that dim A = 0 andAcl is simple Then A is special and (A|Acl ιcl-op) sim= (A|Z(A) e) as Cardy algebras

Proof By Proposition 316 A is simple and special Since Acl is simple the algebramap ιcl-op Acl rarr R(A) is either zero or a monomorphism But ιcl-op ηcl = ηR(A)and so ιcl-op = 0 Thus ιcl-op is monic By Lemma 218 (ii) ζminus1

A ιlowastcl-op is epi The Cardycondition (314) implies

ιcl-op ζminus1A ιlowastcl-op = ζminus1

A PlR(A) = e r (339)

Composing this with e r from the left yields e r ιcl-op ζminus1A ιlowastcl-op = e r =

ιcl-op ζminus1A ιlowastcl-op Since ζminus1

A ιlowastcl-op is epi we have

e r ιcl-op = ιcl-op (340)

Actually (340) also follows from (313) and specialness of R(A) We will prove that

( fop fcl) (A|Acl ιcl-op) minusrarr (A|Z(A) e) where fop = idA fcl = r ιcl-op

(341)

is an isomorphism of Cardy algebrasfcl is an algebra map Compatibility with the units follows since ιcl-op is an algebramap

fcl ηcl = r ιcl-op ηcl = r ηR(A) = ηZ(A) (342)

Compatibility with the multiplication also follows since ιcl-op is an algebra map

m Z(A) ( fcl otimes fcl) = r m R(A) (e otimes e) (r otimes r) (ιcl-op otimes ιcl-op)

= r m R(A) (ιcl-op otimes ιcl-op) = r ιcl-op mcl = fcl mcl (343)

where in the second step we used (340)

Cardy Algebras and Sewing Constraints I 903

fcl is an isomorphism As above since fcl is an algebra map and since Acl is simple fcl

has to be monic By Lemma 319 rlowast = ζminus1A e Thus f lowast

cl = ιlowastcl-op rlowast = ζminus1A ιlowastcl-op e and

fcl f lowastcl = r ιcl-op ζminus1

A ιlowastcl-op e = r e r e = idZ(A) (344)

and so fcl is also epi and hence isofcl is a coalgebra map Since fcl is an algebra map so is f minus1

cl By (344) f minus1cl = f lowast

cland by Lemma 218 (iii) this implies that fcl is a also coalgebra mapThe diagram (315) commutes Commutativity of (315) is equivalent to e fcl = ιcl-opwhich holds by (340)

Let A be a special symmetric Frobenius algebra So far we have seen that (A|Z(A) e)is a Cardy algebra and that all Cardy algebras with Aop = A and simple Acl are of thisform It is now natural to ask if every simple Acl does occur as part of a Cardy algebraThe following theorem provides an affirmative answer Recall that for an A-left moduleM the object Mor otimesA M is an algebra (see eg [KR Lem 42])

Theorem 322 If Acl is a simple modular invariant commutative symmetric FrobeniusC2plusmn-algebra then there exist a simple special symmetric Frobenius C-algebra A and amorphism ιcl-op Acl rarr R(A) such that

(i) Acl sim= Z(A) as Frobenius algebras(ii) (A|Acl ιcl-op) is a Cardy C|C2plusmn-algebra

(iii) T (Acl) sim= oplusκisinJ Morκ otimesA Mκ as algebras where MκκisinJ is a set of representa-

tives of the isomorphism classes of simple A-left modules

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis haploid It then follows from Theorem 34 that Acl is special By Proposition 224T (Acl) is a special symmetric Frobenius algebra in C Thus T (Acl) = oplusi Ai where theAi are simple symmetric Frobenius algebras We will show that at least one of the Aiis special Since T (Acl) is special we have mT (Acl) T (Acl) = ζ idT (Acl) for someζ isin C

times Restricting this to the summand Ai shows mi i = ζ idAi FurthermoreεT (Acl) ηT (Acl) = ξ id1 for some ξ isin C

times But εT (Acl) ηT (Acl) = sumi εi ηi and so

at least one of the εi ηi has to be nonzero Therefore at least one of the Ai is speciallet A equiv Ai be this summand We denote the embedding A rarr T (Acl) by e0 and therestriction T (Acl) A by r0 Notice that r0 is an algebra homomorphism Define

ιcl-op = χ (r0) Acl minusrarr R(A) (345)

By Proposition 216 ιcl-op is an algebra homomorphism Next we verify the centre con-dition (313) or rather its equivalent form (320) By substituting the definitions onecan convince oneself that the commutativity mcl cAclAcl = mcl of Acl in C2plusmn implies thecondition mT (Acl) = mT (Acl) in C see [K2 Prop 36] Here T (Acl)otimes T (Acl) rarrT (Acl)otimes T (Acl) is given by

=oplusmn

(idCl

notimes cCl

m Crnotimes idCr

m

)

(cCl

m Clnotimes cminus1

Crn C

rm

)

(idCl

motimes cminus1

Cln C

rm

otimes idCrn

)

(346)

and we decomposed Acl as Acl = oplusnCln times Cr

n As a consequence we obtain the identity

r0 mT (Acl) (idAcl otimes e0) = r0 mT (Acl) (idAcl otimes e0) (347)

904 L Kong I Runkel

Using that r0 is an algebra map and that by definition ιcl-op = r0 we obtain (320)In order to show that (A|Acl ιcl-op) is a Cardy algebra it remains to show that the

Cardy condition (314) is satisfied We will demonstrate this via a detour by first provingthat Acl sim= Z(A) as Frobenius algebras

Recall the notations e and r given in (331) Using the centre condition (320) onecan check that Pl

R(A) ιcl-op = m R(A) R(A) ιcl-op By specialness of A we have

m A A = ζA idA and so together with e r = ζminus1A Pl

R(A) we get

e r ιcl-op = ιcl-op (348)

Next consider the morphism

fcl = r ιcl-op Acl minusrarr Z(A) (349)

By the same derivation as in (342) and (343) one sees that fcl is an algebra map Inparticular fcl ηcl = ηZ(A) = 0 and so fcl = 0 Since Acl is simple fcl has to be amonomorphism

By the same argument as used in the proof of Theorem 34 (ii) up to isomorphismAcl is the unique simple local Acl-(left-)module The algebra monomorphism fcl turnsZ(A) into an Acl-module Since Z(A) is commutative it is local as an Acl-module andso Z(A) sim= AoplusN

cl for some N ge 1 By construction A is a simple special symmetricFrobenius algebra Proposition 225 and Corollary 227 show that R(A) inherits all theseproperties and thus Z(A) is simple (see the comment below Eq (332)) By Theorem318 Z(A) is modular invariant and then by Theorem 34 (i) dim Z(A) = Dim C Thisimplies that N = 1 in Z(A) sim= AoplusN

cl and so fcl is in fact an isomorphismSince Acl and Z(A) are both haploid we have εZ(A) fcl = ξ εcl for some ξ isin C

timesThe counit uniquely determines the Frobenius structure on Acl and Z(A) (see eg [FRSLemma 37]) so that fcl is a coalgebra isomorphism iff ξ = 1 To compute ξ we composethe above identity with ηcl from the right Defining ζcl via εcl ηcl = ζminus1

cl Dim C middot id1 andusing (333) gives ξ = dim A ζclζ

2A By rescaling the comultiplication and the counit

of A and consequently changing ζA we can always achieve ξ = 1 This proves part (i)of the theorem

Equation (348) implies that ιcl-op = e fcl Since fcl is an isomorphism of Frobeniusalgebras by Lemmas 218 and 319 we have

ιcl-op ιlowastcl-op = e fcl f lowastcl elowast = ζA e r = Pl

R(A) (350)

Thus (A|Acl ιcl-op) is a Cardy algebra This proves part (ii) of the theoremPart (iii) can be seen as follows By [KR Prop 43] T Z(A) sim= oplusκisinJ Mor

κ otimesA Mκ asalgebras Together with the observation that T ( fcl) T (Acl) rarr T Z(A) is an isomor-phism of algebras this proves part (iii) Remark 323 Part (i) of Theorem 322 was announced by Muumlger [Muuml2] We provide anindependent proof in the setting of Cardy algebras

The above theorem together with Lemma 211 and Theorem 34 shows that a sim-ple commutative symmetric Frobenius C2plusmn-algebra Acl with dim Acl = Dim C is alwayspart of a Cardy algebra (Aop|Acl ιcl-op) for some simple special symmetric Frobeniusalgebra Aop in C However the above proof also illustrates that Aop is not unique Thisraises the question how two Cardy algebras with a given Acl can differ This question isanswered by [KR Thm 11] which in the present framework can be restated as follows

Cardy Algebras and Sewing Constraints I 905

Theorem 324 If (A(i)op |A(i)cl ι(i)cl-op) i = 1 2 are two Cardy C|C2plusmn-algebras such that

A(i)cl is simple and dim A(i)op = 0 for i = 1 2 then A(1)clsim= A(2)cl as algebras if and only

if A(1)op and A(2)op are Morita equivalent

Proof Theorem 11 in [KR] is stated for A(i)op being non-degenerate algebras and A(i)cl =Z(A(i)op ) for i = 1 2 By Proposition 316 A(i)op are simple and special for i = 1 2

Then by [KR Lem 21] A(i)op are non-degenerate algebras By Theorem 321 we have

A(i)clsim= Z(A(i)op ) as Frobenius algebras Finally by [KR Thm 11] Z(A(1)op ) sim= Z(A(2)op )

as algebras iff A(1)op and A(2)op are Morita equivalent Let Cmax(C2plusmn) be the set of equivalence classes [B] of simple modular invariant

commutative symmetric Frobenius algebras B in C2plusmn Two such algebras B and B prime areequivalent if B and B prime are isomorphic as algebras (but not necessarily as Frobenius alge-bras) Let Msimp(C) be the set of Morita classes of simple special symmetric Frobeniusalgebras in C Define the map z Msimp(C) rarr Cmax(C2plusmn) by z A rarr [Z(A)] whereA denotes the Morita class of A From Theorem 322 (i) and [KR Thm 11] we learn

Corollary 325 The map z Msimp(C) rarr Cmax(C2plusmn) is a bijection

A Appendix

A1 Proof of Lemma 27 We will show that if (F ψ F2 ψ

F0 ) is a colax tensor functor

from C1 to C2 then (G φG2 φ

G0 ) is a lax tensor functor from C2 to C1 Applying this

result to the opposed categories then gives the converse statementWe need to show that φG

0 and φG2 make the diagrams (21) and (22) commute We

first prove the commutativity of (21) Consider the following diagram

G(A)otimes (G(B)otimes G(C))

GψF2 δ

idG(A)otimesφG2 G(A)otimes G(B otimes C)

GψF2 δ

G(FG(A)otimes F(G(B)otimes G(C)))

G(F(idG(A))otimesF(φG2 ))

G(idFG(A)otimesψ F2 )

G(FG(A)otimes FG(B otimes C))

G(ρAotimesρBotimesC )

G(FG(A)otimes (FG(B)otimes FG(C)))

G(ρAotimes(ρBotimesρC )) G(A otimes (B otimes C))

(A1)

The top subdiagram is commutative because of the naturality of Gψ F2 δ The commu-

tativity of the bottom subdiagram follows from the following identities

(ρB otimes ρC ) ψ F2 = (ρB otimes ρC ) ψ F

2 ρF Fδ

= ρBotimesC FG(ρB otimes ρC ) FG(ψ F2 ) Fδ

= ρBotimesC F(φG2 ) (A2)

as a map F(G(B) otimes G(C)) rarr B otimes C The commutativity of (A1) implies that thecomposition of maps in the left column in (21) can be replaced by

G(ρA otimes (ρB otimes ρC )) G((idFG(A) otimes ψ F

2 ) ψ F2

) δ (A3)

906 L Kong I Runkel

Similarly we can show that the composition of maps in the right column in (21) can bereplaced by

G((ρA otimes ρB)otimes ρC ) G((ψ F

2 otimes idFG(C)) ψ F2

) δ (A4)

Using the commutativity of (23) it is easy to see that (21) with the left and right col-umns of (21) replaced by (A3) and (A4) respectively is commutative Hence (21) iscommutative

Now we prove the commutativity of the first diagram in (22)

φG2 (φG

0 otimes idG(A))

(1)= G(ρ12 otimes ρA) Gψ F2 δ

[(Gψ F

0 δ11)otimes idG(A)

]

(2)= G(ρ12 otimes ρA) Gψ F2 G F(Gψ0 otimes idG(A)) G F(δ11 otimes idG(A)) δ

(3)= G(ρ12 otimes ρA) G(FG(ψ F0 )otimes idFG(A)) Gψ F

2 G F(δ11 otimes idG(A)) δ(4)= G(id12 otimes ρA) G

([ψ F

0 ρF(11) (Fδ)11] otimes idFG(A)

) Gψ F

2 δ(5)= G(id12 otimes ρA) G(ψ F

0 otimes idFG(A)) Gψ F2 δ

(6)= G(id12 otimes ρA) G(lminus1FG(A)) G F(lG(A)) δ

(7)= G(lminus1A ) GρA δG lG(A)

(8)= G(lminus1A ) lG(A) (A5)

where in step (1) we substituted the definition of φG0 φ

G2 given in (210) in step (2)

we used the naturality of δ in step (3) we used the naturality of Gψ F2 in step (4) we

switched the position between Gρ12 and G FG(ψ F0 ) and the position between Gψ2 and

G F(δ12 otimes idG(A)) using the naturality of ρ and FψG2 respectively in step (5) we applied

the second identity in (28) in step (6) we used (24) in step (7) we used the naturalityof lminus1 and δ in step (8) we used the first identity in (28)

The proof of the commutativity of the second diagram in (22) is similar Thus wehave shown that G is a lax tensor functor

A2 Proof of Lemma 320 To prepare the proof recall that for a given object B isin Cthe modular group P SL(2Z) acts on the space oplusi HomC(B otimes Ui Ui ) see eg [BKSect 31] and [K3 Eq (455)] We will only need the action of S and Sminus1 Let f isinoplusi HomC(B otimes Ui Ui ) Then

S oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A6)

Cardy Algebras and Sewing Constraints I 907

Sminus1 oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A7)

By Lemma 32 to establish that (337) is S-invariant it is enough to prove the identity(34) when f is given by (337) Using (A6) and (A7) we can see that Eq (34) simplysays that oplusi j [RHS of (34)] is invariant under the action of S times S Consider the elementg of oplus jkisinIHomC2plusmn(R(A)otimes (Uor

j times Uk)Uorj times Uk) given by

g =oplusjkisinI

sumα

R(A)

R(A)

R(A)

R(A)

Uorj times Uk

Uorj times Uk

α

α

(A8)

By the above arguments proving S-invariance of (337) is equivalent to proving invari-ance of g under the action of S times S

For i isin I we denote by gi the component of g in(oplus jisinIHomC(A otimes Uor

i otimes Uorj U

orj )

)otimes (opluskisinIHomC(Ui otimes UkUk))

We view the second Hom-space in the above tensor product as a Hom-space in C+ insteadof Cminus It is enough to show that gi is invariant under the action of S times Sminus1 Note thatthe action of Sminus1 in C+ is equivalent to that of S in Cminus

The morphism gi can be canonically identified with a bilinear pairing

( middot middot )i ⎛⎝oplus

jisinIHomC(Uor

j A otimes Uori otimes Uor

j )

⎞⎠

times(oplus

kisinIHomC(UkUi otimes Uk)

)minusrarr C (A9)

as follows For h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) we set

(h1 h2)i = (dim U j dim Uk)minus1 trUor

j timesUk[gi (h1 times h2)] (A10)

908 L Kong I Runkel

When substituting the explicit form of the product m R(A) of R(A) = (A times 1) otimes R(1)after a short calculation one finds

(h1 h2)i =sumα

1

dim U j

A

A

A

A

A A

A α

α

h1

h2

U jUk

Uorj

Ui

Uk (A11)

Here the top morphism PlR(A) has been simplified with the help of the identity

PR(A) m R(A) (PR(A) otimes PR(A))

=(oplus

iisinI((m A A)otimes idUor

i)times idUi

) m R(A) (PR(A) otimes PR(A)) (A12)

which can be checked by direct calculation along the same lines as in the proof of [FrLem 310]

The action of the modular transformation S on oplusiisinI(B otimes Ui Ui ) for B isin C nat-urally induces an action on oplusiisinI(Ui B otimes Ui ) [K3 Prop 514] which we denote bySlowast In the present case we get an action of Slowast on oplus jisinIHomC(Uor

j A otimes Uori otimes Uor

j ) and

opluskisinIHomC(UkUi otimes Uk) Then to show gi is invariant under the action of S times Sminus1

amounts to showing that

(h1 h2)i = ((Sminus1)lowast h1 Slowast h2)i (A13)

Cardy Algebras and Sewing Constraints I 909

for all h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) We have

((Sminus1)lowast h1 Slowast h2)i =sum

mnα

dim Un

DimC

A

A

A

A

A A

A

Uk

α

α

h1

h2

Ui

U j

UmUn

(A14)

Now drag the upper vertex indexed by α in the above graph along its Uorm -leg until it

meets the lower vertex also indexed by α then sum over α and m This gives

((Sminus1)lowast h1 Slowast h2)i =sum

n

dim Un

DimC

A

A

A

A

A A

A

Uk

Un

h1

h2

Ui

U j

(A15)

910 L Kong I Runkel

If we just look at the neighbourhood of the Un-loop in the above graph we see thefollowing subgraphs

sumn

dim Un

DimC

A Aor

Uorj U j

Uk Uork

Un=

sumα

1

dim U j

A Aor Uk Uork

Uorj U j

αα

(A16)

where we have applied [BK Cor 3111] Substituting this subgraph back to the originalgraph in (A15) we obtain

((Sminus1)lowast h1 Slowast h2)i =sumα

1

dim U j

A

A

A

A

A A

A

A

A

Uorj

α

α

h1

h2m A

Ui

U jUk

Uk

(A17)

The graph in (A17) is equal to that in (A11) In order to see this we first drag theldquobubblerdquo (m A A) along A lines and through the m A vertex (because m A A is abimodule map) until it reaches the lower-left leg of the upper vertex indexed by α Thendrag the m A vertex along the (red) dotted line in the above graph Finally we applythe associativity of A (A12) and [Fr Lem 311] Then we see that the graph in (A17)exactly matches with the one in (A11)

Acknowledgements We would like to thank the organisers of the Oberwolfach Arbeitsgemeinschaft ldquoAlge-braic structures in conformal field theoriesrdquo (April 2007) where this work was started for an inspiring meetingWe would further like to thank the Hausdorff Institute for Mathematics in Bonn and the organisers of the stim-ulating meeting ldquoGeometry and Physicsrdquo (May 2008) We are indebted to Alexei Davydov Jens FjelstadJuumlrgen Fuchs Yi-Zhi Huang Alexei Kitaev Urs Schreiber Christoph Schweigert Stephan Stolz and PeterTeichner for helpful discussions andor comments on a draft of this paper The research of IR was partiallysupported by the EPSRC First Grant EPE0050471 the PPARC rolling grant PPC5071451 and the MarieCurie network lsquoSuperstring Theoryrsquo (MRTN-CT-2004-512194)

Cardy Algebras and Sewing Constraints I 911

References

[AN] Alexeevski A Natanzon SM Noncommutative two-dimensional topological field theories andhurwitz numbers for real algebraic curves Sel Math New Ser 12 307ndash377 (2006)

[Bi] Bichon J Cosovereign hopf algebras J Pure Appl Alg 157 121ndash133 (2001)[BK] Bakalov B Kirillov AA Lectures on Tensor Categories and Modular Functors Providence

RI Amer Math Soc 2001[C1] Cardy JL Operator content of two-dimensional conformal invariant theories Nucl Phys

B 270 186ndash204 (1986)[C2] Cardy JL Boundary conditions fusion rules and the verlinde formula Nucl Phys B 324

581ndash596 (1989)[CL] Cardy JL Lewellen DC Bulk and boundary operators in conformal field theory Phys Lett

B 259 274ndash278 (1991)[DMZ] Dong C-Y Mason G Zhu Y-C Discrete series of the Virasoro algebra and the moonshine

module In Algebraic Groups and Their Generalizations Quantum and infinite-dimensional Meth-ods Proc Symp Pure Math 56 Part 2 Providence RI Amer MathSoc 1994 pp 295ndash316

[DP] Day B Pastro C Note on frobenius monoidal functors New York J Math 14 733ndash742 (2008)[ENO1] Etingof PI Nikshych D Ostrik V On fusion categories Ann Math 162 581ndash642 (2005)[ENO2] Etingof PI Nikshych D Ostrik V An analogue of radfordrsquos s4 formula for finite tensor

categories Int Math Research Notices 54 2915ndash2933 (2004)[Fe] Felder G Froumlhlich J Fuchs J Schweigert C Correlation functions and boundary conditions in

rational conformal field theory and three-dimensional topology Comp Math 131 189ndash237 (2002)[FHL] Frenkel IB Huang Y-Z Lepowsky J On axiomatic approaches to vertex operator algebras

and modules Mem Amer Math Soc 104 (1993)[Fj] Fjelstad J Fuchs J Runkel I Schweigert C Uniqueness of openclosed rational cft with

given algebra of open states Adv Theor Math Phys 12 1283ndash1375 (2008)[FK] Froumlhlich J King C The chern-simons theory and knot polynomials Commun Math

Phys 126 167ndash199 (1989)[Fr] Froumlhlich J Fuchs J Runkel I Schweigert C Correspondences of ribbon categories Adv

Math 199 192ndash329 (2006)[FRS] Fuchs J Runkel I Schweigert C Tft construction of rcft correlators I partition functions Nucl

Phys B 646 353ndash497 (2002)[FS] Fuchs J Schweigert C Category theory for conformal boundary conditions Fields Inst Com-

mun 39 25ndash71 (2003)[H1] Huang Y-Z Two-dimensional conformal geometry and vertex operator algebras Progress in

Mathematics Vol 148 Boston Birkhaumluser 1997[H2] Huang Y-Z Rigidity and modularity of vertex tensor categories Comm Contemp Math 10

871ndash911 (2008)[HK1] Huang Y-Z Kong L Open-string vertex algebra category and operads Commun Math

Phys 250 433ndash471 (2004)[HK2] Huang Y-Z Kong L Full field algebras Commun Math Phys 272 345ndash396 (2007)[HK3] Huang Y-Z Kong L Modular invariance for conformal full field algebras httparxivorgabs

math0609570v2[mathQA] 2006[HL] Huang Y-Z Lepowsky J Tensor products of modules for a vertex operator algebra and vertex

tensor categories In Lie Theory and Geometry in honor of Bertram Kostant ed R BrylinskiJ-L Brylinski V Guillemin V Kac Boston Birkhaumluser 1994 pp 349ndash383

[JS] Joyal A Street R Braided tensor categories Adv Math 102 20ndash78 (1993)[K1] Kong L Full field algebras operads and tensor categories Adv Math 213 271ndash340 (2007)[K2] Kong L Open-closed field algebras Commun Math Phys 280 207ndash261 (2008)[K3] Kong L Cardy condition for open-closed field algebras Commun Math Phys 283 25ndash92 (2008)[Ki] Kitaev A Private communication[KO] Kirillov AA Ostrik V On q-analog of mckay correspondence and ade classification of sl(2)

conformal field theories Adv Math 171 183ndash227 (2002)[KR] Kong L Runkel I Morita classes of algebras in modular tensor categories Adv Math 219

1548ndash1576 (2008)[Ld] Lauda AD Frobenius algebras and ambidextrous adjunctions Theo Appl Cat 16 84ndash

122 (2006)[Li1] Li H-S Regular representations of vertex operator algebras Commun Contemp Math 4

639ndash683 (2002)[Li2] Li H-S Regular representations and huang-lepowsky tensor functors for vertex operator

algebras J Alge 255 423ndash462 (2002)

890 L Kong I Runkel

Lemma 226 For a C-algebra A the map

R HomA|A(A A) rarr HomR(A)|R(A)(R(A) R(A)) (261)

given by f rarr R( f ) is well-defined and an isomorphism

Proof Since R is a lax tensor functor R(A) is naturally a R(A)-bimodule It is easy tosee that R in (261) is a well-defined map R(A) is also naturally a R(1)-bimodule whichcan be identified with the induced R(1)-bimodule structure on (A times 1)otimes R(1) wherethe left R(1) action on (Atimes1)otimes R(1) is given by (idAtimes1 otimesm R(1))(cminus1

Atimes1R(1)otimes idR(1))We have the following natural isomorphisms

HomR(1)|R(1)(R(A) R(A))sim=minusrarr HomC2plusmn(A times 1 R(A))

χminus1

minusminusrarr HomC(A A) (262)

which by (253) are given by for f isin HomR(1)|R(1)(R(A) R(A))

f rarr f prime = f (idAtimes1 otimes ηR(1)) rarr f primeprime = ρ T ( f prime) (263)

and its inverse is given by for g isin HomC(A A)

g rarr gprime = R(g) δ rarr gprimeprime = (idAtimes1 otimes m R(1)) (gprime otimes idR(1)) (264)

where gprimeprime is indeed a R(1)-bimodule map due to the commutativity of R(1) It is easy tocheck that gprimeprime = R(g) in (264) Therefore R gives an isomorphism from HomC(A A)to HomR(1)|R(1)(R(A) R(A)) Moreover one verifies that R(g) is an R(A)-bimodulemap iff g is an A-bimodule map In other words R g rarr R(g) gives an isomorphism

HomA|A(A A)sim=minusrarr HomR(A)|R(A)(R(A) R(A))

Corollary 227 Let A be a C-algebra

(i) A is absolutely simple iff R(A) is absolutely simple(ii) Let A be in addition Frobenius Then A is simple and special iff R(A) is simple and

special

Proof Part (i) immediately follows from Lemma 226 The statement of part (ii) withoutthe qualifier lsquosimplersquo is proved in Proposition 225 But as in the proof of (iii)rArr(i) inLemma 211 a special Frobenius algebra in a semi-simple category has a semi-simplecategory of bimodules and for a semi-simple C-linear category simple and absolutelysimple are equivalent Part (ii) then follows from part (i)

The following lemma will be needed in Sect 32 below to discuss the properties ofCardy algebras

Lemma 228 Let A be a Frobenius algebra in C Then (δA)lowast = ρA

Proof Recall from (250) that δA is a morphism A rarr T R(A) Since T and R are bothFrobenius functors T R(A) is a Frobenius algebra in C Substituting the definitions aftera short calculation one finds

Cardy Algebras and Sewing Constraints I 891

εT R(A) mT R(A) = Dim(C)oplusiisinI

A Uori Ui A U or

ı Uı

(265)

Substituting this in the definition of (δA)lowast gives again after a short calculation the mor-

phism ρA At an intermediate step one uses that the part of the morphism (δA)lowast which

is made up of Ui and Uı ribbons their duals and the basis morphisms λ(iı)0 andϒ(iı)0can be replaced by 1

dim Uimiddot dUi

3 Cardy Algebras

In this section we start by investigating the properties of Frobenius algebras which sat-isfy the so-called modular invariance condition We then give two definitions of a Cardyalgebra and prove their equivalence Finally in Sect 33 we study the properties of thesealgebras and state our main results

We fix a modular tensor category C Recall that C2plusmn is an abbreviation for C+ Cminus

31 Modular invariance In C2plusmn we define the object K and the morphism ω K rarr Kas

K =oplus

i jisinIUi times U j ω =

sumi jisinI

dim Ui dim U j

Dim C idUi timesU j (31)

They have the property (see eg [BK Cor 3111])

ω

K

(Uk times Ul )or

(Uk times Ul )or

Ui times U j

Ui times U j

= δik δ jlDim C

dim Ui dim U jbUi timesU j dUi timesU j (32)

892 L Kong I Runkel

Definition 31

(i) Let A B be objects of C2plusmn A morphism f A otimes B rarr B is called S-invariant iff

A

B

W

W

f =

A

B

W

W

f

K

ω(33)

holds for all for W isin C2plusmn(ii) A C2plusmn-algebra (Aclmcl ηcl) is called modular invariant iff θAcl = idAcl and mcl

is S-invariant

Lemma 32 The morphism f A otimes B rarr B is S-invariant if and only if

A

B

Ui times U j

Ui times U j

f = Dim Cdim Ui dim U j

sumα

A

B

B

f

Ui times U j

Ui times U j

α

α

(34)

holds for all i j isin I

Proof Condition (33) holds for all W iff it holds for all W = Ui times U j i j isin I so itis enough to show that the right-hand side of (33) with W = Ui times U j is equal to theright-hand side of (34) Recall the notation for basis morphisms in (236) Starting from(33) write

idBor otimes idUi timesU j =⎛⎝sum

klα

(b(ktimeslα)

B

)or (

bB(ktimeslα)

)or⎞⎠ otimes idUi timesU j (35)

and then apply (32) The graphical representation of the resulting morphism can bedeformed to give (34) Remark 33 As shown in [K3 Sect 61] the modular invariance condition of a C2plusmn-alge-bra exactly coincides with the modular invariance condition for torus 1-point correlationfunctions of a genus-01 closed CFT In particular the condition θAcl = idAcl is equiv-alent to invariance under the modular transformation T τ rarr τ + 1 and the condition(34) with f = mcl is equivalent to invariance under S τ rarr minus 1

τ Combining the

modular invariance condition with the genus zero properties of a genus-01 closed CFTresults in a modular invariant commutative symmetric Frobenius algebra in C2plusmn

Cardy Algebras and Sewing Constraints I 893

Let Acl be a modular invariant C2plusmn-algebra Evaluating (34) for f = mcl composingit with ηcl otimes idUi timesU j and taking the trace implies the following identity

Zi j = 1

DimCAclUi timesU j where

(36)

Zi j = dimC HomC2plusmn(UitimesU j Acl)

Decomposing Acl into simple objects this gives

Zi j =sum

klisinISik Zkl Sminus1

l j where Si j = si jradic

DimC (37)

which in CFT terms is of course nothing but the invariance of the torus partition functionunder the modular S-transformation

The following theorem gives a simple criterion for modular invariance

Theorem 34 Let Acl be a haploid commutative symmetric Frobenius algebra in C2plusmn

(i) If Acl is modular invariant then dim Acl = Dim C(ii) If dim Acl = Dim C then Acl is special and modular invariant

Proof Part (i) Since Acl is haploid for i = j = 0 Equation (36) reduces to 1 =dim AclDimCPart (ii) By the same reasoning as in the proof of (iii)rArr(i) in Lemma 211 one showsthat Acl is special Thus mcl cl = ζclidAcl for some ζcl = 0

By [KO Thm 45] the category (C2plusmn)locAcl

of local Acl-modules is again a mod-

ular tensor category and Dim (C2plusmn)locAcl

= (Dim C dim Acl)2 (see [Fr Prop 321 amp

Rem 323] for the same statement in the notation used here) Thus by assumption wehave Dim(C2plusmn)loc

Acl= 1 It then follows from [ENO1 Thm 23] that up to isomorphism

(C2plusmn)locAcl

has a unique simple object (namely the tensor unit) In other words every simplelocal Acl-module is isomorphic to Acl (seen as a left-module over itself)

We have the following isomorphisms between morphism spaces [FS Prop 47 amp 411]

HomAcl(Acl otimes (Ui times U j ) Acl) sim= HomC2plusmn(Ui times U j Acl)

HomAcl(Acl Acl otimes (Ui times U j )) sim= HomC2plusmn(AclUi times U j )

Using these to transport the bases (236) from the right to the left we obtain bases bα(i j)αof HomAcl(Acl otimes (Ui times U j ) Acl) and b(i j)

β β of HomAcl(Acl Acl otimes (Ui times U j )) Thesecan be expressed graphically as

894 L Kong I Runkel

bα(i j) =

Acl

Acl

Acl

Ui times U j

α

b(i j)β = DimC

dim Ui dim U j

1

ζcl

Acl

Ui times U jAcl

Acl

β

(38)

where the nonzero factor in b(i j)β is included for convenience Notice that bα(i j) b(i j)

β isa left Acl-module map Since Acl is simple as a left module over itself we have

bα(i j) b(i j)β = λαβ idAcl (39)

for some λαβ isin C By computing tr(bα(i j) b(i j)β ) it is easy to verify that λαβ = δαβ

We will now prove the following identity

1

ζcl

Acl

Acl

Ui times U j

Ui times U j

Acl

=sumα

DimCdim Ui dim U j

1

ζcl

Acl

Ui times U j

Ui times U j

Acl

Acl

Acl

α

α

(310)

One checks that the left-hand side of this equation is an idempotent which we denote byζminus1

cl PlAcl(Ui timesU j ) cf [Fr Sect 31] By [Fr Prop 41] the image Im(ζminus1

cl PlAcl(Ui timesU j ))

is a local Acl-module and hence isomorphic to AoplusNcl for some N isin Zge0

All left-module morphisms from Acl to Aclotimes(Ui timesU j ) are linear combinations of the

b(i j)β Furthermore one verifies that ζminus1

cl PlAcl(Ui timesU j )b(i j)

β = b(i j)β Therefore the b(i j)

β

describe precisely the image of the idempotent ie ζminus1cl Pl

Acl(Ui timesU j ) = sum

α b(i j)α bα(i j)

which is nothing but (310) Composing (310) with ζcl middot εcl otimes idUi timesU j from the left (iefrom the top) produces (34)

In addition since Acl is commutative symmetric Frobenius it satisfies θAcl = idAcl

[Fr Prop 225] Altogether this shows that Acl is modular invariant Remark 35 As we were writing this paper we heard that the results in Theorem 34were obtained independently by Kitaev and Muumlger [Ki]

Remark 36 Setting i = j = 0 in (36) gives the identity dim Acl = Z00 Dim C [KRProp 23] Combining this with Theorem 34 (ii) one may wonder if a general modularinvariant commutative symmetric Frobenius algebra Acl in C2plusmn is isomorphic to a directsum of simple such algebras However this is not so For example one can take the com-mutative symmetric Frobenius algebra Acl = C[x]x2 in the category of vector spacesequipped with the non-degenerate trace ε(ax + b) = a In this case the modular invari-ance condition holds automatically but Acl is clearly not a direct sum of two algebras

Cardy Algebras and Sewing Constraints I 895

For a general modular tensor category C the algebra C[x]x2 R(1) understood as analgebra in C2plusmn via the braided monoidal isomorphism Vect f (C) C2plusmn rarr C2plusmn providesanother counter-example

32 Two definitions Define a morphism PlA A rarr A for a Frobenius algebra A in C

or C2plusmn as follows [Fr Sect 24]

PlA =

A

A

A

m

(311)

If A is also commutative and obeys m A A = ζA idA we have PlA = ζA idA In partic-

ular this holds if A is commutative and special Using the fact that the Frobenius algebraR(1) is commutative and normalised-special one can check that Pl

R(A) R(A) rarr R(A)takes the following form

PlR(A) =

oplusiisinI

A

A

Uori

Uori

A

m

times

Ui

Ui

(312)

With these ingredients we can now give the first definition of a Cardy C|C2plusmn-algebrawhich was introduced in [K3 Def 514] cf Remark 315 below

Definition 37 (CardyC|C2plusmn-algebra I) A CardyC|C2plusmn-algebra is a triple (Aop|Acl ιcl-op)where (Aclmcl ηclcl εcl) is a modular invariant commutative symmetric Frobe-nius C2plusmn-algebra (Aopmop ηopop εop) is a symmetric Frobenius C-algebra andιcl-op Acl rarr R(Aop) an algebra homomorphism such that the following conditionsare satisfied

(i) Centre condition

Acl R(Aop)

R(Aop)

R(Aop)

ιcl-op

m R(Aop)

=

Acl

ιcl-op

R(Aop)

R(Aop)

R(Aop)

m R(Aop)

(313)

896 L Kong I Runkel

(ii) Cardy condition

ιcl-op ιlowastcl-op =

R(Aop)

R(Aop)

R(Aop)

(314)

Remark 38

(i) The name ldquoCardy C|C2plusmn-algebrardquo in Definition 37 was chosen because many ofthe important ingredients were first studied by Cardy the modular invariance ofthe closed theory [C1] the consistency of the annulus amplitude [C2] and thebulk-boundary OPE [CL] On the other hand the boundary-boundary OPE andthe OPE analogue of the centre condition were first considered in [Lw]

(ii) One can easily see that in the special case that C is the category Vect f (C) of finite-dimensional C-vector spaces a Cardy C|C2plusmn-algebra gives exactly the algebraicformulation of two-dimensional open-closed topological field theory over C (cfRemark 614 in [K3]) see [Lz Sect 48] [Mo Thm 11] [AN Thm 45] [LPCor 43] [MS Sect 22] When passing to a general modular tensor category Cthere are two important differences to the two-dimensional topological field the-ory Firstly the algebras Acl and Aop now live in different categories which inparticular affects the formulation of the centre condition and the Cardy conditionSecondly the modular invariance condition has to be imposed on Acl In the caseC = Vect f (C) modular invariance holds automatically

Definition 39 A homomorphism of Cardy C|C2plusmn-algebras (A(1)op |A(1)cl ι(1)cl-op) rarr (A(2)op

|A(2)cl ι(2)cl-op) is a pair ( fop fcl) of Frobenius algebra homomorphisms fop A(1)op rarr A(2)op

and fcl A(1)cl rarr A(2)cl such that the diagram

A(1)cl

fcl

ι(1)cl-op

A(2)cl

ι(2)cl-op

R(A(1)op )

R( fop) R(A(2)op )

(315)

commutes

Remark 310 Since a homomorphism of Frobenius algebras is invertible (cf Lemma218 (iv)) a homomorphism of Cardy algebras is always an isomorphism

For a homomorphism ( fop fcl) of Cardy C|C2plusmn-algebras using the commutativity of(315) and the fact that fcl and fop are both algebra and coalgebra homomorphisms itis easy to show that (315) commutes iff

R(A(1)op )R( fop)

(ι(1)cl-op)

lowast

R(A(2)op )

(ι(2)cl-op)

lowast

A(1)cl

fcl A(2)cl

(316)

Cardy Algebras and Sewing Constraints I 897

commutesLet (Aop|Acl ιcl-op) be a Cardy C|C2plusmn-algebra Define the morphism

ιcl-op = χminus1(ιcl-op) T (Acl) minusrarr Aop (317)

Decompose Acl as Acl = oplusNn=1Cl

n times Crn such that Cl

1 times Cr1 = ηcl(1 times 1) We use ι(n)cl-op

to denote the restriction of ιcl-op to Cln times Cr

n and ι(n)cl-op to denote the restriction of ιcl-op

to Cln otimes Cr

n We introduce the following graphical notation

ι(n)cl-op =

Cln Cr

n

Aop

(318)

By (243) ιcl-op can be expressed in terms of ιcl-op as follows

ιcl-op =Noplus

n=1

oplusiisinI

sumα

Cln

Crn

Aop

α

Uori

times α

Crn

Ui

(319)

Lemma 311 The centre condition (313) is equivalent to the following condition in C

Cln Cr

n

Aop

Aop

Aop

=

Cln Cr

n

Aop

Aop

Aop

for n = 1 N (320)

Proof First insert (319) and the definition (217) of m R(Aop) into (313) Then apply thecommutativity of R(1) to the left hand side of (313) The equivalence between (313)and (320) follows immediately Remark 312 The centre condition (320) is very natural from the open-closed confor-mal field theory point of view Correlators on the upper half plane are expressed in termsof conformal blocks on the full complex plane The objects Cl

n and Crn are associated to

the field insertion at a point z in the upper half plane and at the complex conjugate pointz in the lower half plane respectively The object Aop corresponds to a field inserted ata point r on the real axis The centre condition (320) simply says that the correlationfunctions in the disjoint domains |z| gt r gt 0 and r gt |z| gt 0 are analytic continuationsof each other see [K2 Prop 118]

898 L Kong I Runkel

Recall that we define ιlowastcl-op Aop rarr T (Acl) as in (230) We introduce the graphicalnotation

ιlowastcl-op =Noplus

n=1

Cln Cr

n

Aop

=Noplus

n=1

Cln Cr

n

Aop

Aop T (minus1Acl) (321)

where the second equality follows from (215) (221) and (259)

Lemma 313 The Cardy condition (314) is equivalent to the following identity in C

Noplusn=1

DimCdim Ui

sumα

Cln

Crn

Crn

α

α

Aop

Aop Uori

Uori

=Aop

Aop

Aop

Uori

Uori

op

mop for all i isin I (322)

Proof By (244) (312) and (319) it is easy to see that (322) is equivalent to thefollowing identity

ιcl-op χ(ιlowastcl-op) = PlR(Aop)

(323)

Therefore it is enough to show that

χ(ιlowastcl-op) = ιlowastcl-op (324)

We have

χminus1(ιlowastcl-op)(1)= T (ιlowastcl-op) δ (2)= T (ιcl-op)

lowast δlowastlowast (3)= (ρ T (ιcl-op)

)lowast (4)= (ιcl-op

)lowast (325)

In step (1) we use the expression (253) for χminus1 step (2) follows from Lemma 218 (v)and Lemma 219 Step (3) is Lemma 218 (i) and Lemma 228 and finally step (4)amounts to substituting (253) and (317) Acting with χ on both sides of the aboveequality produces (324)

Combining Lemmas 311 and 313 and Proposition 216 we obtain the followingequivalent definition of Cardy C|C2plusmn-algebra (recall the graphical notation (318) forιcl-op and (321) for ιlowastcl-op)

Cardy Algebras and Sewing Constraints I 899

Definition 314 (CardyC|C2plusmn-algebra II) A CardyC-algebra is a triple (Aop|Acl ιcl-op)where Acl is a commutative symmetric Frobenius C2plusmn-algebra satisfying property (34)with f = mcl Aop is a symmetric Frobenius C-algebra and ιcl-op T (Acl) rarr Aop isan algebra homomorphism satisfying the conditions (320) and (322)

Remark 315 Up to a choice of normalisation Definition 314 is the same as the orig-inal one in [K3 Def 613] The difference between the two definitions is the factorDim C dim Ui on the left hand side of (322) which in [K3 Def 613] is given byradic

Dim C dim Ui The two definitions are related by rescaling the coproduct cl andcounit εcl of Acl by 1

radicDim C and

radicDim C respectively We chose the convention in

(322) to remove all dimension factors from the expression (314) for the Cardy condition

33 Uniqueness and existence theorems In this subsection we investigate the structureof Cardy algebras We start with the following proposition which when combined withthe results of part II provides an alternative proof of [Fj Prop 422]

Proposition 316 Let (Aop|Acl ιcl-op) be a Cardy C|C2plusmn-algebra If Acl is simple anddim Aop = 0 then Aop is simple and special

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis therefore haploid Restricting the Cardy condition (322) to the case Ui = 1 andcomposing both sides with εop from the left we see that εop kills all terms associated toU j times 1 isin Acl in the sum except for a single 1 times 1 term Thus we obtain the followingidentity

β εop = dAop (mop otimes idAorop) (idAop otimes bAop) (326)

where β isin C Composing with ηop from the right in turn implies that βεop ηop =dim Aop which is nonzero by assumption Thus also β = 0 and εop is a nonzero multipleof the morphism on the right hand side of (326) By [FRS Lem 311] Aop is special

Since Aop is a special Frobenius algebra Aop is semi-simple as an Aop-bimodule(apply [FS Prop 524] to Aop tensored with its opposite algebra) Suppose Aop is not

simple so that we can write Aop = A(1)op oplus A(2)op for nonzero Aop-bimodules A(1)op and A(2)op We denote the canonical embeddings and projections associated to this decompositionas ι12 and π12 We have the identities

mop (ι1 otimes ι2) = 0 εop ηop =2sum

i=1

εop ιi πi ηop (327)

The first identity follows since π1 mop (ι1 otimes ι2) = 0 (as mop gives the left action of

Aop on Aop and hence it preserves A(2)op ) and similarly π2 mop (ι1 otimes ι2) = 0 Thesecond identity is just the completeness of ι12 π12

Since εop ηop = 0 without losing generality we can assume εop ι1 π1 ηop = 0Using that π2 is a bimodule map we compute

π2 [LHS of (322)]Ui =1 ι1 π1 ηop = π2 [RHS of (322)]Ui =1 ι1 π1 ηop

= PlA(2)op

π2 ι1 π1 ηop = 0 (328)

900 L Kong I Runkel

On the other hand using that Acl is haploid that ιcl-op is an algebra map and thatι lowastcl-op is a coalgebra map one can check that the left-hand side of (328) is equal toλ(εop ι1 π1 ηop) π2 ηop for some λ = 0 This implies that π2 ηop = 0 Thusηop = sum

i ιi πi ηop = ι1 π1 ηop Hence we have

0 = π2 ι2 = π2 mop (ηop otimes ι2) = π2 mop ((ι1 π1 ηop)otimes ι2) (329)

However the right-hand side is zero by (327) This is a contradiction and hence Aopmust be simple

To formulate the next theorem we need the notion of the full centre of an algebra [FjDef 49] Recall that an algebra A in a braided tensor category has a left centre and aright centre [VZO] both of which are sub-algebras of A Of these two we will onlyneed the left centre The following definition is [Fr Def 231] which in our setting isequivalent to that of [VZO]

Definition 317 Let A be a symmetric special Frobenius algebra such that m A A =ζA idA

(i) The left centre Cl(A) of A is the image of the idempotent ζminus1A Pl

A(ii) The full centre Z(A) is Cl(R(A))

That ζminus1A Pl

A is an idempotent follows from [FRS Lem 52] when keeping track ofthe factors ζA ([FRS] assumes normalised-special ie ζA = 1) Note that Cl(A) is againan object of C while Z(A) is an object of C2plusmn Let el

A Cl(A) rarr A be the embeddingof Cl(A) into A The left centre is in fact the maximal subobject of A such that

m A cAA (elA otimes idA) = m A (330)

see [Fr Lem 232] This observation explains the name left centre and also makes theconnection to [O Def 15]

The full centre is by definition the image of the idempotent ζminus1A Pl

R(A) R(A) rarrR(A) Since C2plusmn is abelian the idempotent splits and we obtain the embedding andrestriction morphisms

e Z(A) rarr R(A) and r R(A) Z(A) (331)

which obey r e = idZ(A) and e r = ζminus1A Pl

R(A) It follows from Proposition 225 and

[Fr Prop 237] that Z(A) is a commutative symmetric Frobenius algebra in C2plusmn withstructure morphisms5

m Z(A) = r m R(A) (e otimes e) ηZ(A) = r ηR(A)

Z(A) = ζA middot (r otimes r) R(A) e εZ(A) = ζminus1A middot εR(A) e

(332)

Moreover if A is simple then Z(A) is simple and if A is simple and dim A = 0 thenZ(A) is simple and special The normalisation of the counit is such that

εZ(A) ηZ(A) = ζminus2A dim A DimC (333)

5 The normalisation of product and unit is the standard one The factors in the coproduct and counit haveto be included in order for (A|Z(A) e) to be a Cardy algebra see Theorem 318 below The normalisation ofthe counit enters the Cardy condition (314) through the definition of ( middot )lowast

Cardy Algebras and Sewing Constraints I 901

Theorem 318 Let A be a special symmetric Frobenius C-algebra Then (A|Z(A) e)is a Cardy C|C2plusmn-algebra

The proof of this theorem makes use the following two lemmas

Lemma 319 e Z(A) rarr R(A) is an algebra map and elowast = ζA middot r

Proof It follows from [Fr Lem 229] (or by direct calculation using in particularm R(A) R(A) = ζAidR(A)) that

m R(A) (e otimes e) = ζminus1A middot Pl

R(A) m R(A) (e otimes e) (334)

Substituting e r = ζminus1A Pl

R(A) shows that e is compatible with multiplication For theunit one finds

e ηZ(A) = e r ηR(A) = ζminus1A Pl

R(A) ηR(A) = ηR(A) (335)

Thus e is an algebra map For the second statement one computes

elowast (1)= ζA

R(A)

r

R(A)

R(A)

Z(A)

R(A)

Z(A)

r

e

e

(2)=

R(A)

r

R(A)

R(A)

R(A)

Z(A)

(3)=

R(A)

r

R(A)

Z(A)

(4)= ζA middot r (336)

where in (1) the definitions (230) and (332) have been substituted step (2) is e r = ζminus1

A PlR(A) step (3) uses that R(A) is symmetric Frobenius and step (4) is again

e r = ζminus1A Pl

R(A) Lemma 320 Let A be a symmetric Frobenius algebra in C The morphism

PlR(A) m R(A)

(Pl

R(A) otimes PlR(A)

) R(A)otimes R(A) minusrarr R(A) (337)

is S-invariant

The proof of this lemma is a slightly lengthy explicit calculation and has been deferredto Appendix A2

902 L Kong I Runkel

Proof of Theorem 318 That e is an algebra map was proved in Lemma 319 The centrecondition (313) holds by property (330) of the left centre The Cardy condition (314)also is an immediate consequence of Lemma 319

ιcl-op ιlowastcl-op = e (ζA r) = PlR(A) (338)

The full centre Z(A) is a commutative symmetric Frobenius algebra It remains to provemodular invariance That θZ(A) = idZ(A) is implied by commutativity and symmetryof Z(A) [Fr Prop 225] The S-invariance condition (33) follows from Lemma 320In (337) substitute Pl

R(A) = ζA e r and then put the resulting morphism into (33)Compose the resulting equation with e otimes idW from the right (ie from the bottom) andsubstitute the definition (332) of m Z(A) This results in the statement that m Z(A) isS-invariant

The following theorem is analogous to [LR Prop 29] and [Fj Thm 426] whichroughly speaking answer the question under which circumstances the restriction of atwo-dimensional conformal field theory to the boundary already determines the entireconformal field theory The first work is set in Minkowski space and uses operator alge-bras and subfactors while the second work is set in Euclidean space and uses modulartensor categories

Theorem 321 Let (A|Acl ιcl-op) be a Cardy C|C2plusmn-algebra such that dim A = 0 andAcl is simple Then A is special and (A|Acl ιcl-op) sim= (A|Z(A) e) as Cardy algebras

Proof By Proposition 316 A is simple and special Since Acl is simple the algebramap ιcl-op Acl rarr R(A) is either zero or a monomorphism But ιcl-op ηcl = ηR(A)and so ιcl-op = 0 Thus ιcl-op is monic By Lemma 218 (ii) ζminus1

A ιlowastcl-op is epi The Cardycondition (314) implies

ιcl-op ζminus1A ιlowastcl-op = ζminus1

A PlR(A) = e r (339)

Composing this with e r from the left yields e r ιcl-op ζminus1A ιlowastcl-op = e r =

ιcl-op ζminus1A ιlowastcl-op Since ζminus1

A ιlowastcl-op is epi we have

e r ιcl-op = ιcl-op (340)

Actually (340) also follows from (313) and specialness of R(A) We will prove that

( fop fcl) (A|Acl ιcl-op) minusrarr (A|Z(A) e) where fop = idA fcl = r ιcl-op

(341)

is an isomorphism of Cardy algebrasfcl is an algebra map Compatibility with the units follows since ιcl-op is an algebramap

fcl ηcl = r ιcl-op ηcl = r ηR(A) = ηZ(A) (342)

Compatibility with the multiplication also follows since ιcl-op is an algebra map

m Z(A) ( fcl otimes fcl) = r m R(A) (e otimes e) (r otimes r) (ιcl-op otimes ιcl-op)

= r m R(A) (ιcl-op otimes ιcl-op) = r ιcl-op mcl = fcl mcl (343)

where in the second step we used (340)

Cardy Algebras and Sewing Constraints I 903

fcl is an isomorphism As above since fcl is an algebra map and since Acl is simple fcl

has to be monic By Lemma 319 rlowast = ζminus1A e Thus f lowast

cl = ιlowastcl-op rlowast = ζminus1A ιlowastcl-op e and

fcl f lowastcl = r ιcl-op ζminus1

A ιlowastcl-op e = r e r e = idZ(A) (344)

and so fcl is also epi and hence isofcl is a coalgebra map Since fcl is an algebra map so is f minus1

cl By (344) f minus1cl = f lowast

cland by Lemma 218 (iii) this implies that fcl is a also coalgebra mapThe diagram (315) commutes Commutativity of (315) is equivalent to e fcl = ιcl-opwhich holds by (340)

Let A be a special symmetric Frobenius algebra So far we have seen that (A|Z(A) e)is a Cardy algebra and that all Cardy algebras with Aop = A and simple Acl are of thisform It is now natural to ask if every simple Acl does occur as part of a Cardy algebraThe following theorem provides an affirmative answer Recall that for an A-left moduleM the object Mor otimesA M is an algebra (see eg [KR Lem 42])

Theorem 322 If Acl is a simple modular invariant commutative symmetric FrobeniusC2plusmn-algebra then there exist a simple special symmetric Frobenius C-algebra A and amorphism ιcl-op Acl rarr R(A) such that

(i) Acl sim= Z(A) as Frobenius algebras(ii) (A|Acl ιcl-op) is a Cardy C|C2plusmn-algebra

(iii) T (Acl) sim= oplusκisinJ Morκ otimesA Mκ as algebras where MκκisinJ is a set of representa-

tives of the isomorphism classes of simple A-left modules

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis haploid It then follows from Theorem 34 that Acl is special By Proposition 224T (Acl) is a special symmetric Frobenius algebra in C Thus T (Acl) = oplusi Ai where theAi are simple symmetric Frobenius algebras We will show that at least one of the Aiis special Since T (Acl) is special we have mT (Acl) T (Acl) = ζ idT (Acl) for someζ isin C

times Restricting this to the summand Ai shows mi i = ζ idAi FurthermoreεT (Acl) ηT (Acl) = ξ id1 for some ξ isin C

times But εT (Acl) ηT (Acl) = sumi εi ηi and so

at least one of the εi ηi has to be nonzero Therefore at least one of the Ai is speciallet A equiv Ai be this summand We denote the embedding A rarr T (Acl) by e0 and therestriction T (Acl) A by r0 Notice that r0 is an algebra homomorphism Define

ιcl-op = χ (r0) Acl minusrarr R(A) (345)

By Proposition 216 ιcl-op is an algebra homomorphism Next we verify the centre con-dition (313) or rather its equivalent form (320) By substituting the definitions onecan convince oneself that the commutativity mcl cAclAcl = mcl of Acl in C2plusmn implies thecondition mT (Acl) = mT (Acl) in C see [K2 Prop 36] Here T (Acl)otimes T (Acl) rarrT (Acl)otimes T (Acl) is given by

=oplusmn

(idCl

notimes cCl

m Crnotimes idCr

m

)

(cCl

m Clnotimes cminus1

Crn C

rm

)

(idCl

motimes cminus1

Cln C

rm

otimes idCrn

)

(346)

and we decomposed Acl as Acl = oplusnCln times Cr

n As a consequence we obtain the identity

r0 mT (Acl) (idAcl otimes e0) = r0 mT (Acl) (idAcl otimes e0) (347)

904 L Kong I Runkel

Using that r0 is an algebra map and that by definition ιcl-op = r0 we obtain (320)In order to show that (A|Acl ιcl-op) is a Cardy algebra it remains to show that the

Cardy condition (314) is satisfied We will demonstrate this via a detour by first provingthat Acl sim= Z(A) as Frobenius algebras

Recall the notations e and r given in (331) Using the centre condition (320) onecan check that Pl

R(A) ιcl-op = m R(A) R(A) ιcl-op By specialness of A we have

m A A = ζA idA and so together with e r = ζminus1A Pl

R(A) we get

e r ιcl-op = ιcl-op (348)

Next consider the morphism

fcl = r ιcl-op Acl minusrarr Z(A) (349)

By the same derivation as in (342) and (343) one sees that fcl is an algebra map Inparticular fcl ηcl = ηZ(A) = 0 and so fcl = 0 Since Acl is simple fcl has to be amonomorphism

By the same argument as used in the proof of Theorem 34 (ii) up to isomorphismAcl is the unique simple local Acl-(left-)module The algebra monomorphism fcl turnsZ(A) into an Acl-module Since Z(A) is commutative it is local as an Acl-module andso Z(A) sim= AoplusN

cl for some N ge 1 By construction A is a simple special symmetricFrobenius algebra Proposition 225 and Corollary 227 show that R(A) inherits all theseproperties and thus Z(A) is simple (see the comment below Eq (332)) By Theorem318 Z(A) is modular invariant and then by Theorem 34 (i) dim Z(A) = Dim C Thisimplies that N = 1 in Z(A) sim= AoplusN

cl and so fcl is in fact an isomorphismSince Acl and Z(A) are both haploid we have εZ(A) fcl = ξ εcl for some ξ isin C

timesThe counit uniquely determines the Frobenius structure on Acl and Z(A) (see eg [FRSLemma 37]) so that fcl is a coalgebra isomorphism iff ξ = 1 To compute ξ we composethe above identity with ηcl from the right Defining ζcl via εcl ηcl = ζminus1

cl Dim C middot id1 andusing (333) gives ξ = dim A ζclζ

2A By rescaling the comultiplication and the counit

of A and consequently changing ζA we can always achieve ξ = 1 This proves part (i)of the theorem

Equation (348) implies that ιcl-op = e fcl Since fcl is an isomorphism of Frobeniusalgebras by Lemmas 218 and 319 we have

ιcl-op ιlowastcl-op = e fcl f lowastcl elowast = ζA e r = Pl

R(A) (350)

Thus (A|Acl ιcl-op) is a Cardy algebra This proves part (ii) of the theoremPart (iii) can be seen as follows By [KR Prop 43] T Z(A) sim= oplusκisinJ Mor

κ otimesA Mκ asalgebras Together with the observation that T ( fcl) T (Acl) rarr T Z(A) is an isomor-phism of algebras this proves part (iii) Remark 323 Part (i) of Theorem 322 was announced by Muumlger [Muuml2] We provide anindependent proof in the setting of Cardy algebras

The above theorem together with Lemma 211 and Theorem 34 shows that a sim-ple commutative symmetric Frobenius C2plusmn-algebra Acl with dim Acl = Dim C is alwayspart of a Cardy algebra (Aop|Acl ιcl-op) for some simple special symmetric Frobeniusalgebra Aop in C However the above proof also illustrates that Aop is not unique Thisraises the question how two Cardy algebras with a given Acl can differ This question isanswered by [KR Thm 11] which in the present framework can be restated as follows

Cardy Algebras and Sewing Constraints I 905

Theorem 324 If (A(i)op |A(i)cl ι(i)cl-op) i = 1 2 are two Cardy C|C2plusmn-algebras such that

A(i)cl is simple and dim A(i)op = 0 for i = 1 2 then A(1)clsim= A(2)cl as algebras if and only

if A(1)op and A(2)op are Morita equivalent

Proof Theorem 11 in [KR] is stated for A(i)op being non-degenerate algebras and A(i)cl =Z(A(i)op ) for i = 1 2 By Proposition 316 A(i)op are simple and special for i = 1 2

Then by [KR Lem 21] A(i)op are non-degenerate algebras By Theorem 321 we have

A(i)clsim= Z(A(i)op ) as Frobenius algebras Finally by [KR Thm 11] Z(A(1)op ) sim= Z(A(2)op )

as algebras iff A(1)op and A(2)op are Morita equivalent Let Cmax(C2plusmn) be the set of equivalence classes [B] of simple modular invariant

commutative symmetric Frobenius algebras B in C2plusmn Two such algebras B and B prime areequivalent if B and B prime are isomorphic as algebras (but not necessarily as Frobenius alge-bras) Let Msimp(C) be the set of Morita classes of simple special symmetric Frobeniusalgebras in C Define the map z Msimp(C) rarr Cmax(C2plusmn) by z A rarr [Z(A)] whereA denotes the Morita class of A From Theorem 322 (i) and [KR Thm 11] we learn

Corollary 325 The map z Msimp(C) rarr Cmax(C2plusmn) is a bijection

A Appendix

A1 Proof of Lemma 27 We will show that if (F ψ F2 ψ

F0 ) is a colax tensor functor

from C1 to C2 then (G φG2 φ

G0 ) is a lax tensor functor from C2 to C1 Applying this

result to the opposed categories then gives the converse statementWe need to show that φG

0 and φG2 make the diagrams (21) and (22) commute We

first prove the commutativity of (21) Consider the following diagram

G(A)otimes (G(B)otimes G(C))

GψF2 δ

idG(A)otimesφG2 G(A)otimes G(B otimes C)

GψF2 δ

G(FG(A)otimes F(G(B)otimes G(C)))

G(F(idG(A))otimesF(φG2 ))

G(idFG(A)otimesψ F2 )

G(FG(A)otimes FG(B otimes C))

G(ρAotimesρBotimesC )

G(FG(A)otimes (FG(B)otimes FG(C)))

G(ρAotimes(ρBotimesρC )) G(A otimes (B otimes C))

(A1)

The top subdiagram is commutative because of the naturality of Gψ F2 δ The commu-

tativity of the bottom subdiagram follows from the following identities

(ρB otimes ρC ) ψ F2 = (ρB otimes ρC ) ψ F

2 ρF Fδ

= ρBotimesC FG(ρB otimes ρC ) FG(ψ F2 ) Fδ

= ρBotimesC F(φG2 ) (A2)

as a map F(G(B) otimes G(C)) rarr B otimes C The commutativity of (A1) implies that thecomposition of maps in the left column in (21) can be replaced by

G(ρA otimes (ρB otimes ρC )) G((idFG(A) otimes ψ F

2 ) ψ F2

) δ (A3)

906 L Kong I Runkel

Similarly we can show that the composition of maps in the right column in (21) can bereplaced by

G((ρA otimes ρB)otimes ρC ) G((ψ F

2 otimes idFG(C)) ψ F2

) δ (A4)

Using the commutativity of (23) it is easy to see that (21) with the left and right col-umns of (21) replaced by (A3) and (A4) respectively is commutative Hence (21) iscommutative

Now we prove the commutativity of the first diagram in (22)

φG2 (φG

0 otimes idG(A))

(1)= G(ρ12 otimes ρA) Gψ F2 δ

[(Gψ F

0 δ11)otimes idG(A)

]

(2)= G(ρ12 otimes ρA) Gψ F2 G F(Gψ0 otimes idG(A)) G F(δ11 otimes idG(A)) δ

(3)= G(ρ12 otimes ρA) G(FG(ψ F0 )otimes idFG(A)) Gψ F

2 G F(δ11 otimes idG(A)) δ(4)= G(id12 otimes ρA) G

([ψ F

0 ρF(11) (Fδ)11] otimes idFG(A)

) Gψ F

2 δ(5)= G(id12 otimes ρA) G(ψ F

0 otimes idFG(A)) Gψ F2 δ

(6)= G(id12 otimes ρA) G(lminus1FG(A)) G F(lG(A)) δ

(7)= G(lminus1A ) GρA δG lG(A)

(8)= G(lminus1A ) lG(A) (A5)

where in step (1) we substituted the definition of φG0 φ

G2 given in (210) in step (2)

we used the naturality of δ in step (3) we used the naturality of Gψ F2 in step (4) we

switched the position between Gρ12 and G FG(ψ F0 ) and the position between Gψ2 and

G F(δ12 otimes idG(A)) using the naturality of ρ and FψG2 respectively in step (5) we applied

the second identity in (28) in step (6) we used (24) in step (7) we used the naturalityof lminus1 and δ in step (8) we used the first identity in (28)

The proof of the commutativity of the second diagram in (22) is similar Thus wehave shown that G is a lax tensor functor

A2 Proof of Lemma 320 To prepare the proof recall that for a given object B isin Cthe modular group P SL(2Z) acts on the space oplusi HomC(B otimes Ui Ui ) see eg [BKSect 31] and [K3 Eq (455)] We will only need the action of S and Sminus1 Let f isinoplusi HomC(B otimes Ui Ui ) Then

S oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A6)

Cardy Algebras and Sewing Constraints I 907

Sminus1 oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A7)

By Lemma 32 to establish that (337) is S-invariant it is enough to prove the identity(34) when f is given by (337) Using (A6) and (A7) we can see that Eq (34) simplysays that oplusi j [RHS of (34)] is invariant under the action of S times S Consider the elementg of oplus jkisinIHomC2plusmn(R(A)otimes (Uor

j times Uk)Uorj times Uk) given by

g =oplusjkisinI

sumα

R(A)

R(A)

R(A)

R(A)

Uorj times Uk

Uorj times Uk

α

α

(A8)

By the above arguments proving S-invariance of (337) is equivalent to proving invari-ance of g under the action of S times S

For i isin I we denote by gi the component of g in(oplus jisinIHomC(A otimes Uor

i otimes Uorj U

orj )

)otimes (opluskisinIHomC(Ui otimes UkUk))

We view the second Hom-space in the above tensor product as a Hom-space in C+ insteadof Cminus It is enough to show that gi is invariant under the action of S times Sminus1 Note thatthe action of Sminus1 in C+ is equivalent to that of S in Cminus

The morphism gi can be canonically identified with a bilinear pairing

( middot middot )i ⎛⎝oplus

jisinIHomC(Uor

j A otimes Uori otimes Uor

j )

⎞⎠

times(oplus

kisinIHomC(UkUi otimes Uk)

)minusrarr C (A9)

as follows For h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) we set

(h1 h2)i = (dim U j dim Uk)minus1 trUor

j timesUk[gi (h1 times h2)] (A10)

908 L Kong I Runkel

When substituting the explicit form of the product m R(A) of R(A) = (A times 1) otimes R(1)after a short calculation one finds

(h1 h2)i =sumα

1

dim U j

A

A

A

A

A A

A α

α

h1

h2

U jUk

Uorj

Ui

Uk (A11)

Here the top morphism PlR(A) has been simplified with the help of the identity

PR(A) m R(A) (PR(A) otimes PR(A))

=(oplus

iisinI((m A A)otimes idUor

i)times idUi

) m R(A) (PR(A) otimes PR(A)) (A12)

which can be checked by direct calculation along the same lines as in the proof of [FrLem 310]

The action of the modular transformation S on oplusiisinI(B otimes Ui Ui ) for B isin C nat-urally induces an action on oplusiisinI(Ui B otimes Ui ) [K3 Prop 514] which we denote bySlowast In the present case we get an action of Slowast on oplus jisinIHomC(Uor

j A otimes Uori otimes Uor

j ) and

opluskisinIHomC(UkUi otimes Uk) Then to show gi is invariant under the action of S times Sminus1

amounts to showing that

(h1 h2)i = ((Sminus1)lowast h1 Slowast h2)i (A13)

Cardy Algebras and Sewing Constraints I 909

for all h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) We have

((Sminus1)lowast h1 Slowast h2)i =sum

mnα

dim Un

DimC

A

A

A

A

A A

A

Uk

α

α

h1

h2

Ui

U j

UmUn

(A14)

Now drag the upper vertex indexed by α in the above graph along its Uorm -leg until it

meets the lower vertex also indexed by α then sum over α and m This gives

((Sminus1)lowast h1 Slowast h2)i =sum

n

dim Un

DimC

A

A

A

A

A A

A

Uk

Un

h1

h2

Ui

U j

(A15)

910 L Kong I Runkel

If we just look at the neighbourhood of the Un-loop in the above graph we see thefollowing subgraphs

sumn

dim Un

DimC

A Aor

Uorj U j

Uk Uork

Un=

sumα

1

dim U j

A Aor Uk Uork

Uorj U j

αα

(A16)

where we have applied [BK Cor 3111] Substituting this subgraph back to the originalgraph in (A15) we obtain

((Sminus1)lowast h1 Slowast h2)i =sumα

1

dim U j

A

A

A

A

A A

A

A

A

Uorj

α

α

h1

h2m A

Ui

U jUk

Uk

(A17)

The graph in (A17) is equal to that in (A11) In order to see this we first drag theldquobubblerdquo (m A A) along A lines and through the m A vertex (because m A A is abimodule map) until it reaches the lower-left leg of the upper vertex indexed by α Thendrag the m A vertex along the (red) dotted line in the above graph Finally we applythe associativity of A (A12) and [Fr Lem 311] Then we see that the graph in (A17)exactly matches with the one in (A11)

Acknowledgements We would like to thank the organisers of the Oberwolfach Arbeitsgemeinschaft ldquoAlge-braic structures in conformal field theoriesrdquo (April 2007) where this work was started for an inspiring meetingWe would further like to thank the Hausdorff Institute for Mathematics in Bonn and the organisers of the stim-ulating meeting ldquoGeometry and Physicsrdquo (May 2008) We are indebted to Alexei Davydov Jens FjelstadJuumlrgen Fuchs Yi-Zhi Huang Alexei Kitaev Urs Schreiber Christoph Schweigert Stephan Stolz and PeterTeichner for helpful discussions andor comments on a draft of this paper The research of IR was partiallysupported by the EPSRC First Grant EPE0050471 the PPARC rolling grant PPC5071451 and the MarieCurie network lsquoSuperstring Theoryrsquo (MRTN-CT-2004-512194)

Cardy Algebras and Sewing Constraints I 911

References

[AN] Alexeevski A Natanzon SM Noncommutative two-dimensional topological field theories andhurwitz numbers for real algebraic curves Sel Math New Ser 12 307ndash377 (2006)

[Bi] Bichon J Cosovereign hopf algebras J Pure Appl Alg 157 121ndash133 (2001)[BK] Bakalov B Kirillov AA Lectures on Tensor Categories and Modular Functors Providence

RI Amer Math Soc 2001[C1] Cardy JL Operator content of two-dimensional conformal invariant theories Nucl Phys

B 270 186ndash204 (1986)[C2] Cardy JL Boundary conditions fusion rules and the verlinde formula Nucl Phys B 324

581ndash596 (1989)[CL] Cardy JL Lewellen DC Bulk and boundary operators in conformal field theory Phys Lett

B 259 274ndash278 (1991)[DMZ] Dong C-Y Mason G Zhu Y-C Discrete series of the Virasoro algebra and the moonshine

module In Algebraic Groups and Their Generalizations Quantum and infinite-dimensional Meth-ods Proc Symp Pure Math 56 Part 2 Providence RI Amer MathSoc 1994 pp 295ndash316

[DP] Day B Pastro C Note on frobenius monoidal functors New York J Math 14 733ndash742 (2008)[ENO1] Etingof PI Nikshych D Ostrik V On fusion categories Ann Math 162 581ndash642 (2005)[ENO2] Etingof PI Nikshych D Ostrik V An analogue of radfordrsquos s4 formula for finite tensor

categories Int Math Research Notices 54 2915ndash2933 (2004)[Fe] Felder G Froumlhlich J Fuchs J Schweigert C Correlation functions and boundary conditions in

rational conformal field theory and three-dimensional topology Comp Math 131 189ndash237 (2002)[FHL] Frenkel IB Huang Y-Z Lepowsky J On axiomatic approaches to vertex operator algebras

and modules Mem Amer Math Soc 104 (1993)[Fj] Fjelstad J Fuchs J Runkel I Schweigert C Uniqueness of openclosed rational cft with

given algebra of open states Adv Theor Math Phys 12 1283ndash1375 (2008)[FK] Froumlhlich J King C The chern-simons theory and knot polynomials Commun Math

Phys 126 167ndash199 (1989)[Fr] Froumlhlich J Fuchs J Runkel I Schweigert C Correspondences of ribbon categories Adv

Math 199 192ndash329 (2006)[FRS] Fuchs J Runkel I Schweigert C Tft construction of rcft correlators I partition functions Nucl

Phys B 646 353ndash497 (2002)[FS] Fuchs J Schweigert C Category theory for conformal boundary conditions Fields Inst Com-

mun 39 25ndash71 (2003)[H1] Huang Y-Z Two-dimensional conformal geometry and vertex operator algebras Progress in

Mathematics Vol 148 Boston Birkhaumluser 1997[H2] Huang Y-Z Rigidity and modularity of vertex tensor categories Comm Contemp Math 10

871ndash911 (2008)[HK1] Huang Y-Z Kong L Open-string vertex algebra category and operads Commun Math

Phys 250 433ndash471 (2004)[HK2] Huang Y-Z Kong L Full field algebras Commun Math Phys 272 345ndash396 (2007)[HK3] Huang Y-Z Kong L Modular invariance for conformal full field algebras httparxivorgabs

math0609570v2[mathQA] 2006[HL] Huang Y-Z Lepowsky J Tensor products of modules for a vertex operator algebra and vertex

tensor categories In Lie Theory and Geometry in honor of Bertram Kostant ed R BrylinskiJ-L Brylinski V Guillemin V Kac Boston Birkhaumluser 1994 pp 349ndash383

[JS] Joyal A Street R Braided tensor categories Adv Math 102 20ndash78 (1993)[K1] Kong L Full field algebras operads and tensor categories Adv Math 213 271ndash340 (2007)[K2] Kong L Open-closed field algebras Commun Math Phys 280 207ndash261 (2008)[K3] Kong L Cardy condition for open-closed field algebras Commun Math Phys 283 25ndash92 (2008)[Ki] Kitaev A Private communication[KO] Kirillov AA Ostrik V On q-analog of mckay correspondence and ade classification of sl(2)

conformal field theories Adv Math 171 183ndash227 (2002)[KR] Kong L Runkel I Morita classes of algebras in modular tensor categories Adv Math 219

1548ndash1576 (2008)[Ld] Lauda AD Frobenius algebras and ambidextrous adjunctions Theo Appl Cat 16 84ndash

122 (2006)[Li1] Li H-S Regular representations of vertex operator algebras Commun Contemp Math 4

639ndash683 (2002)[Li2] Li H-S Regular representations and huang-lepowsky tensor functors for vertex operator

algebras J Alge 255 423ndash462 (2002)

Cardy Algebras and Sewing Constraints I 891

εT R(A) mT R(A) = Dim(C)oplusiisinI

A Uori Ui A U or

ı Uı

(265)

Substituting this in the definition of (δA)lowast gives again after a short calculation the mor-

phism ρA At an intermediate step one uses that the part of the morphism (δA)lowast which

is made up of Ui and Uı ribbons their duals and the basis morphisms λ(iı)0 andϒ(iı)0can be replaced by 1

dim Uimiddot dUi

3 Cardy Algebras

In this section we start by investigating the properties of Frobenius algebras which sat-isfy the so-called modular invariance condition We then give two definitions of a Cardyalgebra and prove their equivalence Finally in Sect 33 we study the properties of thesealgebras and state our main results

We fix a modular tensor category C Recall that C2plusmn is an abbreviation for C+ Cminus

31 Modular invariance In C2plusmn we define the object K and the morphism ω K rarr Kas

K =oplus

i jisinIUi times U j ω =

sumi jisinI

dim Ui dim U j

Dim C idUi timesU j (31)

They have the property (see eg [BK Cor 3111])

ω

K

(Uk times Ul )or

(Uk times Ul )or

Ui times U j

Ui times U j

= δik δ jlDim C

dim Ui dim U jbUi timesU j dUi timesU j (32)

892 L Kong I Runkel

Definition 31

(i) Let A B be objects of C2plusmn A morphism f A otimes B rarr B is called S-invariant iff

A

B

W

W

f =

A

B

W

W

f

K

ω(33)

holds for all for W isin C2plusmn(ii) A C2plusmn-algebra (Aclmcl ηcl) is called modular invariant iff θAcl = idAcl and mcl

is S-invariant

Lemma 32 The morphism f A otimes B rarr B is S-invariant if and only if

A

B

Ui times U j

Ui times U j

f = Dim Cdim Ui dim U j

sumα

A

B

B

f

Ui times U j

Ui times U j

α

α

(34)

holds for all i j isin I

Proof Condition (33) holds for all W iff it holds for all W = Ui times U j i j isin I so itis enough to show that the right-hand side of (33) with W = Ui times U j is equal to theright-hand side of (34) Recall the notation for basis morphisms in (236) Starting from(33) write

idBor otimes idUi timesU j =⎛⎝sum

klα

(b(ktimeslα)

B

)or (

bB(ktimeslα)

)or⎞⎠ otimes idUi timesU j (35)

and then apply (32) The graphical representation of the resulting morphism can bedeformed to give (34) Remark 33 As shown in [K3 Sect 61] the modular invariance condition of a C2plusmn-alge-bra exactly coincides with the modular invariance condition for torus 1-point correlationfunctions of a genus-01 closed CFT In particular the condition θAcl = idAcl is equiv-alent to invariance under the modular transformation T τ rarr τ + 1 and the condition(34) with f = mcl is equivalent to invariance under S τ rarr minus 1

τ Combining the

modular invariance condition with the genus zero properties of a genus-01 closed CFTresults in a modular invariant commutative symmetric Frobenius algebra in C2plusmn

Cardy Algebras and Sewing Constraints I 893

Let Acl be a modular invariant C2plusmn-algebra Evaluating (34) for f = mcl composingit with ηcl otimes idUi timesU j and taking the trace implies the following identity

Zi j = 1

DimCAclUi timesU j where

(36)

Zi j = dimC HomC2plusmn(UitimesU j Acl)

Decomposing Acl into simple objects this gives

Zi j =sum

klisinISik Zkl Sminus1

l j where Si j = si jradic

DimC (37)

which in CFT terms is of course nothing but the invariance of the torus partition functionunder the modular S-transformation

The following theorem gives a simple criterion for modular invariance

Theorem 34 Let Acl be a haploid commutative symmetric Frobenius algebra in C2plusmn

(i) If Acl is modular invariant then dim Acl = Dim C(ii) If dim Acl = Dim C then Acl is special and modular invariant

Proof Part (i) Since Acl is haploid for i = j = 0 Equation (36) reduces to 1 =dim AclDimCPart (ii) By the same reasoning as in the proof of (iii)rArr(i) in Lemma 211 one showsthat Acl is special Thus mcl cl = ζclidAcl for some ζcl = 0

By [KO Thm 45] the category (C2plusmn)locAcl

of local Acl-modules is again a mod-

ular tensor category and Dim (C2plusmn)locAcl

= (Dim C dim Acl)2 (see [Fr Prop 321 amp

Rem 323] for the same statement in the notation used here) Thus by assumption wehave Dim(C2plusmn)loc

Acl= 1 It then follows from [ENO1 Thm 23] that up to isomorphism

(C2plusmn)locAcl

has a unique simple object (namely the tensor unit) In other words every simplelocal Acl-module is isomorphic to Acl (seen as a left-module over itself)

We have the following isomorphisms between morphism spaces [FS Prop 47 amp 411]

HomAcl(Acl otimes (Ui times U j ) Acl) sim= HomC2plusmn(Ui times U j Acl)

HomAcl(Acl Acl otimes (Ui times U j )) sim= HomC2plusmn(AclUi times U j )

Using these to transport the bases (236) from the right to the left we obtain bases bα(i j)αof HomAcl(Acl otimes (Ui times U j ) Acl) and b(i j)

β β of HomAcl(Acl Acl otimes (Ui times U j )) Thesecan be expressed graphically as

894 L Kong I Runkel

bα(i j) =

Acl

Acl

Acl

Ui times U j

α

b(i j)β = DimC

dim Ui dim U j

1

ζcl

Acl

Ui times U jAcl

Acl

β

(38)

where the nonzero factor in b(i j)β is included for convenience Notice that bα(i j) b(i j)

β isa left Acl-module map Since Acl is simple as a left module over itself we have

bα(i j) b(i j)β = λαβ idAcl (39)

for some λαβ isin C By computing tr(bα(i j) b(i j)β ) it is easy to verify that λαβ = δαβ

We will now prove the following identity

1

ζcl

Acl

Acl

Ui times U j

Ui times U j

Acl

=sumα

DimCdim Ui dim U j

1

ζcl

Acl

Ui times U j

Ui times U j

Acl

Acl

Acl

α

α

(310)

One checks that the left-hand side of this equation is an idempotent which we denote byζminus1

cl PlAcl(Ui timesU j ) cf [Fr Sect 31] By [Fr Prop 41] the image Im(ζminus1

cl PlAcl(Ui timesU j ))

is a local Acl-module and hence isomorphic to AoplusNcl for some N isin Zge0

All left-module morphisms from Acl to Aclotimes(Ui timesU j ) are linear combinations of the

b(i j)β Furthermore one verifies that ζminus1

cl PlAcl(Ui timesU j )b(i j)

β = b(i j)β Therefore the b(i j)

β

describe precisely the image of the idempotent ie ζminus1cl Pl

Acl(Ui timesU j ) = sum

α b(i j)α bα(i j)

which is nothing but (310) Composing (310) with ζcl middot εcl otimes idUi timesU j from the left (iefrom the top) produces (34)

In addition since Acl is commutative symmetric Frobenius it satisfies θAcl = idAcl

[Fr Prop 225] Altogether this shows that Acl is modular invariant Remark 35 As we were writing this paper we heard that the results in Theorem 34were obtained independently by Kitaev and Muumlger [Ki]

Remark 36 Setting i = j = 0 in (36) gives the identity dim Acl = Z00 Dim C [KRProp 23] Combining this with Theorem 34 (ii) one may wonder if a general modularinvariant commutative symmetric Frobenius algebra Acl in C2plusmn is isomorphic to a directsum of simple such algebras However this is not so For example one can take the com-mutative symmetric Frobenius algebra Acl = C[x]x2 in the category of vector spacesequipped with the non-degenerate trace ε(ax + b) = a In this case the modular invari-ance condition holds automatically but Acl is clearly not a direct sum of two algebras

Cardy Algebras and Sewing Constraints I 895

For a general modular tensor category C the algebra C[x]x2 R(1) understood as analgebra in C2plusmn via the braided monoidal isomorphism Vect f (C) C2plusmn rarr C2plusmn providesanother counter-example

32 Two definitions Define a morphism PlA A rarr A for a Frobenius algebra A in C

or C2plusmn as follows [Fr Sect 24]

PlA =

A

A

A

m

(311)

If A is also commutative and obeys m A A = ζA idA we have PlA = ζA idA In partic-

ular this holds if A is commutative and special Using the fact that the Frobenius algebraR(1) is commutative and normalised-special one can check that Pl

R(A) R(A) rarr R(A)takes the following form

PlR(A) =

oplusiisinI

A

A

Uori

Uori

A

m

times

Ui

Ui

(312)

With these ingredients we can now give the first definition of a Cardy C|C2plusmn-algebrawhich was introduced in [K3 Def 514] cf Remark 315 below

Definition 37 (CardyC|C2plusmn-algebra I) A CardyC|C2plusmn-algebra is a triple (Aop|Acl ιcl-op)where (Aclmcl ηclcl εcl) is a modular invariant commutative symmetric Frobe-nius C2plusmn-algebra (Aopmop ηopop εop) is a symmetric Frobenius C-algebra andιcl-op Acl rarr R(Aop) an algebra homomorphism such that the following conditionsare satisfied

(i) Centre condition

Acl R(Aop)

R(Aop)

R(Aop)

ιcl-op

m R(Aop)

=

Acl

ιcl-op

R(Aop)

R(Aop)

R(Aop)

m R(Aop)

(313)

896 L Kong I Runkel

(ii) Cardy condition

ιcl-op ιlowastcl-op =

R(Aop)

R(Aop)

R(Aop)

(314)

Remark 38

(i) The name ldquoCardy C|C2plusmn-algebrardquo in Definition 37 was chosen because many ofthe important ingredients were first studied by Cardy the modular invariance ofthe closed theory [C1] the consistency of the annulus amplitude [C2] and thebulk-boundary OPE [CL] On the other hand the boundary-boundary OPE andthe OPE analogue of the centre condition were first considered in [Lw]

(ii) One can easily see that in the special case that C is the category Vect f (C) of finite-dimensional C-vector spaces a Cardy C|C2plusmn-algebra gives exactly the algebraicformulation of two-dimensional open-closed topological field theory over C (cfRemark 614 in [K3]) see [Lz Sect 48] [Mo Thm 11] [AN Thm 45] [LPCor 43] [MS Sect 22] When passing to a general modular tensor category Cthere are two important differences to the two-dimensional topological field the-ory Firstly the algebras Acl and Aop now live in different categories which inparticular affects the formulation of the centre condition and the Cardy conditionSecondly the modular invariance condition has to be imposed on Acl In the caseC = Vect f (C) modular invariance holds automatically

Definition 39 A homomorphism of Cardy C|C2plusmn-algebras (A(1)op |A(1)cl ι(1)cl-op) rarr (A(2)op

|A(2)cl ι(2)cl-op) is a pair ( fop fcl) of Frobenius algebra homomorphisms fop A(1)op rarr A(2)op

and fcl A(1)cl rarr A(2)cl such that the diagram

A(1)cl

fcl

ι(1)cl-op

A(2)cl

ι(2)cl-op

R(A(1)op )

R( fop) R(A(2)op )

(315)

commutes

Remark 310 Since a homomorphism of Frobenius algebras is invertible (cf Lemma218 (iv)) a homomorphism of Cardy algebras is always an isomorphism

For a homomorphism ( fop fcl) of Cardy C|C2plusmn-algebras using the commutativity of(315) and the fact that fcl and fop are both algebra and coalgebra homomorphisms itis easy to show that (315) commutes iff

R(A(1)op )R( fop)

(ι(1)cl-op)

lowast

R(A(2)op )

(ι(2)cl-op)

lowast

A(1)cl

fcl A(2)cl

(316)

Cardy Algebras and Sewing Constraints I 897

commutesLet (Aop|Acl ιcl-op) be a Cardy C|C2plusmn-algebra Define the morphism

ιcl-op = χminus1(ιcl-op) T (Acl) minusrarr Aop (317)

Decompose Acl as Acl = oplusNn=1Cl

n times Crn such that Cl

1 times Cr1 = ηcl(1 times 1) We use ι(n)cl-op

to denote the restriction of ιcl-op to Cln times Cr

n and ι(n)cl-op to denote the restriction of ιcl-op

to Cln otimes Cr

n We introduce the following graphical notation

ι(n)cl-op =

Cln Cr

n

Aop

(318)

By (243) ιcl-op can be expressed in terms of ιcl-op as follows

ιcl-op =Noplus

n=1

oplusiisinI

sumα

Cln

Crn

Aop

α

Uori

times α

Crn

Ui

(319)

Lemma 311 The centre condition (313) is equivalent to the following condition in C

Cln Cr

n

Aop

Aop

Aop

=

Cln Cr

n

Aop

Aop

Aop

for n = 1 N (320)

Proof First insert (319) and the definition (217) of m R(Aop) into (313) Then apply thecommutativity of R(1) to the left hand side of (313) The equivalence between (313)and (320) follows immediately Remark 312 The centre condition (320) is very natural from the open-closed confor-mal field theory point of view Correlators on the upper half plane are expressed in termsof conformal blocks on the full complex plane The objects Cl

n and Crn are associated to

the field insertion at a point z in the upper half plane and at the complex conjugate pointz in the lower half plane respectively The object Aop corresponds to a field inserted ata point r on the real axis The centre condition (320) simply says that the correlationfunctions in the disjoint domains |z| gt r gt 0 and r gt |z| gt 0 are analytic continuationsof each other see [K2 Prop 118]

898 L Kong I Runkel

Recall that we define ιlowastcl-op Aop rarr T (Acl) as in (230) We introduce the graphicalnotation

ιlowastcl-op =Noplus

n=1

Cln Cr

n

Aop

=Noplus

n=1

Cln Cr

n

Aop

Aop T (minus1Acl) (321)

where the second equality follows from (215) (221) and (259)

Lemma 313 The Cardy condition (314) is equivalent to the following identity in C

Noplusn=1

DimCdim Ui

sumα

Cln

Crn

Crn

α

α

Aop

Aop Uori

Uori

=Aop

Aop

Aop

Uori

Uori

op

mop for all i isin I (322)

Proof By (244) (312) and (319) it is easy to see that (322) is equivalent to thefollowing identity

ιcl-op χ(ιlowastcl-op) = PlR(Aop)

(323)

Therefore it is enough to show that

χ(ιlowastcl-op) = ιlowastcl-op (324)

We have

χminus1(ιlowastcl-op)(1)= T (ιlowastcl-op) δ (2)= T (ιcl-op)

lowast δlowastlowast (3)= (ρ T (ιcl-op)

)lowast (4)= (ιcl-op

)lowast (325)

In step (1) we use the expression (253) for χminus1 step (2) follows from Lemma 218 (v)and Lemma 219 Step (3) is Lemma 218 (i) and Lemma 228 and finally step (4)amounts to substituting (253) and (317) Acting with χ on both sides of the aboveequality produces (324)

Combining Lemmas 311 and 313 and Proposition 216 we obtain the followingequivalent definition of Cardy C|C2plusmn-algebra (recall the graphical notation (318) forιcl-op and (321) for ιlowastcl-op)

Cardy Algebras and Sewing Constraints I 899

Definition 314 (CardyC|C2plusmn-algebra II) A CardyC-algebra is a triple (Aop|Acl ιcl-op)where Acl is a commutative symmetric Frobenius C2plusmn-algebra satisfying property (34)with f = mcl Aop is a symmetric Frobenius C-algebra and ιcl-op T (Acl) rarr Aop isan algebra homomorphism satisfying the conditions (320) and (322)

Remark 315 Up to a choice of normalisation Definition 314 is the same as the orig-inal one in [K3 Def 613] The difference between the two definitions is the factorDim C dim Ui on the left hand side of (322) which in [K3 Def 613] is given byradic

Dim C dim Ui The two definitions are related by rescaling the coproduct cl andcounit εcl of Acl by 1

radicDim C and

radicDim C respectively We chose the convention in

(322) to remove all dimension factors from the expression (314) for the Cardy condition

33 Uniqueness and existence theorems In this subsection we investigate the structureof Cardy algebras We start with the following proposition which when combined withthe results of part II provides an alternative proof of [Fj Prop 422]

Proposition 316 Let (Aop|Acl ιcl-op) be a Cardy C|C2plusmn-algebra If Acl is simple anddim Aop = 0 then Aop is simple and special

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis therefore haploid Restricting the Cardy condition (322) to the case Ui = 1 andcomposing both sides with εop from the left we see that εop kills all terms associated toU j times 1 isin Acl in the sum except for a single 1 times 1 term Thus we obtain the followingidentity

β εop = dAop (mop otimes idAorop) (idAop otimes bAop) (326)

where β isin C Composing with ηop from the right in turn implies that βεop ηop =dim Aop which is nonzero by assumption Thus also β = 0 and εop is a nonzero multipleof the morphism on the right hand side of (326) By [FRS Lem 311] Aop is special

Since Aop is a special Frobenius algebra Aop is semi-simple as an Aop-bimodule(apply [FS Prop 524] to Aop tensored with its opposite algebra) Suppose Aop is not

simple so that we can write Aop = A(1)op oplus A(2)op for nonzero Aop-bimodules A(1)op and A(2)op We denote the canonical embeddings and projections associated to this decompositionas ι12 and π12 We have the identities

mop (ι1 otimes ι2) = 0 εop ηop =2sum

i=1

εop ιi πi ηop (327)

The first identity follows since π1 mop (ι1 otimes ι2) = 0 (as mop gives the left action of

Aop on Aop and hence it preserves A(2)op ) and similarly π2 mop (ι1 otimes ι2) = 0 Thesecond identity is just the completeness of ι12 π12

Since εop ηop = 0 without losing generality we can assume εop ι1 π1 ηop = 0Using that π2 is a bimodule map we compute

π2 [LHS of (322)]Ui =1 ι1 π1 ηop = π2 [RHS of (322)]Ui =1 ι1 π1 ηop

= PlA(2)op

π2 ι1 π1 ηop = 0 (328)

900 L Kong I Runkel

On the other hand using that Acl is haploid that ιcl-op is an algebra map and thatι lowastcl-op is a coalgebra map one can check that the left-hand side of (328) is equal toλ(εop ι1 π1 ηop) π2 ηop for some λ = 0 This implies that π2 ηop = 0 Thusηop = sum

i ιi πi ηop = ι1 π1 ηop Hence we have

0 = π2 ι2 = π2 mop (ηop otimes ι2) = π2 mop ((ι1 π1 ηop)otimes ι2) (329)

However the right-hand side is zero by (327) This is a contradiction and hence Aopmust be simple

To formulate the next theorem we need the notion of the full centre of an algebra [FjDef 49] Recall that an algebra A in a braided tensor category has a left centre and aright centre [VZO] both of which are sub-algebras of A Of these two we will onlyneed the left centre The following definition is [Fr Def 231] which in our setting isequivalent to that of [VZO]

Definition 317 Let A be a symmetric special Frobenius algebra such that m A A =ζA idA

(i) The left centre Cl(A) of A is the image of the idempotent ζminus1A Pl

A(ii) The full centre Z(A) is Cl(R(A))

That ζminus1A Pl

A is an idempotent follows from [FRS Lem 52] when keeping track ofthe factors ζA ([FRS] assumes normalised-special ie ζA = 1) Note that Cl(A) is againan object of C while Z(A) is an object of C2plusmn Let el

A Cl(A) rarr A be the embeddingof Cl(A) into A The left centre is in fact the maximal subobject of A such that

m A cAA (elA otimes idA) = m A (330)

see [Fr Lem 232] This observation explains the name left centre and also makes theconnection to [O Def 15]

The full centre is by definition the image of the idempotent ζminus1A Pl

R(A) R(A) rarrR(A) Since C2plusmn is abelian the idempotent splits and we obtain the embedding andrestriction morphisms

e Z(A) rarr R(A) and r R(A) Z(A) (331)

which obey r e = idZ(A) and e r = ζminus1A Pl

R(A) It follows from Proposition 225 and

[Fr Prop 237] that Z(A) is a commutative symmetric Frobenius algebra in C2plusmn withstructure morphisms5

m Z(A) = r m R(A) (e otimes e) ηZ(A) = r ηR(A)

Z(A) = ζA middot (r otimes r) R(A) e εZ(A) = ζminus1A middot εR(A) e

(332)

Moreover if A is simple then Z(A) is simple and if A is simple and dim A = 0 thenZ(A) is simple and special The normalisation of the counit is such that

εZ(A) ηZ(A) = ζminus2A dim A DimC (333)

5 The normalisation of product and unit is the standard one The factors in the coproduct and counit haveto be included in order for (A|Z(A) e) to be a Cardy algebra see Theorem 318 below The normalisation ofthe counit enters the Cardy condition (314) through the definition of ( middot )lowast

Cardy Algebras and Sewing Constraints I 901

Theorem 318 Let A be a special symmetric Frobenius C-algebra Then (A|Z(A) e)is a Cardy C|C2plusmn-algebra

The proof of this theorem makes use the following two lemmas

Lemma 319 e Z(A) rarr R(A) is an algebra map and elowast = ζA middot r

Proof It follows from [Fr Lem 229] (or by direct calculation using in particularm R(A) R(A) = ζAidR(A)) that

m R(A) (e otimes e) = ζminus1A middot Pl

R(A) m R(A) (e otimes e) (334)

Substituting e r = ζminus1A Pl

R(A) shows that e is compatible with multiplication For theunit one finds

e ηZ(A) = e r ηR(A) = ζminus1A Pl

R(A) ηR(A) = ηR(A) (335)

Thus e is an algebra map For the second statement one computes

elowast (1)= ζA

R(A)

r

R(A)

R(A)

Z(A)

R(A)

Z(A)

r

e

e

(2)=

R(A)

r

R(A)

R(A)

R(A)

Z(A)

(3)=

R(A)

r

R(A)

Z(A)

(4)= ζA middot r (336)

where in (1) the definitions (230) and (332) have been substituted step (2) is e r = ζminus1

A PlR(A) step (3) uses that R(A) is symmetric Frobenius and step (4) is again

e r = ζminus1A Pl

R(A) Lemma 320 Let A be a symmetric Frobenius algebra in C The morphism

PlR(A) m R(A)

(Pl

R(A) otimes PlR(A)

) R(A)otimes R(A) minusrarr R(A) (337)

is S-invariant

The proof of this lemma is a slightly lengthy explicit calculation and has been deferredto Appendix A2

902 L Kong I Runkel

Proof of Theorem 318 That e is an algebra map was proved in Lemma 319 The centrecondition (313) holds by property (330) of the left centre The Cardy condition (314)also is an immediate consequence of Lemma 319

ιcl-op ιlowastcl-op = e (ζA r) = PlR(A) (338)

The full centre Z(A) is a commutative symmetric Frobenius algebra It remains to provemodular invariance That θZ(A) = idZ(A) is implied by commutativity and symmetryof Z(A) [Fr Prop 225] The S-invariance condition (33) follows from Lemma 320In (337) substitute Pl

R(A) = ζA e r and then put the resulting morphism into (33)Compose the resulting equation with e otimes idW from the right (ie from the bottom) andsubstitute the definition (332) of m Z(A) This results in the statement that m Z(A) isS-invariant

The following theorem is analogous to [LR Prop 29] and [Fj Thm 426] whichroughly speaking answer the question under which circumstances the restriction of atwo-dimensional conformal field theory to the boundary already determines the entireconformal field theory The first work is set in Minkowski space and uses operator alge-bras and subfactors while the second work is set in Euclidean space and uses modulartensor categories

Theorem 321 Let (A|Acl ιcl-op) be a Cardy C|C2plusmn-algebra such that dim A = 0 andAcl is simple Then A is special and (A|Acl ιcl-op) sim= (A|Z(A) e) as Cardy algebras

Proof By Proposition 316 A is simple and special Since Acl is simple the algebramap ιcl-op Acl rarr R(A) is either zero or a monomorphism But ιcl-op ηcl = ηR(A)and so ιcl-op = 0 Thus ιcl-op is monic By Lemma 218 (ii) ζminus1

A ιlowastcl-op is epi The Cardycondition (314) implies

ιcl-op ζminus1A ιlowastcl-op = ζminus1

A PlR(A) = e r (339)

Composing this with e r from the left yields e r ιcl-op ζminus1A ιlowastcl-op = e r =

ιcl-op ζminus1A ιlowastcl-op Since ζminus1

A ιlowastcl-op is epi we have

e r ιcl-op = ιcl-op (340)

Actually (340) also follows from (313) and specialness of R(A) We will prove that

( fop fcl) (A|Acl ιcl-op) minusrarr (A|Z(A) e) where fop = idA fcl = r ιcl-op

(341)

is an isomorphism of Cardy algebrasfcl is an algebra map Compatibility with the units follows since ιcl-op is an algebramap

fcl ηcl = r ιcl-op ηcl = r ηR(A) = ηZ(A) (342)

Compatibility with the multiplication also follows since ιcl-op is an algebra map

m Z(A) ( fcl otimes fcl) = r m R(A) (e otimes e) (r otimes r) (ιcl-op otimes ιcl-op)

= r m R(A) (ιcl-op otimes ιcl-op) = r ιcl-op mcl = fcl mcl (343)

where in the second step we used (340)

Cardy Algebras and Sewing Constraints I 903

fcl is an isomorphism As above since fcl is an algebra map and since Acl is simple fcl

has to be monic By Lemma 319 rlowast = ζminus1A e Thus f lowast

cl = ιlowastcl-op rlowast = ζminus1A ιlowastcl-op e and

fcl f lowastcl = r ιcl-op ζminus1

A ιlowastcl-op e = r e r e = idZ(A) (344)

and so fcl is also epi and hence isofcl is a coalgebra map Since fcl is an algebra map so is f minus1

cl By (344) f minus1cl = f lowast

cland by Lemma 218 (iii) this implies that fcl is a also coalgebra mapThe diagram (315) commutes Commutativity of (315) is equivalent to e fcl = ιcl-opwhich holds by (340)

Let A be a special symmetric Frobenius algebra So far we have seen that (A|Z(A) e)is a Cardy algebra and that all Cardy algebras with Aop = A and simple Acl are of thisform It is now natural to ask if every simple Acl does occur as part of a Cardy algebraThe following theorem provides an affirmative answer Recall that for an A-left moduleM the object Mor otimesA M is an algebra (see eg [KR Lem 42])

Theorem 322 If Acl is a simple modular invariant commutative symmetric FrobeniusC2plusmn-algebra then there exist a simple special symmetric Frobenius C-algebra A and amorphism ιcl-op Acl rarr R(A) such that

(i) Acl sim= Z(A) as Frobenius algebras(ii) (A|Acl ιcl-op) is a Cardy C|C2plusmn-algebra

(iii) T (Acl) sim= oplusκisinJ Morκ otimesA Mκ as algebras where MκκisinJ is a set of representa-

tives of the isomorphism classes of simple A-left modules

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis haploid It then follows from Theorem 34 that Acl is special By Proposition 224T (Acl) is a special symmetric Frobenius algebra in C Thus T (Acl) = oplusi Ai where theAi are simple symmetric Frobenius algebras We will show that at least one of the Aiis special Since T (Acl) is special we have mT (Acl) T (Acl) = ζ idT (Acl) for someζ isin C

times Restricting this to the summand Ai shows mi i = ζ idAi FurthermoreεT (Acl) ηT (Acl) = ξ id1 for some ξ isin C

times But εT (Acl) ηT (Acl) = sumi εi ηi and so

at least one of the εi ηi has to be nonzero Therefore at least one of the Ai is speciallet A equiv Ai be this summand We denote the embedding A rarr T (Acl) by e0 and therestriction T (Acl) A by r0 Notice that r0 is an algebra homomorphism Define

ιcl-op = χ (r0) Acl minusrarr R(A) (345)

By Proposition 216 ιcl-op is an algebra homomorphism Next we verify the centre con-dition (313) or rather its equivalent form (320) By substituting the definitions onecan convince oneself that the commutativity mcl cAclAcl = mcl of Acl in C2plusmn implies thecondition mT (Acl) = mT (Acl) in C see [K2 Prop 36] Here T (Acl)otimes T (Acl) rarrT (Acl)otimes T (Acl) is given by

=oplusmn

(idCl

notimes cCl

m Crnotimes idCr

m

)

(cCl

m Clnotimes cminus1

Crn C

rm

)

(idCl

motimes cminus1

Cln C

rm

otimes idCrn

)

(346)

and we decomposed Acl as Acl = oplusnCln times Cr

n As a consequence we obtain the identity

r0 mT (Acl) (idAcl otimes e0) = r0 mT (Acl) (idAcl otimes e0) (347)

904 L Kong I Runkel

Using that r0 is an algebra map and that by definition ιcl-op = r0 we obtain (320)In order to show that (A|Acl ιcl-op) is a Cardy algebra it remains to show that the

Cardy condition (314) is satisfied We will demonstrate this via a detour by first provingthat Acl sim= Z(A) as Frobenius algebras

Recall the notations e and r given in (331) Using the centre condition (320) onecan check that Pl

R(A) ιcl-op = m R(A) R(A) ιcl-op By specialness of A we have

m A A = ζA idA and so together with e r = ζminus1A Pl

R(A) we get

e r ιcl-op = ιcl-op (348)

Next consider the morphism

fcl = r ιcl-op Acl minusrarr Z(A) (349)

By the same derivation as in (342) and (343) one sees that fcl is an algebra map Inparticular fcl ηcl = ηZ(A) = 0 and so fcl = 0 Since Acl is simple fcl has to be amonomorphism

By the same argument as used in the proof of Theorem 34 (ii) up to isomorphismAcl is the unique simple local Acl-(left-)module The algebra monomorphism fcl turnsZ(A) into an Acl-module Since Z(A) is commutative it is local as an Acl-module andso Z(A) sim= AoplusN

cl for some N ge 1 By construction A is a simple special symmetricFrobenius algebra Proposition 225 and Corollary 227 show that R(A) inherits all theseproperties and thus Z(A) is simple (see the comment below Eq (332)) By Theorem318 Z(A) is modular invariant and then by Theorem 34 (i) dim Z(A) = Dim C Thisimplies that N = 1 in Z(A) sim= AoplusN

cl and so fcl is in fact an isomorphismSince Acl and Z(A) are both haploid we have εZ(A) fcl = ξ εcl for some ξ isin C

timesThe counit uniquely determines the Frobenius structure on Acl and Z(A) (see eg [FRSLemma 37]) so that fcl is a coalgebra isomorphism iff ξ = 1 To compute ξ we composethe above identity with ηcl from the right Defining ζcl via εcl ηcl = ζminus1

cl Dim C middot id1 andusing (333) gives ξ = dim A ζclζ

2A By rescaling the comultiplication and the counit

of A and consequently changing ζA we can always achieve ξ = 1 This proves part (i)of the theorem

Equation (348) implies that ιcl-op = e fcl Since fcl is an isomorphism of Frobeniusalgebras by Lemmas 218 and 319 we have

ιcl-op ιlowastcl-op = e fcl f lowastcl elowast = ζA e r = Pl

R(A) (350)

Thus (A|Acl ιcl-op) is a Cardy algebra This proves part (ii) of the theoremPart (iii) can be seen as follows By [KR Prop 43] T Z(A) sim= oplusκisinJ Mor

κ otimesA Mκ asalgebras Together with the observation that T ( fcl) T (Acl) rarr T Z(A) is an isomor-phism of algebras this proves part (iii) Remark 323 Part (i) of Theorem 322 was announced by Muumlger [Muuml2] We provide anindependent proof in the setting of Cardy algebras

The above theorem together with Lemma 211 and Theorem 34 shows that a sim-ple commutative symmetric Frobenius C2plusmn-algebra Acl with dim Acl = Dim C is alwayspart of a Cardy algebra (Aop|Acl ιcl-op) for some simple special symmetric Frobeniusalgebra Aop in C However the above proof also illustrates that Aop is not unique Thisraises the question how two Cardy algebras with a given Acl can differ This question isanswered by [KR Thm 11] which in the present framework can be restated as follows

Cardy Algebras and Sewing Constraints I 905

Theorem 324 If (A(i)op |A(i)cl ι(i)cl-op) i = 1 2 are two Cardy C|C2plusmn-algebras such that

A(i)cl is simple and dim A(i)op = 0 for i = 1 2 then A(1)clsim= A(2)cl as algebras if and only

if A(1)op and A(2)op are Morita equivalent

Proof Theorem 11 in [KR] is stated for A(i)op being non-degenerate algebras and A(i)cl =Z(A(i)op ) for i = 1 2 By Proposition 316 A(i)op are simple and special for i = 1 2

Then by [KR Lem 21] A(i)op are non-degenerate algebras By Theorem 321 we have

A(i)clsim= Z(A(i)op ) as Frobenius algebras Finally by [KR Thm 11] Z(A(1)op ) sim= Z(A(2)op )

as algebras iff A(1)op and A(2)op are Morita equivalent Let Cmax(C2plusmn) be the set of equivalence classes [B] of simple modular invariant

commutative symmetric Frobenius algebras B in C2plusmn Two such algebras B and B prime areequivalent if B and B prime are isomorphic as algebras (but not necessarily as Frobenius alge-bras) Let Msimp(C) be the set of Morita classes of simple special symmetric Frobeniusalgebras in C Define the map z Msimp(C) rarr Cmax(C2plusmn) by z A rarr [Z(A)] whereA denotes the Morita class of A From Theorem 322 (i) and [KR Thm 11] we learn

Corollary 325 The map z Msimp(C) rarr Cmax(C2plusmn) is a bijection

A Appendix

A1 Proof of Lemma 27 We will show that if (F ψ F2 ψ

F0 ) is a colax tensor functor

from C1 to C2 then (G φG2 φ

G0 ) is a lax tensor functor from C2 to C1 Applying this

result to the opposed categories then gives the converse statementWe need to show that φG

0 and φG2 make the diagrams (21) and (22) commute We

first prove the commutativity of (21) Consider the following diagram

G(A)otimes (G(B)otimes G(C))

GψF2 δ

idG(A)otimesφG2 G(A)otimes G(B otimes C)

GψF2 δ

G(FG(A)otimes F(G(B)otimes G(C)))

G(F(idG(A))otimesF(φG2 ))

G(idFG(A)otimesψ F2 )

G(FG(A)otimes FG(B otimes C))

G(ρAotimesρBotimesC )

G(FG(A)otimes (FG(B)otimes FG(C)))

G(ρAotimes(ρBotimesρC )) G(A otimes (B otimes C))

(A1)

The top subdiagram is commutative because of the naturality of Gψ F2 δ The commu-

tativity of the bottom subdiagram follows from the following identities

(ρB otimes ρC ) ψ F2 = (ρB otimes ρC ) ψ F

2 ρF Fδ

= ρBotimesC FG(ρB otimes ρC ) FG(ψ F2 ) Fδ

= ρBotimesC F(φG2 ) (A2)

as a map F(G(B) otimes G(C)) rarr B otimes C The commutativity of (A1) implies that thecomposition of maps in the left column in (21) can be replaced by

G(ρA otimes (ρB otimes ρC )) G((idFG(A) otimes ψ F

2 ) ψ F2

) δ (A3)

906 L Kong I Runkel

Similarly we can show that the composition of maps in the right column in (21) can bereplaced by

G((ρA otimes ρB)otimes ρC ) G((ψ F

2 otimes idFG(C)) ψ F2

) δ (A4)

Using the commutativity of (23) it is easy to see that (21) with the left and right col-umns of (21) replaced by (A3) and (A4) respectively is commutative Hence (21) iscommutative

Now we prove the commutativity of the first diagram in (22)

φG2 (φG

0 otimes idG(A))

(1)= G(ρ12 otimes ρA) Gψ F2 δ

[(Gψ F

0 δ11)otimes idG(A)

]

(2)= G(ρ12 otimes ρA) Gψ F2 G F(Gψ0 otimes idG(A)) G F(δ11 otimes idG(A)) δ

(3)= G(ρ12 otimes ρA) G(FG(ψ F0 )otimes idFG(A)) Gψ F

2 G F(δ11 otimes idG(A)) δ(4)= G(id12 otimes ρA) G

([ψ F

0 ρF(11) (Fδ)11] otimes idFG(A)

) Gψ F

2 δ(5)= G(id12 otimes ρA) G(ψ F

0 otimes idFG(A)) Gψ F2 δ

(6)= G(id12 otimes ρA) G(lminus1FG(A)) G F(lG(A)) δ

(7)= G(lminus1A ) GρA δG lG(A)

(8)= G(lminus1A ) lG(A) (A5)

where in step (1) we substituted the definition of φG0 φ

G2 given in (210) in step (2)

we used the naturality of δ in step (3) we used the naturality of Gψ F2 in step (4) we

switched the position between Gρ12 and G FG(ψ F0 ) and the position between Gψ2 and

G F(δ12 otimes idG(A)) using the naturality of ρ and FψG2 respectively in step (5) we applied

the second identity in (28) in step (6) we used (24) in step (7) we used the naturalityof lminus1 and δ in step (8) we used the first identity in (28)

The proof of the commutativity of the second diagram in (22) is similar Thus wehave shown that G is a lax tensor functor

A2 Proof of Lemma 320 To prepare the proof recall that for a given object B isin Cthe modular group P SL(2Z) acts on the space oplusi HomC(B otimes Ui Ui ) see eg [BKSect 31] and [K3 Eq (455)] We will only need the action of S and Sminus1 Let f isinoplusi HomC(B otimes Ui Ui ) Then

S oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A6)

Cardy Algebras and Sewing Constraints I 907

Sminus1 oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A7)

By Lemma 32 to establish that (337) is S-invariant it is enough to prove the identity(34) when f is given by (337) Using (A6) and (A7) we can see that Eq (34) simplysays that oplusi j [RHS of (34)] is invariant under the action of S times S Consider the elementg of oplus jkisinIHomC2plusmn(R(A)otimes (Uor

j times Uk)Uorj times Uk) given by

g =oplusjkisinI

sumα

R(A)

R(A)

R(A)

R(A)

Uorj times Uk

Uorj times Uk

α

α

(A8)

By the above arguments proving S-invariance of (337) is equivalent to proving invari-ance of g under the action of S times S

For i isin I we denote by gi the component of g in(oplus jisinIHomC(A otimes Uor

i otimes Uorj U

orj )

)otimes (opluskisinIHomC(Ui otimes UkUk))

We view the second Hom-space in the above tensor product as a Hom-space in C+ insteadof Cminus It is enough to show that gi is invariant under the action of S times Sminus1 Note thatthe action of Sminus1 in C+ is equivalent to that of S in Cminus

The morphism gi can be canonically identified with a bilinear pairing

( middot middot )i ⎛⎝oplus

jisinIHomC(Uor

j A otimes Uori otimes Uor

j )

⎞⎠

times(oplus

kisinIHomC(UkUi otimes Uk)

)minusrarr C (A9)

as follows For h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) we set

(h1 h2)i = (dim U j dim Uk)minus1 trUor

j timesUk[gi (h1 times h2)] (A10)

908 L Kong I Runkel

When substituting the explicit form of the product m R(A) of R(A) = (A times 1) otimes R(1)after a short calculation one finds

(h1 h2)i =sumα

1

dim U j

A

A

A

A

A A

A α

α

h1

h2

U jUk

Uorj

Ui

Uk (A11)

Here the top morphism PlR(A) has been simplified with the help of the identity

PR(A) m R(A) (PR(A) otimes PR(A))

=(oplus

iisinI((m A A)otimes idUor

i)times idUi

) m R(A) (PR(A) otimes PR(A)) (A12)

which can be checked by direct calculation along the same lines as in the proof of [FrLem 310]

The action of the modular transformation S on oplusiisinI(B otimes Ui Ui ) for B isin C nat-urally induces an action on oplusiisinI(Ui B otimes Ui ) [K3 Prop 514] which we denote bySlowast In the present case we get an action of Slowast on oplus jisinIHomC(Uor

j A otimes Uori otimes Uor

j ) and

opluskisinIHomC(UkUi otimes Uk) Then to show gi is invariant under the action of S times Sminus1

amounts to showing that

(h1 h2)i = ((Sminus1)lowast h1 Slowast h2)i (A13)

Cardy Algebras and Sewing Constraints I 909

for all h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) We have

((Sminus1)lowast h1 Slowast h2)i =sum

mnα

dim Un

DimC

A

A

A

A

A A

A

Uk

α

α

h1

h2

Ui

U j

UmUn

(A14)

Now drag the upper vertex indexed by α in the above graph along its Uorm -leg until it

meets the lower vertex also indexed by α then sum over α and m This gives

((Sminus1)lowast h1 Slowast h2)i =sum

n

dim Un

DimC

A

A

A

A

A A

A

Uk

Un

h1

h2

Ui

U j

(A15)

910 L Kong I Runkel

If we just look at the neighbourhood of the Un-loop in the above graph we see thefollowing subgraphs

sumn

dim Un

DimC

A Aor

Uorj U j

Uk Uork

Un=

sumα

1

dim U j

A Aor Uk Uork

Uorj U j

αα

(A16)

where we have applied [BK Cor 3111] Substituting this subgraph back to the originalgraph in (A15) we obtain

((Sminus1)lowast h1 Slowast h2)i =sumα

1

dim U j

A

A

A

A

A A

A

A

A

Uorj

α

α

h1

h2m A

Ui

U jUk

Uk

(A17)

The graph in (A17) is equal to that in (A11) In order to see this we first drag theldquobubblerdquo (m A A) along A lines and through the m A vertex (because m A A is abimodule map) until it reaches the lower-left leg of the upper vertex indexed by α Thendrag the m A vertex along the (red) dotted line in the above graph Finally we applythe associativity of A (A12) and [Fr Lem 311] Then we see that the graph in (A17)exactly matches with the one in (A11)

Acknowledgements We would like to thank the organisers of the Oberwolfach Arbeitsgemeinschaft ldquoAlge-braic structures in conformal field theoriesrdquo (April 2007) where this work was started for an inspiring meetingWe would further like to thank the Hausdorff Institute for Mathematics in Bonn and the organisers of the stim-ulating meeting ldquoGeometry and Physicsrdquo (May 2008) We are indebted to Alexei Davydov Jens FjelstadJuumlrgen Fuchs Yi-Zhi Huang Alexei Kitaev Urs Schreiber Christoph Schweigert Stephan Stolz and PeterTeichner for helpful discussions andor comments on a draft of this paper The research of IR was partiallysupported by the EPSRC First Grant EPE0050471 the PPARC rolling grant PPC5071451 and the MarieCurie network lsquoSuperstring Theoryrsquo (MRTN-CT-2004-512194)

Cardy Algebras and Sewing Constraints I 911

References

[AN] Alexeevski A Natanzon SM Noncommutative two-dimensional topological field theories andhurwitz numbers for real algebraic curves Sel Math New Ser 12 307ndash377 (2006)

[Bi] Bichon J Cosovereign hopf algebras J Pure Appl Alg 157 121ndash133 (2001)[BK] Bakalov B Kirillov AA Lectures on Tensor Categories and Modular Functors Providence

RI Amer Math Soc 2001[C1] Cardy JL Operator content of two-dimensional conformal invariant theories Nucl Phys

B 270 186ndash204 (1986)[C2] Cardy JL Boundary conditions fusion rules and the verlinde formula Nucl Phys B 324

581ndash596 (1989)[CL] Cardy JL Lewellen DC Bulk and boundary operators in conformal field theory Phys Lett

B 259 274ndash278 (1991)[DMZ] Dong C-Y Mason G Zhu Y-C Discrete series of the Virasoro algebra and the moonshine

module In Algebraic Groups and Their Generalizations Quantum and infinite-dimensional Meth-ods Proc Symp Pure Math 56 Part 2 Providence RI Amer MathSoc 1994 pp 295ndash316

[DP] Day B Pastro C Note on frobenius monoidal functors New York J Math 14 733ndash742 (2008)[ENO1] Etingof PI Nikshych D Ostrik V On fusion categories Ann Math 162 581ndash642 (2005)[ENO2] Etingof PI Nikshych D Ostrik V An analogue of radfordrsquos s4 formula for finite tensor

categories Int Math Research Notices 54 2915ndash2933 (2004)[Fe] Felder G Froumlhlich J Fuchs J Schweigert C Correlation functions and boundary conditions in

rational conformal field theory and three-dimensional topology Comp Math 131 189ndash237 (2002)[FHL] Frenkel IB Huang Y-Z Lepowsky J On axiomatic approaches to vertex operator algebras

and modules Mem Amer Math Soc 104 (1993)[Fj] Fjelstad J Fuchs J Runkel I Schweigert C Uniqueness of openclosed rational cft with

given algebra of open states Adv Theor Math Phys 12 1283ndash1375 (2008)[FK] Froumlhlich J King C The chern-simons theory and knot polynomials Commun Math

Phys 126 167ndash199 (1989)[Fr] Froumlhlich J Fuchs J Runkel I Schweigert C Correspondences of ribbon categories Adv

Math 199 192ndash329 (2006)[FRS] Fuchs J Runkel I Schweigert C Tft construction of rcft correlators I partition functions Nucl

Phys B 646 353ndash497 (2002)[FS] Fuchs J Schweigert C Category theory for conformal boundary conditions Fields Inst Com-

mun 39 25ndash71 (2003)[H1] Huang Y-Z Two-dimensional conformal geometry and vertex operator algebras Progress in

Mathematics Vol 148 Boston Birkhaumluser 1997[H2] Huang Y-Z Rigidity and modularity of vertex tensor categories Comm Contemp Math 10

871ndash911 (2008)[HK1] Huang Y-Z Kong L Open-string vertex algebra category and operads Commun Math

Phys 250 433ndash471 (2004)[HK2] Huang Y-Z Kong L Full field algebras Commun Math Phys 272 345ndash396 (2007)[HK3] Huang Y-Z Kong L Modular invariance for conformal full field algebras httparxivorgabs

math0609570v2[mathQA] 2006[HL] Huang Y-Z Lepowsky J Tensor products of modules for a vertex operator algebra and vertex

tensor categories In Lie Theory and Geometry in honor of Bertram Kostant ed R BrylinskiJ-L Brylinski V Guillemin V Kac Boston Birkhaumluser 1994 pp 349ndash383

[JS] Joyal A Street R Braided tensor categories Adv Math 102 20ndash78 (1993)[K1] Kong L Full field algebras operads and tensor categories Adv Math 213 271ndash340 (2007)[K2] Kong L Open-closed field algebras Commun Math Phys 280 207ndash261 (2008)[K3] Kong L Cardy condition for open-closed field algebras Commun Math Phys 283 25ndash92 (2008)[Ki] Kitaev A Private communication[KO] Kirillov AA Ostrik V On q-analog of mckay correspondence and ade classification of sl(2)

conformal field theories Adv Math 171 183ndash227 (2002)[KR] Kong L Runkel I Morita classes of algebras in modular tensor categories Adv Math 219

1548ndash1576 (2008)[Ld] Lauda AD Frobenius algebras and ambidextrous adjunctions Theo Appl Cat 16 84ndash

122 (2006)[Li1] Li H-S Regular representations of vertex operator algebras Commun Contemp Math 4

639ndash683 (2002)[Li2] Li H-S Regular representations and huang-lepowsky tensor functors for vertex operator

algebras J Alge 255 423ndash462 (2002)

892 L Kong I Runkel

Definition 31

(i) Let A B be objects of C2plusmn A morphism f A otimes B rarr B is called S-invariant iff

A

B

W

W

f =

A

B

W

W

f

K

ω(33)

holds for all for W isin C2plusmn(ii) A C2plusmn-algebra (Aclmcl ηcl) is called modular invariant iff θAcl = idAcl and mcl

is S-invariant

Lemma 32 The morphism f A otimes B rarr B is S-invariant if and only if

A

B

Ui times U j

Ui times U j

f = Dim Cdim Ui dim U j

sumα

A

B

B

f

Ui times U j

Ui times U j

α

α

(34)

holds for all i j isin I

Proof Condition (33) holds for all W iff it holds for all W = Ui times U j i j isin I so itis enough to show that the right-hand side of (33) with W = Ui times U j is equal to theright-hand side of (34) Recall the notation for basis morphisms in (236) Starting from(33) write

idBor otimes idUi timesU j =⎛⎝sum

klα

(b(ktimeslα)

B

)or (

bB(ktimeslα)

)or⎞⎠ otimes idUi timesU j (35)

and then apply (32) The graphical representation of the resulting morphism can bedeformed to give (34) Remark 33 As shown in [K3 Sect 61] the modular invariance condition of a C2plusmn-alge-bra exactly coincides with the modular invariance condition for torus 1-point correlationfunctions of a genus-01 closed CFT In particular the condition θAcl = idAcl is equiv-alent to invariance under the modular transformation T τ rarr τ + 1 and the condition(34) with f = mcl is equivalent to invariance under S τ rarr minus 1

τ Combining the

modular invariance condition with the genus zero properties of a genus-01 closed CFTresults in a modular invariant commutative symmetric Frobenius algebra in C2plusmn

Cardy Algebras and Sewing Constraints I 893

Let Acl be a modular invariant C2plusmn-algebra Evaluating (34) for f = mcl composingit with ηcl otimes idUi timesU j and taking the trace implies the following identity

Zi j = 1

DimCAclUi timesU j where

(36)

Zi j = dimC HomC2plusmn(UitimesU j Acl)

Decomposing Acl into simple objects this gives

Zi j =sum

klisinISik Zkl Sminus1

l j where Si j = si jradic

DimC (37)

which in CFT terms is of course nothing but the invariance of the torus partition functionunder the modular S-transformation

The following theorem gives a simple criterion for modular invariance

Theorem 34 Let Acl be a haploid commutative symmetric Frobenius algebra in C2plusmn

(i) If Acl is modular invariant then dim Acl = Dim C(ii) If dim Acl = Dim C then Acl is special and modular invariant

Proof Part (i) Since Acl is haploid for i = j = 0 Equation (36) reduces to 1 =dim AclDimCPart (ii) By the same reasoning as in the proof of (iii)rArr(i) in Lemma 211 one showsthat Acl is special Thus mcl cl = ζclidAcl for some ζcl = 0

By [KO Thm 45] the category (C2plusmn)locAcl

of local Acl-modules is again a mod-

ular tensor category and Dim (C2plusmn)locAcl

= (Dim C dim Acl)2 (see [Fr Prop 321 amp

Rem 323] for the same statement in the notation used here) Thus by assumption wehave Dim(C2plusmn)loc

Acl= 1 It then follows from [ENO1 Thm 23] that up to isomorphism

(C2plusmn)locAcl

has a unique simple object (namely the tensor unit) In other words every simplelocal Acl-module is isomorphic to Acl (seen as a left-module over itself)

We have the following isomorphisms between morphism spaces [FS Prop 47 amp 411]

HomAcl(Acl otimes (Ui times U j ) Acl) sim= HomC2plusmn(Ui times U j Acl)

HomAcl(Acl Acl otimes (Ui times U j )) sim= HomC2plusmn(AclUi times U j )

Using these to transport the bases (236) from the right to the left we obtain bases bα(i j)αof HomAcl(Acl otimes (Ui times U j ) Acl) and b(i j)

β β of HomAcl(Acl Acl otimes (Ui times U j )) Thesecan be expressed graphically as

894 L Kong I Runkel

bα(i j) =

Acl

Acl

Acl

Ui times U j

α

b(i j)β = DimC

dim Ui dim U j

1

ζcl

Acl

Ui times U jAcl

Acl

β

(38)

where the nonzero factor in b(i j)β is included for convenience Notice that bα(i j) b(i j)

β isa left Acl-module map Since Acl is simple as a left module over itself we have

bα(i j) b(i j)β = λαβ idAcl (39)

for some λαβ isin C By computing tr(bα(i j) b(i j)β ) it is easy to verify that λαβ = δαβ

We will now prove the following identity

1

ζcl

Acl

Acl

Ui times U j

Ui times U j

Acl

=sumα

DimCdim Ui dim U j

1

ζcl

Acl

Ui times U j

Ui times U j

Acl

Acl

Acl

α

α

(310)

One checks that the left-hand side of this equation is an idempotent which we denote byζminus1

cl PlAcl(Ui timesU j ) cf [Fr Sect 31] By [Fr Prop 41] the image Im(ζminus1

cl PlAcl(Ui timesU j ))

is a local Acl-module and hence isomorphic to AoplusNcl for some N isin Zge0

All left-module morphisms from Acl to Aclotimes(Ui timesU j ) are linear combinations of the

b(i j)β Furthermore one verifies that ζminus1

cl PlAcl(Ui timesU j )b(i j)

β = b(i j)β Therefore the b(i j)

β

describe precisely the image of the idempotent ie ζminus1cl Pl

Acl(Ui timesU j ) = sum

α b(i j)α bα(i j)

which is nothing but (310) Composing (310) with ζcl middot εcl otimes idUi timesU j from the left (iefrom the top) produces (34)

In addition since Acl is commutative symmetric Frobenius it satisfies θAcl = idAcl

[Fr Prop 225] Altogether this shows that Acl is modular invariant Remark 35 As we were writing this paper we heard that the results in Theorem 34were obtained independently by Kitaev and Muumlger [Ki]

Remark 36 Setting i = j = 0 in (36) gives the identity dim Acl = Z00 Dim C [KRProp 23] Combining this with Theorem 34 (ii) one may wonder if a general modularinvariant commutative symmetric Frobenius algebra Acl in C2plusmn is isomorphic to a directsum of simple such algebras However this is not so For example one can take the com-mutative symmetric Frobenius algebra Acl = C[x]x2 in the category of vector spacesequipped with the non-degenerate trace ε(ax + b) = a In this case the modular invari-ance condition holds automatically but Acl is clearly not a direct sum of two algebras

Cardy Algebras and Sewing Constraints I 895

For a general modular tensor category C the algebra C[x]x2 R(1) understood as analgebra in C2plusmn via the braided monoidal isomorphism Vect f (C) C2plusmn rarr C2plusmn providesanother counter-example

32 Two definitions Define a morphism PlA A rarr A for a Frobenius algebra A in C

or C2plusmn as follows [Fr Sect 24]

PlA =

A

A

A

m

(311)

If A is also commutative and obeys m A A = ζA idA we have PlA = ζA idA In partic-

ular this holds if A is commutative and special Using the fact that the Frobenius algebraR(1) is commutative and normalised-special one can check that Pl

R(A) R(A) rarr R(A)takes the following form

PlR(A) =

oplusiisinI

A

A

Uori

Uori

A

m

times

Ui

Ui

(312)

With these ingredients we can now give the first definition of a Cardy C|C2plusmn-algebrawhich was introduced in [K3 Def 514] cf Remark 315 below

Definition 37 (CardyC|C2plusmn-algebra I) A CardyC|C2plusmn-algebra is a triple (Aop|Acl ιcl-op)where (Aclmcl ηclcl εcl) is a modular invariant commutative symmetric Frobe-nius C2plusmn-algebra (Aopmop ηopop εop) is a symmetric Frobenius C-algebra andιcl-op Acl rarr R(Aop) an algebra homomorphism such that the following conditionsare satisfied

(i) Centre condition

Acl R(Aop)

R(Aop)

R(Aop)

ιcl-op

m R(Aop)

=

Acl

ιcl-op

R(Aop)

R(Aop)

R(Aop)

m R(Aop)

(313)

896 L Kong I Runkel

(ii) Cardy condition

ιcl-op ιlowastcl-op =

R(Aop)

R(Aop)

R(Aop)

(314)

Remark 38

(i) The name ldquoCardy C|C2plusmn-algebrardquo in Definition 37 was chosen because many ofthe important ingredients were first studied by Cardy the modular invariance ofthe closed theory [C1] the consistency of the annulus amplitude [C2] and thebulk-boundary OPE [CL] On the other hand the boundary-boundary OPE andthe OPE analogue of the centre condition were first considered in [Lw]

(ii) One can easily see that in the special case that C is the category Vect f (C) of finite-dimensional C-vector spaces a Cardy C|C2plusmn-algebra gives exactly the algebraicformulation of two-dimensional open-closed topological field theory over C (cfRemark 614 in [K3]) see [Lz Sect 48] [Mo Thm 11] [AN Thm 45] [LPCor 43] [MS Sect 22] When passing to a general modular tensor category Cthere are two important differences to the two-dimensional topological field the-ory Firstly the algebras Acl and Aop now live in different categories which inparticular affects the formulation of the centre condition and the Cardy conditionSecondly the modular invariance condition has to be imposed on Acl In the caseC = Vect f (C) modular invariance holds automatically

Definition 39 A homomorphism of Cardy C|C2plusmn-algebras (A(1)op |A(1)cl ι(1)cl-op) rarr (A(2)op

|A(2)cl ι(2)cl-op) is a pair ( fop fcl) of Frobenius algebra homomorphisms fop A(1)op rarr A(2)op

and fcl A(1)cl rarr A(2)cl such that the diagram

A(1)cl

fcl

ι(1)cl-op

A(2)cl

ι(2)cl-op

R(A(1)op )

R( fop) R(A(2)op )

(315)

commutes

Remark 310 Since a homomorphism of Frobenius algebras is invertible (cf Lemma218 (iv)) a homomorphism of Cardy algebras is always an isomorphism

For a homomorphism ( fop fcl) of Cardy C|C2plusmn-algebras using the commutativity of(315) and the fact that fcl and fop are both algebra and coalgebra homomorphisms itis easy to show that (315) commutes iff

R(A(1)op )R( fop)

(ι(1)cl-op)

lowast

R(A(2)op )

(ι(2)cl-op)

lowast

A(1)cl

fcl A(2)cl

(316)

Cardy Algebras and Sewing Constraints I 897

commutesLet (Aop|Acl ιcl-op) be a Cardy C|C2plusmn-algebra Define the morphism

ιcl-op = χminus1(ιcl-op) T (Acl) minusrarr Aop (317)

Decompose Acl as Acl = oplusNn=1Cl

n times Crn such that Cl

1 times Cr1 = ηcl(1 times 1) We use ι(n)cl-op

to denote the restriction of ιcl-op to Cln times Cr

n and ι(n)cl-op to denote the restriction of ιcl-op

to Cln otimes Cr

n We introduce the following graphical notation

ι(n)cl-op =

Cln Cr

n

Aop

(318)

By (243) ιcl-op can be expressed in terms of ιcl-op as follows

ιcl-op =Noplus

n=1

oplusiisinI

sumα

Cln

Crn

Aop

α

Uori

times α

Crn

Ui

(319)

Lemma 311 The centre condition (313) is equivalent to the following condition in C

Cln Cr

n

Aop

Aop

Aop

=

Cln Cr

n

Aop

Aop

Aop

for n = 1 N (320)

Proof First insert (319) and the definition (217) of m R(Aop) into (313) Then apply thecommutativity of R(1) to the left hand side of (313) The equivalence between (313)and (320) follows immediately Remark 312 The centre condition (320) is very natural from the open-closed confor-mal field theory point of view Correlators on the upper half plane are expressed in termsof conformal blocks on the full complex plane The objects Cl

n and Crn are associated to

the field insertion at a point z in the upper half plane and at the complex conjugate pointz in the lower half plane respectively The object Aop corresponds to a field inserted ata point r on the real axis The centre condition (320) simply says that the correlationfunctions in the disjoint domains |z| gt r gt 0 and r gt |z| gt 0 are analytic continuationsof each other see [K2 Prop 118]

898 L Kong I Runkel

Recall that we define ιlowastcl-op Aop rarr T (Acl) as in (230) We introduce the graphicalnotation

ιlowastcl-op =Noplus

n=1

Cln Cr

n

Aop

=Noplus

n=1

Cln Cr

n

Aop

Aop T (minus1Acl) (321)

where the second equality follows from (215) (221) and (259)

Lemma 313 The Cardy condition (314) is equivalent to the following identity in C

Noplusn=1

DimCdim Ui

sumα

Cln

Crn

Crn

α

α

Aop

Aop Uori

Uori

=Aop

Aop

Aop

Uori

Uori

op

mop for all i isin I (322)

Proof By (244) (312) and (319) it is easy to see that (322) is equivalent to thefollowing identity

ιcl-op χ(ιlowastcl-op) = PlR(Aop)

(323)

Therefore it is enough to show that

χ(ιlowastcl-op) = ιlowastcl-op (324)

We have

χminus1(ιlowastcl-op)(1)= T (ιlowastcl-op) δ (2)= T (ιcl-op)

lowast δlowastlowast (3)= (ρ T (ιcl-op)

)lowast (4)= (ιcl-op

)lowast (325)

In step (1) we use the expression (253) for χminus1 step (2) follows from Lemma 218 (v)and Lemma 219 Step (3) is Lemma 218 (i) and Lemma 228 and finally step (4)amounts to substituting (253) and (317) Acting with χ on both sides of the aboveequality produces (324)

Combining Lemmas 311 and 313 and Proposition 216 we obtain the followingequivalent definition of Cardy C|C2plusmn-algebra (recall the graphical notation (318) forιcl-op and (321) for ιlowastcl-op)

Cardy Algebras and Sewing Constraints I 899

Definition 314 (CardyC|C2plusmn-algebra II) A CardyC-algebra is a triple (Aop|Acl ιcl-op)where Acl is a commutative symmetric Frobenius C2plusmn-algebra satisfying property (34)with f = mcl Aop is a symmetric Frobenius C-algebra and ιcl-op T (Acl) rarr Aop isan algebra homomorphism satisfying the conditions (320) and (322)

Remark 315 Up to a choice of normalisation Definition 314 is the same as the orig-inal one in [K3 Def 613] The difference between the two definitions is the factorDim C dim Ui on the left hand side of (322) which in [K3 Def 613] is given byradic

Dim C dim Ui The two definitions are related by rescaling the coproduct cl andcounit εcl of Acl by 1

radicDim C and

radicDim C respectively We chose the convention in

(322) to remove all dimension factors from the expression (314) for the Cardy condition

33 Uniqueness and existence theorems In this subsection we investigate the structureof Cardy algebras We start with the following proposition which when combined withthe results of part II provides an alternative proof of [Fj Prop 422]

Proposition 316 Let (Aop|Acl ιcl-op) be a Cardy C|C2plusmn-algebra If Acl is simple anddim Aop = 0 then Aop is simple and special

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis therefore haploid Restricting the Cardy condition (322) to the case Ui = 1 andcomposing both sides with εop from the left we see that εop kills all terms associated toU j times 1 isin Acl in the sum except for a single 1 times 1 term Thus we obtain the followingidentity

β εop = dAop (mop otimes idAorop) (idAop otimes bAop) (326)

where β isin C Composing with ηop from the right in turn implies that βεop ηop =dim Aop which is nonzero by assumption Thus also β = 0 and εop is a nonzero multipleof the morphism on the right hand side of (326) By [FRS Lem 311] Aop is special

Since Aop is a special Frobenius algebra Aop is semi-simple as an Aop-bimodule(apply [FS Prop 524] to Aop tensored with its opposite algebra) Suppose Aop is not

simple so that we can write Aop = A(1)op oplus A(2)op for nonzero Aop-bimodules A(1)op and A(2)op We denote the canonical embeddings and projections associated to this decompositionas ι12 and π12 We have the identities

mop (ι1 otimes ι2) = 0 εop ηop =2sum

i=1

εop ιi πi ηop (327)

The first identity follows since π1 mop (ι1 otimes ι2) = 0 (as mop gives the left action of

Aop on Aop and hence it preserves A(2)op ) and similarly π2 mop (ι1 otimes ι2) = 0 Thesecond identity is just the completeness of ι12 π12

Since εop ηop = 0 without losing generality we can assume εop ι1 π1 ηop = 0Using that π2 is a bimodule map we compute

π2 [LHS of (322)]Ui =1 ι1 π1 ηop = π2 [RHS of (322)]Ui =1 ι1 π1 ηop

= PlA(2)op

π2 ι1 π1 ηop = 0 (328)

900 L Kong I Runkel

On the other hand using that Acl is haploid that ιcl-op is an algebra map and thatι lowastcl-op is a coalgebra map one can check that the left-hand side of (328) is equal toλ(εop ι1 π1 ηop) π2 ηop for some λ = 0 This implies that π2 ηop = 0 Thusηop = sum

i ιi πi ηop = ι1 π1 ηop Hence we have

0 = π2 ι2 = π2 mop (ηop otimes ι2) = π2 mop ((ι1 π1 ηop)otimes ι2) (329)

However the right-hand side is zero by (327) This is a contradiction and hence Aopmust be simple

To formulate the next theorem we need the notion of the full centre of an algebra [FjDef 49] Recall that an algebra A in a braided tensor category has a left centre and aright centre [VZO] both of which are sub-algebras of A Of these two we will onlyneed the left centre The following definition is [Fr Def 231] which in our setting isequivalent to that of [VZO]

Definition 317 Let A be a symmetric special Frobenius algebra such that m A A =ζA idA

(i) The left centre Cl(A) of A is the image of the idempotent ζminus1A Pl

A(ii) The full centre Z(A) is Cl(R(A))

That ζminus1A Pl

A is an idempotent follows from [FRS Lem 52] when keeping track ofthe factors ζA ([FRS] assumes normalised-special ie ζA = 1) Note that Cl(A) is againan object of C while Z(A) is an object of C2plusmn Let el

A Cl(A) rarr A be the embeddingof Cl(A) into A The left centre is in fact the maximal subobject of A such that

m A cAA (elA otimes idA) = m A (330)

see [Fr Lem 232] This observation explains the name left centre and also makes theconnection to [O Def 15]

The full centre is by definition the image of the idempotent ζminus1A Pl

R(A) R(A) rarrR(A) Since C2plusmn is abelian the idempotent splits and we obtain the embedding andrestriction morphisms

e Z(A) rarr R(A) and r R(A) Z(A) (331)

which obey r e = idZ(A) and e r = ζminus1A Pl

R(A) It follows from Proposition 225 and

[Fr Prop 237] that Z(A) is a commutative symmetric Frobenius algebra in C2plusmn withstructure morphisms5

m Z(A) = r m R(A) (e otimes e) ηZ(A) = r ηR(A)

Z(A) = ζA middot (r otimes r) R(A) e εZ(A) = ζminus1A middot εR(A) e

(332)

Moreover if A is simple then Z(A) is simple and if A is simple and dim A = 0 thenZ(A) is simple and special The normalisation of the counit is such that

εZ(A) ηZ(A) = ζminus2A dim A DimC (333)

5 The normalisation of product and unit is the standard one The factors in the coproduct and counit haveto be included in order for (A|Z(A) e) to be a Cardy algebra see Theorem 318 below The normalisation ofthe counit enters the Cardy condition (314) through the definition of ( middot )lowast

Cardy Algebras and Sewing Constraints I 901

Theorem 318 Let A be a special symmetric Frobenius C-algebra Then (A|Z(A) e)is a Cardy C|C2plusmn-algebra

The proof of this theorem makes use the following two lemmas

Lemma 319 e Z(A) rarr R(A) is an algebra map and elowast = ζA middot r

Proof It follows from [Fr Lem 229] (or by direct calculation using in particularm R(A) R(A) = ζAidR(A)) that

m R(A) (e otimes e) = ζminus1A middot Pl

R(A) m R(A) (e otimes e) (334)

Substituting e r = ζminus1A Pl

R(A) shows that e is compatible with multiplication For theunit one finds

e ηZ(A) = e r ηR(A) = ζminus1A Pl

R(A) ηR(A) = ηR(A) (335)

Thus e is an algebra map For the second statement one computes

elowast (1)= ζA

R(A)

r

R(A)

R(A)

Z(A)

R(A)

Z(A)

r

e

e

(2)=

R(A)

r

R(A)

R(A)

R(A)

Z(A)

(3)=

R(A)

r

R(A)

Z(A)

(4)= ζA middot r (336)

where in (1) the definitions (230) and (332) have been substituted step (2) is e r = ζminus1

A PlR(A) step (3) uses that R(A) is symmetric Frobenius and step (4) is again

e r = ζminus1A Pl

R(A) Lemma 320 Let A be a symmetric Frobenius algebra in C The morphism

PlR(A) m R(A)

(Pl

R(A) otimes PlR(A)

) R(A)otimes R(A) minusrarr R(A) (337)

is S-invariant

The proof of this lemma is a slightly lengthy explicit calculation and has been deferredto Appendix A2

902 L Kong I Runkel

Proof of Theorem 318 That e is an algebra map was proved in Lemma 319 The centrecondition (313) holds by property (330) of the left centre The Cardy condition (314)also is an immediate consequence of Lemma 319

ιcl-op ιlowastcl-op = e (ζA r) = PlR(A) (338)

The full centre Z(A) is a commutative symmetric Frobenius algebra It remains to provemodular invariance That θZ(A) = idZ(A) is implied by commutativity and symmetryof Z(A) [Fr Prop 225] The S-invariance condition (33) follows from Lemma 320In (337) substitute Pl

R(A) = ζA e r and then put the resulting morphism into (33)Compose the resulting equation with e otimes idW from the right (ie from the bottom) andsubstitute the definition (332) of m Z(A) This results in the statement that m Z(A) isS-invariant

The following theorem is analogous to [LR Prop 29] and [Fj Thm 426] whichroughly speaking answer the question under which circumstances the restriction of atwo-dimensional conformal field theory to the boundary already determines the entireconformal field theory The first work is set in Minkowski space and uses operator alge-bras and subfactors while the second work is set in Euclidean space and uses modulartensor categories

Theorem 321 Let (A|Acl ιcl-op) be a Cardy C|C2plusmn-algebra such that dim A = 0 andAcl is simple Then A is special and (A|Acl ιcl-op) sim= (A|Z(A) e) as Cardy algebras

Proof By Proposition 316 A is simple and special Since Acl is simple the algebramap ιcl-op Acl rarr R(A) is either zero or a monomorphism But ιcl-op ηcl = ηR(A)and so ιcl-op = 0 Thus ιcl-op is monic By Lemma 218 (ii) ζminus1

A ιlowastcl-op is epi The Cardycondition (314) implies

ιcl-op ζminus1A ιlowastcl-op = ζminus1

A PlR(A) = e r (339)

Composing this with e r from the left yields e r ιcl-op ζminus1A ιlowastcl-op = e r =

ιcl-op ζminus1A ιlowastcl-op Since ζminus1

A ιlowastcl-op is epi we have

e r ιcl-op = ιcl-op (340)

Actually (340) also follows from (313) and specialness of R(A) We will prove that

( fop fcl) (A|Acl ιcl-op) minusrarr (A|Z(A) e) where fop = idA fcl = r ιcl-op

(341)

is an isomorphism of Cardy algebrasfcl is an algebra map Compatibility with the units follows since ιcl-op is an algebramap

fcl ηcl = r ιcl-op ηcl = r ηR(A) = ηZ(A) (342)

Compatibility with the multiplication also follows since ιcl-op is an algebra map

m Z(A) ( fcl otimes fcl) = r m R(A) (e otimes e) (r otimes r) (ιcl-op otimes ιcl-op)

= r m R(A) (ιcl-op otimes ιcl-op) = r ιcl-op mcl = fcl mcl (343)

where in the second step we used (340)

Cardy Algebras and Sewing Constraints I 903

fcl is an isomorphism As above since fcl is an algebra map and since Acl is simple fcl

has to be monic By Lemma 319 rlowast = ζminus1A e Thus f lowast

cl = ιlowastcl-op rlowast = ζminus1A ιlowastcl-op e and

fcl f lowastcl = r ιcl-op ζminus1

A ιlowastcl-op e = r e r e = idZ(A) (344)

and so fcl is also epi and hence isofcl is a coalgebra map Since fcl is an algebra map so is f minus1

cl By (344) f minus1cl = f lowast

cland by Lemma 218 (iii) this implies that fcl is a also coalgebra mapThe diagram (315) commutes Commutativity of (315) is equivalent to e fcl = ιcl-opwhich holds by (340)

Let A be a special symmetric Frobenius algebra So far we have seen that (A|Z(A) e)is a Cardy algebra and that all Cardy algebras with Aop = A and simple Acl are of thisform It is now natural to ask if every simple Acl does occur as part of a Cardy algebraThe following theorem provides an affirmative answer Recall that for an A-left moduleM the object Mor otimesA M is an algebra (see eg [KR Lem 42])

Theorem 322 If Acl is a simple modular invariant commutative symmetric FrobeniusC2plusmn-algebra then there exist a simple special symmetric Frobenius C-algebra A and amorphism ιcl-op Acl rarr R(A) such that

(i) Acl sim= Z(A) as Frobenius algebras(ii) (A|Acl ιcl-op) is a Cardy C|C2plusmn-algebra

(iii) T (Acl) sim= oplusκisinJ Morκ otimesA Mκ as algebras where MκκisinJ is a set of representa-

tives of the isomorphism classes of simple A-left modules

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis haploid It then follows from Theorem 34 that Acl is special By Proposition 224T (Acl) is a special symmetric Frobenius algebra in C Thus T (Acl) = oplusi Ai where theAi are simple symmetric Frobenius algebras We will show that at least one of the Aiis special Since T (Acl) is special we have mT (Acl) T (Acl) = ζ idT (Acl) for someζ isin C

times Restricting this to the summand Ai shows mi i = ζ idAi FurthermoreεT (Acl) ηT (Acl) = ξ id1 for some ξ isin C

times But εT (Acl) ηT (Acl) = sumi εi ηi and so

at least one of the εi ηi has to be nonzero Therefore at least one of the Ai is speciallet A equiv Ai be this summand We denote the embedding A rarr T (Acl) by e0 and therestriction T (Acl) A by r0 Notice that r0 is an algebra homomorphism Define

ιcl-op = χ (r0) Acl minusrarr R(A) (345)

By Proposition 216 ιcl-op is an algebra homomorphism Next we verify the centre con-dition (313) or rather its equivalent form (320) By substituting the definitions onecan convince oneself that the commutativity mcl cAclAcl = mcl of Acl in C2plusmn implies thecondition mT (Acl) = mT (Acl) in C see [K2 Prop 36] Here T (Acl)otimes T (Acl) rarrT (Acl)otimes T (Acl) is given by

=oplusmn

(idCl

notimes cCl

m Crnotimes idCr

m

)

(cCl

m Clnotimes cminus1

Crn C

rm

)

(idCl

motimes cminus1

Cln C

rm

otimes idCrn

)

(346)

and we decomposed Acl as Acl = oplusnCln times Cr

n As a consequence we obtain the identity

r0 mT (Acl) (idAcl otimes e0) = r0 mT (Acl) (idAcl otimes e0) (347)

904 L Kong I Runkel

Using that r0 is an algebra map and that by definition ιcl-op = r0 we obtain (320)In order to show that (A|Acl ιcl-op) is a Cardy algebra it remains to show that the

Cardy condition (314) is satisfied We will demonstrate this via a detour by first provingthat Acl sim= Z(A) as Frobenius algebras

Recall the notations e and r given in (331) Using the centre condition (320) onecan check that Pl

R(A) ιcl-op = m R(A) R(A) ιcl-op By specialness of A we have

m A A = ζA idA and so together with e r = ζminus1A Pl

R(A) we get

e r ιcl-op = ιcl-op (348)

Next consider the morphism

fcl = r ιcl-op Acl minusrarr Z(A) (349)

By the same derivation as in (342) and (343) one sees that fcl is an algebra map Inparticular fcl ηcl = ηZ(A) = 0 and so fcl = 0 Since Acl is simple fcl has to be amonomorphism

By the same argument as used in the proof of Theorem 34 (ii) up to isomorphismAcl is the unique simple local Acl-(left-)module The algebra monomorphism fcl turnsZ(A) into an Acl-module Since Z(A) is commutative it is local as an Acl-module andso Z(A) sim= AoplusN

cl for some N ge 1 By construction A is a simple special symmetricFrobenius algebra Proposition 225 and Corollary 227 show that R(A) inherits all theseproperties and thus Z(A) is simple (see the comment below Eq (332)) By Theorem318 Z(A) is modular invariant and then by Theorem 34 (i) dim Z(A) = Dim C Thisimplies that N = 1 in Z(A) sim= AoplusN

cl and so fcl is in fact an isomorphismSince Acl and Z(A) are both haploid we have εZ(A) fcl = ξ εcl for some ξ isin C

timesThe counit uniquely determines the Frobenius structure on Acl and Z(A) (see eg [FRSLemma 37]) so that fcl is a coalgebra isomorphism iff ξ = 1 To compute ξ we composethe above identity with ηcl from the right Defining ζcl via εcl ηcl = ζminus1

cl Dim C middot id1 andusing (333) gives ξ = dim A ζclζ

2A By rescaling the comultiplication and the counit

of A and consequently changing ζA we can always achieve ξ = 1 This proves part (i)of the theorem

Equation (348) implies that ιcl-op = e fcl Since fcl is an isomorphism of Frobeniusalgebras by Lemmas 218 and 319 we have

ιcl-op ιlowastcl-op = e fcl f lowastcl elowast = ζA e r = Pl

R(A) (350)

Thus (A|Acl ιcl-op) is a Cardy algebra This proves part (ii) of the theoremPart (iii) can be seen as follows By [KR Prop 43] T Z(A) sim= oplusκisinJ Mor

κ otimesA Mκ asalgebras Together with the observation that T ( fcl) T (Acl) rarr T Z(A) is an isomor-phism of algebras this proves part (iii) Remark 323 Part (i) of Theorem 322 was announced by Muumlger [Muuml2] We provide anindependent proof in the setting of Cardy algebras

The above theorem together with Lemma 211 and Theorem 34 shows that a sim-ple commutative symmetric Frobenius C2plusmn-algebra Acl with dim Acl = Dim C is alwayspart of a Cardy algebra (Aop|Acl ιcl-op) for some simple special symmetric Frobeniusalgebra Aop in C However the above proof also illustrates that Aop is not unique Thisraises the question how two Cardy algebras with a given Acl can differ This question isanswered by [KR Thm 11] which in the present framework can be restated as follows

Cardy Algebras and Sewing Constraints I 905

Theorem 324 If (A(i)op |A(i)cl ι(i)cl-op) i = 1 2 are two Cardy C|C2plusmn-algebras such that

A(i)cl is simple and dim A(i)op = 0 for i = 1 2 then A(1)clsim= A(2)cl as algebras if and only

if A(1)op and A(2)op are Morita equivalent

Proof Theorem 11 in [KR] is stated for A(i)op being non-degenerate algebras and A(i)cl =Z(A(i)op ) for i = 1 2 By Proposition 316 A(i)op are simple and special for i = 1 2

Then by [KR Lem 21] A(i)op are non-degenerate algebras By Theorem 321 we have

A(i)clsim= Z(A(i)op ) as Frobenius algebras Finally by [KR Thm 11] Z(A(1)op ) sim= Z(A(2)op )

as algebras iff A(1)op and A(2)op are Morita equivalent Let Cmax(C2plusmn) be the set of equivalence classes [B] of simple modular invariant

commutative symmetric Frobenius algebras B in C2plusmn Two such algebras B and B prime areequivalent if B and B prime are isomorphic as algebras (but not necessarily as Frobenius alge-bras) Let Msimp(C) be the set of Morita classes of simple special symmetric Frobeniusalgebras in C Define the map z Msimp(C) rarr Cmax(C2plusmn) by z A rarr [Z(A)] whereA denotes the Morita class of A From Theorem 322 (i) and [KR Thm 11] we learn

Corollary 325 The map z Msimp(C) rarr Cmax(C2plusmn) is a bijection

A Appendix

A1 Proof of Lemma 27 We will show that if (F ψ F2 ψ

F0 ) is a colax tensor functor

from C1 to C2 then (G φG2 φ

G0 ) is a lax tensor functor from C2 to C1 Applying this

result to the opposed categories then gives the converse statementWe need to show that φG

0 and φG2 make the diagrams (21) and (22) commute We

first prove the commutativity of (21) Consider the following diagram

G(A)otimes (G(B)otimes G(C))

GψF2 δ

idG(A)otimesφG2 G(A)otimes G(B otimes C)

GψF2 δ

G(FG(A)otimes F(G(B)otimes G(C)))

G(F(idG(A))otimesF(φG2 ))

G(idFG(A)otimesψ F2 )

G(FG(A)otimes FG(B otimes C))

G(ρAotimesρBotimesC )

G(FG(A)otimes (FG(B)otimes FG(C)))

G(ρAotimes(ρBotimesρC )) G(A otimes (B otimes C))

(A1)

The top subdiagram is commutative because of the naturality of Gψ F2 δ The commu-

tativity of the bottom subdiagram follows from the following identities

(ρB otimes ρC ) ψ F2 = (ρB otimes ρC ) ψ F

2 ρF Fδ

= ρBotimesC FG(ρB otimes ρC ) FG(ψ F2 ) Fδ

= ρBotimesC F(φG2 ) (A2)

as a map F(G(B) otimes G(C)) rarr B otimes C The commutativity of (A1) implies that thecomposition of maps in the left column in (21) can be replaced by

G(ρA otimes (ρB otimes ρC )) G((idFG(A) otimes ψ F

2 ) ψ F2

) δ (A3)

906 L Kong I Runkel

Similarly we can show that the composition of maps in the right column in (21) can bereplaced by

G((ρA otimes ρB)otimes ρC ) G((ψ F

2 otimes idFG(C)) ψ F2

) δ (A4)

Using the commutativity of (23) it is easy to see that (21) with the left and right col-umns of (21) replaced by (A3) and (A4) respectively is commutative Hence (21) iscommutative

Now we prove the commutativity of the first diagram in (22)

φG2 (φG

0 otimes idG(A))

(1)= G(ρ12 otimes ρA) Gψ F2 δ

[(Gψ F

0 δ11)otimes idG(A)

]

(2)= G(ρ12 otimes ρA) Gψ F2 G F(Gψ0 otimes idG(A)) G F(δ11 otimes idG(A)) δ

(3)= G(ρ12 otimes ρA) G(FG(ψ F0 )otimes idFG(A)) Gψ F

2 G F(δ11 otimes idG(A)) δ(4)= G(id12 otimes ρA) G

([ψ F

0 ρF(11) (Fδ)11] otimes idFG(A)

) Gψ F

2 δ(5)= G(id12 otimes ρA) G(ψ F

0 otimes idFG(A)) Gψ F2 δ

(6)= G(id12 otimes ρA) G(lminus1FG(A)) G F(lG(A)) δ

(7)= G(lminus1A ) GρA δG lG(A)

(8)= G(lminus1A ) lG(A) (A5)

where in step (1) we substituted the definition of φG0 φ

G2 given in (210) in step (2)

we used the naturality of δ in step (3) we used the naturality of Gψ F2 in step (4) we

switched the position between Gρ12 and G FG(ψ F0 ) and the position between Gψ2 and

G F(δ12 otimes idG(A)) using the naturality of ρ and FψG2 respectively in step (5) we applied

the second identity in (28) in step (6) we used (24) in step (7) we used the naturalityof lminus1 and δ in step (8) we used the first identity in (28)

The proof of the commutativity of the second diagram in (22) is similar Thus wehave shown that G is a lax tensor functor

A2 Proof of Lemma 320 To prepare the proof recall that for a given object B isin Cthe modular group P SL(2Z) acts on the space oplusi HomC(B otimes Ui Ui ) see eg [BKSect 31] and [K3 Eq (455)] We will only need the action of S and Sminus1 Let f isinoplusi HomC(B otimes Ui Ui ) Then

S oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A6)

Cardy Algebras and Sewing Constraints I 907

Sminus1 oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A7)

By Lemma 32 to establish that (337) is S-invariant it is enough to prove the identity(34) when f is given by (337) Using (A6) and (A7) we can see that Eq (34) simplysays that oplusi j [RHS of (34)] is invariant under the action of S times S Consider the elementg of oplus jkisinIHomC2plusmn(R(A)otimes (Uor

j times Uk)Uorj times Uk) given by

g =oplusjkisinI

sumα

R(A)

R(A)

R(A)

R(A)

Uorj times Uk

Uorj times Uk

α

α

(A8)

By the above arguments proving S-invariance of (337) is equivalent to proving invari-ance of g under the action of S times S

For i isin I we denote by gi the component of g in(oplus jisinIHomC(A otimes Uor

i otimes Uorj U

orj )

)otimes (opluskisinIHomC(Ui otimes UkUk))

We view the second Hom-space in the above tensor product as a Hom-space in C+ insteadof Cminus It is enough to show that gi is invariant under the action of S times Sminus1 Note thatthe action of Sminus1 in C+ is equivalent to that of S in Cminus

The morphism gi can be canonically identified with a bilinear pairing

( middot middot )i ⎛⎝oplus

jisinIHomC(Uor

j A otimes Uori otimes Uor

j )

⎞⎠

times(oplus

kisinIHomC(UkUi otimes Uk)

)minusrarr C (A9)

as follows For h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) we set

(h1 h2)i = (dim U j dim Uk)minus1 trUor

j timesUk[gi (h1 times h2)] (A10)

908 L Kong I Runkel

When substituting the explicit form of the product m R(A) of R(A) = (A times 1) otimes R(1)after a short calculation one finds

(h1 h2)i =sumα

1

dim U j

A

A

A

A

A A

A α

α

h1

h2

U jUk

Uorj

Ui

Uk (A11)

Here the top morphism PlR(A) has been simplified with the help of the identity

PR(A) m R(A) (PR(A) otimes PR(A))

=(oplus

iisinI((m A A)otimes idUor

i)times idUi

) m R(A) (PR(A) otimes PR(A)) (A12)

which can be checked by direct calculation along the same lines as in the proof of [FrLem 310]

The action of the modular transformation S on oplusiisinI(B otimes Ui Ui ) for B isin C nat-urally induces an action on oplusiisinI(Ui B otimes Ui ) [K3 Prop 514] which we denote bySlowast In the present case we get an action of Slowast on oplus jisinIHomC(Uor

j A otimes Uori otimes Uor

j ) and

opluskisinIHomC(UkUi otimes Uk) Then to show gi is invariant under the action of S times Sminus1

amounts to showing that

(h1 h2)i = ((Sminus1)lowast h1 Slowast h2)i (A13)

Cardy Algebras and Sewing Constraints I 909

for all h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) We have

((Sminus1)lowast h1 Slowast h2)i =sum

mnα

dim Un

DimC

A

A

A

A

A A

A

Uk

α

α

h1

h2

Ui

U j

UmUn

(A14)

Now drag the upper vertex indexed by α in the above graph along its Uorm -leg until it

meets the lower vertex also indexed by α then sum over α and m This gives

((Sminus1)lowast h1 Slowast h2)i =sum

n

dim Un

DimC

A

A

A

A

A A

A

Uk

Un

h1

h2

Ui

U j

(A15)

910 L Kong I Runkel

If we just look at the neighbourhood of the Un-loop in the above graph we see thefollowing subgraphs

sumn

dim Un

DimC

A Aor

Uorj U j

Uk Uork

Un=

sumα

1

dim U j

A Aor Uk Uork

Uorj U j

αα

(A16)

where we have applied [BK Cor 3111] Substituting this subgraph back to the originalgraph in (A15) we obtain

((Sminus1)lowast h1 Slowast h2)i =sumα

1

dim U j

A

A

A

A

A A

A

A

A

Uorj

α

α

h1

h2m A

Ui

U jUk

Uk

(A17)

The graph in (A17) is equal to that in (A11) In order to see this we first drag theldquobubblerdquo (m A A) along A lines and through the m A vertex (because m A A is abimodule map) until it reaches the lower-left leg of the upper vertex indexed by α Thendrag the m A vertex along the (red) dotted line in the above graph Finally we applythe associativity of A (A12) and [Fr Lem 311] Then we see that the graph in (A17)exactly matches with the one in (A11)

Acknowledgements We would like to thank the organisers of the Oberwolfach Arbeitsgemeinschaft ldquoAlge-braic structures in conformal field theoriesrdquo (April 2007) where this work was started for an inspiring meetingWe would further like to thank the Hausdorff Institute for Mathematics in Bonn and the organisers of the stim-ulating meeting ldquoGeometry and Physicsrdquo (May 2008) We are indebted to Alexei Davydov Jens FjelstadJuumlrgen Fuchs Yi-Zhi Huang Alexei Kitaev Urs Schreiber Christoph Schweigert Stephan Stolz and PeterTeichner for helpful discussions andor comments on a draft of this paper The research of IR was partiallysupported by the EPSRC First Grant EPE0050471 the PPARC rolling grant PPC5071451 and the MarieCurie network lsquoSuperstring Theoryrsquo (MRTN-CT-2004-512194)

Cardy Algebras and Sewing Constraints I 911

References

[AN] Alexeevski A Natanzon SM Noncommutative two-dimensional topological field theories andhurwitz numbers for real algebraic curves Sel Math New Ser 12 307ndash377 (2006)

[Bi] Bichon J Cosovereign hopf algebras J Pure Appl Alg 157 121ndash133 (2001)[BK] Bakalov B Kirillov AA Lectures on Tensor Categories and Modular Functors Providence

RI Amer Math Soc 2001[C1] Cardy JL Operator content of two-dimensional conformal invariant theories Nucl Phys

B 270 186ndash204 (1986)[C2] Cardy JL Boundary conditions fusion rules and the verlinde formula Nucl Phys B 324

581ndash596 (1989)[CL] Cardy JL Lewellen DC Bulk and boundary operators in conformal field theory Phys Lett

B 259 274ndash278 (1991)[DMZ] Dong C-Y Mason G Zhu Y-C Discrete series of the Virasoro algebra and the moonshine

module In Algebraic Groups and Their Generalizations Quantum and infinite-dimensional Meth-ods Proc Symp Pure Math 56 Part 2 Providence RI Amer MathSoc 1994 pp 295ndash316

[DP] Day B Pastro C Note on frobenius monoidal functors New York J Math 14 733ndash742 (2008)[ENO1] Etingof PI Nikshych D Ostrik V On fusion categories Ann Math 162 581ndash642 (2005)[ENO2] Etingof PI Nikshych D Ostrik V An analogue of radfordrsquos s4 formula for finite tensor

categories Int Math Research Notices 54 2915ndash2933 (2004)[Fe] Felder G Froumlhlich J Fuchs J Schweigert C Correlation functions and boundary conditions in

rational conformal field theory and three-dimensional topology Comp Math 131 189ndash237 (2002)[FHL] Frenkel IB Huang Y-Z Lepowsky J On axiomatic approaches to vertex operator algebras

and modules Mem Amer Math Soc 104 (1993)[Fj] Fjelstad J Fuchs J Runkel I Schweigert C Uniqueness of openclosed rational cft with

given algebra of open states Adv Theor Math Phys 12 1283ndash1375 (2008)[FK] Froumlhlich J King C The chern-simons theory and knot polynomials Commun Math

Phys 126 167ndash199 (1989)[Fr] Froumlhlich J Fuchs J Runkel I Schweigert C Correspondences of ribbon categories Adv

Math 199 192ndash329 (2006)[FRS] Fuchs J Runkel I Schweigert C Tft construction of rcft correlators I partition functions Nucl

Phys B 646 353ndash497 (2002)[FS] Fuchs J Schweigert C Category theory for conformal boundary conditions Fields Inst Com-

mun 39 25ndash71 (2003)[H1] Huang Y-Z Two-dimensional conformal geometry and vertex operator algebras Progress in

Mathematics Vol 148 Boston Birkhaumluser 1997[H2] Huang Y-Z Rigidity and modularity of vertex tensor categories Comm Contemp Math 10

871ndash911 (2008)[HK1] Huang Y-Z Kong L Open-string vertex algebra category and operads Commun Math

Phys 250 433ndash471 (2004)[HK2] Huang Y-Z Kong L Full field algebras Commun Math Phys 272 345ndash396 (2007)[HK3] Huang Y-Z Kong L Modular invariance for conformal full field algebras httparxivorgabs

math0609570v2[mathQA] 2006[HL] Huang Y-Z Lepowsky J Tensor products of modules for a vertex operator algebra and vertex

tensor categories In Lie Theory and Geometry in honor of Bertram Kostant ed R BrylinskiJ-L Brylinski V Guillemin V Kac Boston Birkhaumluser 1994 pp 349ndash383

[JS] Joyal A Street R Braided tensor categories Adv Math 102 20ndash78 (1993)[K1] Kong L Full field algebras operads and tensor categories Adv Math 213 271ndash340 (2007)[K2] Kong L Open-closed field algebras Commun Math Phys 280 207ndash261 (2008)[K3] Kong L Cardy condition for open-closed field algebras Commun Math Phys 283 25ndash92 (2008)[Ki] Kitaev A Private communication[KO] Kirillov AA Ostrik V On q-analog of mckay correspondence and ade classification of sl(2)

conformal field theories Adv Math 171 183ndash227 (2002)[KR] Kong L Runkel I Morita classes of algebras in modular tensor categories Adv Math 219

1548ndash1576 (2008)[Ld] Lauda AD Frobenius algebras and ambidextrous adjunctions Theo Appl Cat 16 84ndash

122 (2006)[Li1] Li H-S Regular representations of vertex operator algebras Commun Contemp Math 4

639ndash683 (2002)[Li2] Li H-S Regular representations and huang-lepowsky tensor functors for vertex operator

algebras J Alge 255 423ndash462 (2002)

Cardy Algebras and Sewing Constraints I 893

Let Acl be a modular invariant C2plusmn-algebra Evaluating (34) for f = mcl composingit with ηcl otimes idUi timesU j and taking the trace implies the following identity

Zi j = 1

DimCAclUi timesU j where

(36)

Zi j = dimC HomC2plusmn(UitimesU j Acl)

Decomposing Acl into simple objects this gives

Zi j =sum

klisinISik Zkl Sminus1

l j where Si j = si jradic

DimC (37)

which in CFT terms is of course nothing but the invariance of the torus partition functionunder the modular S-transformation

The following theorem gives a simple criterion for modular invariance

Theorem 34 Let Acl be a haploid commutative symmetric Frobenius algebra in C2plusmn

(i) If Acl is modular invariant then dim Acl = Dim C(ii) If dim Acl = Dim C then Acl is special and modular invariant

Proof Part (i) Since Acl is haploid for i = j = 0 Equation (36) reduces to 1 =dim AclDimCPart (ii) By the same reasoning as in the proof of (iii)rArr(i) in Lemma 211 one showsthat Acl is special Thus mcl cl = ζclidAcl for some ζcl = 0

By [KO Thm 45] the category (C2plusmn)locAcl

of local Acl-modules is again a mod-

ular tensor category and Dim (C2plusmn)locAcl

= (Dim C dim Acl)2 (see [Fr Prop 321 amp

Rem 323] for the same statement in the notation used here) Thus by assumption wehave Dim(C2plusmn)loc

Acl= 1 It then follows from [ENO1 Thm 23] that up to isomorphism

(C2plusmn)locAcl

has a unique simple object (namely the tensor unit) In other words every simplelocal Acl-module is isomorphic to Acl (seen as a left-module over itself)

We have the following isomorphisms between morphism spaces [FS Prop 47 amp 411]

HomAcl(Acl otimes (Ui times U j ) Acl) sim= HomC2plusmn(Ui times U j Acl)

HomAcl(Acl Acl otimes (Ui times U j )) sim= HomC2plusmn(AclUi times U j )

Using these to transport the bases (236) from the right to the left we obtain bases bα(i j)αof HomAcl(Acl otimes (Ui times U j ) Acl) and b(i j)

β β of HomAcl(Acl Acl otimes (Ui times U j )) Thesecan be expressed graphically as

894 L Kong I Runkel

bα(i j) =

Acl

Acl

Acl

Ui times U j

α

b(i j)β = DimC

dim Ui dim U j

1

ζcl

Acl

Ui times U jAcl

Acl

β

(38)

where the nonzero factor in b(i j)β is included for convenience Notice that bα(i j) b(i j)

β isa left Acl-module map Since Acl is simple as a left module over itself we have

bα(i j) b(i j)β = λαβ idAcl (39)

for some λαβ isin C By computing tr(bα(i j) b(i j)β ) it is easy to verify that λαβ = δαβ

We will now prove the following identity

1

ζcl

Acl

Acl

Ui times U j

Ui times U j

Acl

=sumα

DimCdim Ui dim U j

1

ζcl

Acl

Ui times U j

Ui times U j

Acl

Acl

Acl

α

α

(310)

One checks that the left-hand side of this equation is an idempotent which we denote byζminus1

cl PlAcl(Ui timesU j ) cf [Fr Sect 31] By [Fr Prop 41] the image Im(ζminus1

cl PlAcl(Ui timesU j ))

is a local Acl-module and hence isomorphic to AoplusNcl for some N isin Zge0

All left-module morphisms from Acl to Aclotimes(Ui timesU j ) are linear combinations of the

b(i j)β Furthermore one verifies that ζminus1

cl PlAcl(Ui timesU j )b(i j)

β = b(i j)β Therefore the b(i j)

β

describe precisely the image of the idempotent ie ζminus1cl Pl

Acl(Ui timesU j ) = sum

α b(i j)α bα(i j)

which is nothing but (310) Composing (310) with ζcl middot εcl otimes idUi timesU j from the left (iefrom the top) produces (34)

In addition since Acl is commutative symmetric Frobenius it satisfies θAcl = idAcl

[Fr Prop 225] Altogether this shows that Acl is modular invariant Remark 35 As we were writing this paper we heard that the results in Theorem 34were obtained independently by Kitaev and Muumlger [Ki]

Remark 36 Setting i = j = 0 in (36) gives the identity dim Acl = Z00 Dim C [KRProp 23] Combining this with Theorem 34 (ii) one may wonder if a general modularinvariant commutative symmetric Frobenius algebra Acl in C2plusmn is isomorphic to a directsum of simple such algebras However this is not so For example one can take the com-mutative symmetric Frobenius algebra Acl = C[x]x2 in the category of vector spacesequipped with the non-degenerate trace ε(ax + b) = a In this case the modular invari-ance condition holds automatically but Acl is clearly not a direct sum of two algebras

Cardy Algebras and Sewing Constraints I 895

For a general modular tensor category C the algebra C[x]x2 R(1) understood as analgebra in C2plusmn via the braided monoidal isomorphism Vect f (C) C2plusmn rarr C2plusmn providesanother counter-example

32 Two definitions Define a morphism PlA A rarr A for a Frobenius algebra A in C

or C2plusmn as follows [Fr Sect 24]

PlA =

A

A

A

m

(311)

If A is also commutative and obeys m A A = ζA idA we have PlA = ζA idA In partic-

ular this holds if A is commutative and special Using the fact that the Frobenius algebraR(1) is commutative and normalised-special one can check that Pl

R(A) R(A) rarr R(A)takes the following form

PlR(A) =

oplusiisinI

A

A

Uori

Uori

A

m

times

Ui

Ui

(312)

With these ingredients we can now give the first definition of a Cardy C|C2plusmn-algebrawhich was introduced in [K3 Def 514] cf Remark 315 below

Definition 37 (CardyC|C2plusmn-algebra I) A CardyC|C2plusmn-algebra is a triple (Aop|Acl ιcl-op)where (Aclmcl ηclcl εcl) is a modular invariant commutative symmetric Frobe-nius C2plusmn-algebra (Aopmop ηopop εop) is a symmetric Frobenius C-algebra andιcl-op Acl rarr R(Aop) an algebra homomorphism such that the following conditionsare satisfied

(i) Centre condition

Acl R(Aop)

R(Aop)

R(Aop)

ιcl-op

m R(Aop)

=

Acl

ιcl-op

R(Aop)

R(Aop)

R(Aop)

m R(Aop)

(313)

896 L Kong I Runkel

(ii) Cardy condition

ιcl-op ιlowastcl-op =

R(Aop)

R(Aop)

R(Aop)

(314)

Remark 38

(i) The name ldquoCardy C|C2plusmn-algebrardquo in Definition 37 was chosen because many ofthe important ingredients were first studied by Cardy the modular invariance ofthe closed theory [C1] the consistency of the annulus amplitude [C2] and thebulk-boundary OPE [CL] On the other hand the boundary-boundary OPE andthe OPE analogue of the centre condition were first considered in [Lw]

(ii) One can easily see that in the special case that C is the category Vect f (C) of finite-dimensional C-vector spaces a Cardy C|C2plusmn-algebra gives exactly the algebraicformulation of two-dimensional open-closed topological field theory over C (cfRemark 614 in [K3]) see [Lz Sect 48] [Mo Thm 11] [AN Thm 45] [LPCor 43] [MS Sect 22] When passing to a general modular tensor category Cthere are two important differences to the two-dimensional topological field the-ory Firstly the algebras Acl and Aop now live in different categories which inparticular affects the formulation of the centre condition and the Cardy conditionSecondly the modular invariance condition has to be imposed on Acl In the caseC = Vect f (C) modular invariance holds automatically

Definition 39 A homomorphism of Cardy C|C2plusmn-algebras (A(1)op |A(1)cl ι(1)cl-op) rarr (A(2)op

|A(2)cl ι(2)cl-op) is a pair ( fop fcl) of Frobenius algebra homomorphisms fop A(1)op rarr A(2)op

and fcl A(1)cl rarr A(2)cl such that the diagram

A(1)cl

fcl

ι(1)cl-op

A(2)cl

ι(2)cl-op

R(A(1)op )

R( fop) R(A(2)op )

(315)

commutes

Remark 310 Since a homomorphism of Frobenius algebras is invertible (cf Lemma218 (iv)) a homomorphism of Cardy algebras is always an isomorphism

For a homomorphism ( fop fcl) of Cardy C|C2plusmn-algebras using the commutativity of(315) and the fact that fcl and fop are both algebra and coalgebra homomorphisms itis easy to show that (315) commutes iff

R(A(1)op )R( fop)

(ι(1)cl-op)

lowast

R(A(2)op )

(ι(2)cl-op)

lowast

A(1)cl

fcl A(2)cl

(316)

Cardy Algebras and Sewing Constraints I 897

commutesLet (Aop|Acl ιcl-op) be a Cardy C|C2plusmn-algebra Define the morphism

ιcl-op = χminus1(ιcl-op) T (Acl) minusrarr Aop (317)

Decompose Acl as Acl = oplusNn=1Cl

n times Crn such that Cl

1 times Cr1 = ηcl(1 times 1) We use ι(n)cl-op

to denote the restriction of ιcl-op to Cln times Cr

n and ι(n)cl-op to denote the restriction of ιcl-op

to Cln otimes Cr

n We introduce the following graphical notation

ι(n)cl-op =

Cln Cr

n

Aop

(318)

By (243) ιcl-op can be expressed in terms of ιcl-op as follows

ιcl-op =Noplus

n=1

oplusiisinI

sumα

Cln

Crn

Aop

α

Uori

times α

Crn

Ui

(319)

Lemma 311 The centre condition (313) is equivalent to the following condition in C

Cln Cr

n

Aop

Aop

Aop

=

Cln Cr

n

Aop

Aop

Aop

for n = 1 N (320)

Proof First insert (319) and the definition (217) of m R(Aop) into (313) Then apply thecommutativity of R(1) to the left hand side of (313) The equivalence between (313)and (320) follows immediately Remark 312 The centre condition (320) is very natural from the open-closed confor-mal field theory point of view Correlators on the upper half plane are expressed in termsof conformal blocks on the full complex plane The objects Cl

n and Crn are associated to

the field insertion at a point z in the upper half plane and at the complex conjugate pointz in the lower half plane respectively The object Aop corresponds to a field inserted ata point r on the real axis The centre condition (320) simply says that the correlationfunctions in the disjoint domains |z| gt r gt 0 and r gt |z| gt 0 are analytic continuationsof each other see [K2 Prop 118]

898 L Kong I Runkel

Recall that we define ιlowastcl-op Aop rarr T (Acl) as in (230) We introduce the graphicalnotation

ιlowastcl-op =Noplus

n=1

Cln Cr

n

Aop

=Noplus

n=1

Cln Cr

n

Aop

Aop T (minus1Acl) (321)

where the second equality follows from (215) (221) and (259)

Lemma 313 The Cardy condition (314) is equivalent to the following identity in C

Noplusn=1

DimCdim Ui

sumα

Cln

Crn

Crn

α

α

Aop

Aop Uori

Uori

=Aop

Aop

Aop

Uori

Uori

op

mop for all i isin I (322)

Proof By (244) (312) and (319) it is easy to see that (322) is equivalent to thefollowing identity

ιcl-op χ(ιlowastcl-op) = PlR(Aop)

(323)

Therefore it is enough to show that

χ(ιlowastcl-op) = ιlowastcl-op (324)

We have

χminus1(ιlowastcl-op)(1)= T (ιlowastcl-op) δ (2)= T (ιcl-op)

lowast δlowastlowast (3)= (ρ T (ιcl-op)

)lowast (4)= (ιcl-op

)lowast (325)

In step (1) we use the expression (253) for χminus1 step (2) follows from Lemma 218 (v)and Lemma 219 Step (3) is Lemma 218 (i) and Lemma 228 and finally step (4)amounts to substituting (253) and (317) Acting with χ on both sides of the aboveequality produces (324)

Combining Lemmas 311 and 313 and Proposition 216 we obtain the followingequivalent definition of Cardy C|C2plusmn-algebra (recall the graphical notation (318) forιcl-op and (321) for ιlowastcl-op)

Cardy Algebras and Sewing Constraints I 899

Definition 314 (CardyC|C2plusmn-algebra II) A CardyC-algebra is a triple (Aop|Acl ιcl-op)where Acl is a commutative symmetric Frobenius C2plusmn-algebra satisfying property (34)with f = mcl Aop is a symmetric Frobenius C-algebra and ιcl-op T (Acl) rarr Aop isan algebra homomorphism satisfying the conditions (320) and (322)

Remark 315 Up to a choice of normalisation Definition 314 is the same as the orig-inal one in [K3 Def 613] The difference between the two definitions is the factorDim C dim Ui on the left hand side of (322) which in [K3 Def 613] is given byradic

Dim C dim Ui The two definitions are related by rescaling the coproduct cl andcounit εcl of Acl by 1

radicDim C and

radicDim C respectively We chose the convention in

(322) to remove all dimension factors from the expression (314) for the Cardy condition

33 Uniqueness and existence theorems In this subsection we investigate the structureof Cardy algebras We start with the following proposition which when combined withthe results of part II provides an alternative proof of [Fj Prop 422]

Proposition 316 Let (Aop|Acl ιcl-op) be a Cardy C|C2plusmn-algebra If Acl is simple anddim Aop = 0 then Aop is simple and special

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis therefore haploid Restricting the Cardy condition (322) to the case Ui = 1 andcomposing both sides with εop from the left we see that εop kills all terms associated toU j times 1 isin Acl in the sum except for a single 1 times 1 term Thus we obtain the followingidentity

β εop = dAop (mop otimes idAorop) (idAop otimes bAop) (326)

where β isin C Composing with ηop from the right in turn implies that βεop ηop =dim Aop which is nonzero by assumption Thus also β = 0 and εop is a nonzero multipleof the morphism on the right hand side of (326) By [FRS Lem 311] Aop is special

Since Aop is a special Frobenius algebra Aop is semi-simple as an Aop-bimodule(apply [FS Prop 524] to Aop tensored with its opposite algebra) Suppose Aop is not

simple so that we can write Aop = A(1)op oplus A(2)op for nonzero Aop-bimodules A(1)op and A(2)op We denote the canonical embeddings and projections associated to this decompositionas ι12 and π12 We have the identities

mop (ι1 otimes ι2) = 0 εop ηop =2sum

i=1

εop ιi πi ηop (327)

The first identity follows since π1 mop (ι1 otimes ι2) = 0 (as mop gives the left action of

Aop on Aop and hence it preserves A(2)op ) and similarly π2 mop (ι1 otimes ι2) = 0 Thesecond identity is just the completeness of ι12 π12

Since εop ηop = 0 without losing generality we can assume εop ι1 π1 ηop = 0Using that π2 is a bimodule map we compute

π2 [LHS of (322)]Ui =1 ι1 π1 ηop = π2 [RHS of (322)]Ui =1 ι1 π1 ηop

= PlA(2)op

π2 ι1 π1 ηop = 0 (328)

900 L Kong I Runkel

On the other hand using that Acl is haploid that ιcl-op is an algebra map and thatι lowastcl-op is a coalgebra map one can check that the left-hand side of (328) is equal toλ(εop ι1 π1 ηop) π2 ηop for some λ = 0 This implies that π2 ηop = 0 Thusηop = sum

i ιi πi ηop = ι1 π1 ηop Hence we have

0 = π2 ι2 = π2 mop (ηop otimes ι2) = π2 mop ((ι1 π1 ηop)otimes ι2) (329)

However the right-hand side is zero by (327) This is a contradiction and hence Aopmust be simple

To formulate the next theorem we need the notion of the full centre of an algebra [FjDef 49] Recall that an algebra A in a braided tensor category has a left centre and aright centre [VZO] both of which are sub-algebras of A Of these two we will onlyneed the left centre The following definition is [Fr Def 231] which in our setting isequivalent to that of [VZO]

Definition 317 Let A be a symmetric special Frobenius algebra such that m A A =ζA idA

(i) The left centre Cl(A) of A is the image of the idempotent ζminus1A Pl

A(ii) The full centre Z(A) is Cl(R(A))

That ζminus1A Pl

A is an idempotent follows from [FRS Lem 52] when keeping track ofthe factors ζA ([FRS] assumes normalised-special ie ζA = 1) Note that Cl(A) is againan object of C while Z(A) is an object of C2plusmn Let el

A Cl(A) rarr A be the embeddingof Cl(A) into A The left centre is in fact the maximal subobject of A such that

m A cAA (elA otimes idA) = m A (330)

see [Fr Lem 232] This observation explains the name left centre and also makes theconnection to [O Def 15]

The full centre is by definition the image of the idempotent ζminus1A Pl

R(A) R(A) rarrR(A) Since C2plusmn is abelian the idempotent splits and we obtain the embedding andrestriction morphisms

e Z(A) rarr R(A) and r R(A) Z(A) (331)

which obey r e = idZ(A) and e r = ζminus1A Pl

R(A) It follows from Proposition 225 and

[Fr Prop 237] that Z(A) is a commutative symmetric Frobenius algebra in C2plusmn withstructure morphisms5

m Z(A) = r m R(A) (e otimes e) ηZ(A) = r ηR(A)

Z(A) = ζA middot (r otimes r) R(A) e εZ(A) = ζminus1A middot εR(A) e

(332)

Moreover if A is simple then Z(A) is simple and if A is simple and dim A = 0 thenZ(A) is simple and special The normalisation of the counit is such that

εZ(A) ηZ(A) = ζminus2A dim A DimC (333)

5 The normalisation of product and unit is the standard one The factors in the coproduct and counit haveto be included in order for (A|Z(A) e) to be a Cardy algebra see Theorem 318 below The normalisation ofthe counit enters the Cardy condition (314) through the definition of ( middot )lowast

Cardy Algebras and Sewing Constraints I 901

Theorem 318 Let A be a special symmetric Frobenius C-algebra Then (A|Z(A) e)is a Cardy C|C2plusmn-algebra

The proof of this theorem makes use the following two lemmas

Lemma 319 e Z(A) rarr R(A) is an algebra map and elowast = ζA middot r

Proof It follows from [Fr Lem 229] (or by direct calculation using in particularm R(A) R(A) = ζAidR(A)) that

m R(A) (e otimes e) = ζminus1A middot Pl

R(A) m R(A) (e otimes e) (334)

Substituting e r = ζminus1A Pl

R(A) shows that e is compatible with multiplication For theunit one finds

e ηZ(A) = e r ηR(A) = ζminus1A Pl

R(A) ηR(A) = ηR(A) (335)

Thus e is an algebra map For the second statement one computes

elowast (1)= ζA

R(A)

r

R(A)

R(A)

Z(A)

R(A)

Z(A)

r

e

e

(2)=

R(A)

r

R(A)

R(A)

R(A)

Z(A)

(3)=

R(A)

r

R(A)

Z(A)

(4)= ζA middot r (336)

where in (1) the definitions (230) and (332) have been substituted step (2) is e r = ζminus1

A PlR(A) step (3) uses that R(A) is symmetric Frobenius and step (4) is again

e r = ζminus1A Pl

R(A) Lemma 320 Let A be a symmetric Frobenius algebra in C The morphism

PlR(A) m R(A)

(Pl

R(A) otimes PlR(A)

) R(A)otimes R(A) minusrarr R(A) (337)

is S-invariant

The proof of this lemma is a slightly lengthy explicit calculation and has been deferredto Appendix A2

902 L Kong I Runkel

Proof of Theorem 318 That e is an algebra map was proved in Lemma 319 The centrecondition (313) holds by property (330) of the left centre The Cardy condition (314)also is an immediate consequence of Lemma 319

ιcl-op ιlowastcl-op = e (ζA r) = PlR(A) (338)

The full centre Z(A) is a commutative symmetric Frobenius algebra It remains to provemodular invariance That θZ(A) = idZ(A) is implied by commutativity and symmetryof Z(A) [Fr Prop 225] The S-invariance condition (33) follows from Lemma 320In (337) substitute Pl

R(A) = ζA e r and then put the resulting morphism into (33)Compose the resulting equation with e otimes idW from the right (ie from the bottom) andsubstitute the definition (332) of m Z(A) This results in the statement that m Z(A) isS-invariant

The following theorem is analogous to [LR Prop 29] and [Fj Thm 426] whichroughly speaking answer the question under which circumstances the restriction of atwo-dimensional conformal field theory to the boundary already determines the entireconformal field theory The first work is set in Minkowski space and uses operator alge-bras and subfactors while the second work is set in Euclidean space and uses modulartensor categories

Theorem 321 Let (A|Acl ιcl-op) be a Cardy C|C2plusmn-algebra such that dim A = 0 andAcl is simple Then A is special and (A|Acl ιcl-op) sim= (A|Z(A) e) as Cardy algebras

Proof By Proposition 316 A is simple and special Since Acl is simple the algebramap ιcl-op Acl rarr R(A) is either zero or a monomorphism But ιcl-op ηcl = ηR(A)and so ιcl-op = 0 Thus ιcl-op is monic By Lemma 218 (ii) ζminus1

A ιlowastcl-op is epi The Cardycondition (314) implies

ιcl-op ζminus1A ιlowastcl-op = ζminus1

A PlR(A) = e r (339)

Composing this with e r from the left yields e r ιcl-op ζminus1A ιlowastcl-op = e r =

ιcl-op ζminus1A ιlowastcl-op Since ζminus1

A ιlowastcl-op is epi we have

e r ιcl-op = ιcl-op (340)

Actually (340) also follows from (313) and specialness of R(A) We will prove that

( fop fcl) (A|Acl ιcl-op) minusrarr (A|Z(A) e) where fop = idA fcl = r ιcl-op

(341)

is an isomorphism of Cardy algebrasfcl is an algebra map Compatibility with the units follows since ιcl-op is an algebramap

fcl ηcl = r ιcl-op ηcl = r ηR(A) = ηZ(A) (342)

Compatibility with the multiplication also follows since ιcl-op is an algebra map

m Z(A) ( fcl otimes fcl) = r m R(A) (e otimes e) (r otimes r) (ιcl-op otimes ιcl-op)

= r m R(A) (ιcl-op otimes ιcl-op) = r ιcl-op mcl = fcl mcl (343)

where in the second step we used (340)

Cardy Algebras and Sewing Constraints I 903

fcl is an isomorphism As above since fcl is an algebra map and since Acl is simple fcl

has to be monic By Lemma 319 rlowast = ζminus1A e Thus f lowast

cl = ιlowastcl-op rlowast = ζminus1A ιlowastcl-op e and

fcl f lowastcl = r ιcl-op ζminus1

A ιlowastcl-op e = r e r e = idZ(A) (344)

and so fcl is also epi and hence isofcl is a coalgebra map Since fcl is an algebra map so is f minus1

cl By (344) f minus1cl = f lowast

cland by Lemma 218 (iii) this implies that fcl is a also coalgebra mapThe diagram (315) commutes Commutativity of (315) is equivalent to e fcl = ιcl-opwhich holds by (340)

Let A be a special symmetric Frobenius algebra So far we have seen that (A|Z(A) e)is a Cardy algebra and that all Cardy algebras with Aop = A and simple Acl are of thisform It is now natural to ask if every simple Acl does occur as part of a Cardy algebraThe following theorem provides an affirmative answer Recall that for an A-left moduleM the object Mor otimesA M is an algebra (see eg [KR Lem 42])

Theorem 322 If Acl is a simple modular invariant commutative symmetric FrobeniusC2plusmn-algebra then there exist a simple special symmetric Frobenius C-algebra A and amorphism ιcl-op Acl rarr R(A) such that

(i) Acl sim= Z(A) as Frobenius algebras(ii) (A|Acl ιcl-op) is a Cardy C|C2plusmn-algebra

(iii) T (Acl) sim= oplusκisinJ Morκ otimesA Mκ as algebras where MκκisinJ is a set of representa-

tives of the isomorphism classes of simple A-left modules

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis haploid It then follows from Theorem 34 that Acl is special By Proposition 224T (Acl) is a special symmetric Frobenius algebra in C Thus T (Acl) = oplusi Ai where theAi are simple symmetric Frobenius algebras We will show that at least one of the Aiis special Since T (Acl) is special we have mT (Acl) T (Acl) = ζ idT (Acl) for someζ isin C

times Restricting this to the summand Ai shows mi i = ζ idAi FurthermoreεT (Acl) ηT (Acl) = ξ id1 for some ξ isin C

times But εT (Acl) ηT (Acl) = sumi εi ηi and so

at least one of the εi ηi has to be nonzero Therefore at least one of the Ai is speciallet A equiv Ai be this summand We denote the embedding A rarr T (Acl) by e0 and therestriction T (Acl) A by r0 Notice that r0 is an algebra homomorphism Define

ιcl-op = χ (r0) Acl minusrarr R(A) (345)

By Proposition 216 ιcl-op is an algebra homomorphism Next we verify the centre con-dition (313) or rather its equivalent form (320) By substituting the definitions onecan convince oneself that the commutativity mcl cAclAcl = mcl of Acl in C2plusmn implies thecondition mT (Acl) = mT (Acl) in C see [K2 Prop 36] Here T (Acl)otimes T (Acl) rarrT (Acl)otimes T (Acl) is given by

=oplusmn

(idCl

notimes cCl

m Crnotimes idCr

m

)

(cCl

m Clnotimes cminus1

Crn C

rm

)

(idCl

motimes cminus1

Cln C

rm

otimes idCrn

)

(346)

and we decomposed Acl as Acl = oplusnCln times Cr

n As a consequence we obtain the identity

r0 mT (Acl) (idAcl otimes e0) = r0 mT (Acl) (idAcl otimes e0) (347)

904 L Kong I Runkel

Using that r0 is an algebra map and that by definition ιcl-op = r0 we obtain (320)In order to show that (A|Acl ιcl-op) is a Cardy algebra it remains to show that the

Cardy condition (314) is satisfied We will demonstrate this via a detour by first provingthat Acl sim= Z(A) as Frobenius algebras

Recall the notations e and r given in (331) Using the centre condition (320) onecan check that Pl

R(A) ιcl-op = m R(A) R(A) ιcl-op By specialness of A we have

m A A = ζA idA and so together with e r = ζminus1A Pl

R(A) we get

e r ιcl-op = ιcl-op (348)

Next consider the morphism

fcl = r ιcl-op Acl minusrarr Z(A) (349)

By the same derivation as in (342) and (343) one sees that fcl is an algebra map Inparticular fcl ηcl = ηZ(A) = 0 and so fcl = 0 Since Acl is simple fcl has to be amonomorphism

By the same argument as used in the proof of Theorem 34 (ii) up to isomorphismAcl is the unique simple local Acl-(left-)module The algebra monomorphism fcl turnsZ(A) into an Acl-module Since Z(A) is commutative it is local as an Acl-module andso Z(A) sim= AoplusN

cl for some N ge 1 By construction A is a simple special symmetricFrobenius algebra Proposition 225 and Corollary 227 show that R(A) inherits all theseproperties and thus Z(A) is simple (see the comment below Eq (332)) By Theorem318 Z(A) is modular invariant and then by Theorem 34 (i) dim Z(A) = Dim C Thisimplies that N = 1 in Z(A) sim= AoplusN

cl and so fcl is in fact an isomorphismSince Acl and Z(A) are both haploid we have εZ(A) fcl = ξ εcl for some ξ isin C

timesThe counit uniquely determines the Frobenius structure on Acl and Z(A) (see eg [FRSLemma 37]) so that fcl is a coalgebra isomorphism iff ξ = 1 To compute ξ we composethe above identity with ηcl from the right Defining ζcl via εcl ηcl = ζminus1

cl Dim C middot id1 andusing (333) gives ξ = dim A ζclζ

2A By rescaling the comultiplication and the counit

of A and consequently changing ζA we can always achieve ξ = 1 This proves part (i)of the theorem

Equation (348) implies that ιcl-op = e fcl Since fcl is an isomorphism of Frobeniusalgebras by Lemmas 218 and 319 we have

ιcl-op ιlowastcl-op = e fcl f lowastcl elowast = ζA e r = Pl

R(A) (350)

Thus (A|Acl ιcl-op) is a Cardy algebra This proves part (ii) of the theoremPart (iii) can be seen as follows By [KR Prop 43] T Z(A) sim= oplusκisinJ Mor

κ otimesA Mκ asalgebras Together with the observation that T ( fcl) T (Acl) rarr T Z(A) is an isomor-phism of algebras this proves part (iii) Remark 323 Part (i) of Theorem 322 was announced by Muumlger [Muuml2] We provide anindependent proof in the setting of Cardy algebras

The above theorem together with Lemma 211 and Theorem 34 shows that a sim-ple commutative symmetric Frobenius C2plusmn-algebra Acl with dim Acl = Dim C is alwayspart of a Cardy algebra (Aop|Acl ιcl-op) for some simple special symmetric Frobeniusalgebra Aop in C However the above proof also illustrates that Aop is not unique Thisraises the question how two Cardy algebras with a given Acl can differ This question isanswered by [KR Thm 11] which in the present framework can be restated as follows

Cardy Algebras and Sewing Constraints I 905

Theorem 324 If (A(i)op |A(i)cl ι(i)cl-op) i = 1 2 are two Cardy C|C2plusmn-algebras such that

A(i)cl is simple and dim A(i)op = 0 for i = 1 2 then A(1)clsim= A(2)cl as algebras if and only

if A(1)op and A(2)op are Morita equivalent

Proof Theorem 11 in [KR] is stated for A(i)op being non-degenerate algebras and A(i)cl =Z(A(i)op ) for i = 1 2 By Proposition 316 A(i)op are simple and special for i = 1 2

Then by [KR Lem 21] A(i)op are non-degenerate algebras By Theorem 321 we have

A(i)clsim= Z(A(i)op ) as Frobenius algebras Finally by [KR Thm 11] Z(A(1)op ) sim= Z(A(2)op )

as algebras iff A(1)op and A(2)op are Morita equivalent Let Cmax(C2plusmn) be the set of equivalence classes [B] of simple modular invariant

commutative symmetric Frobenius algebras B in C2plusmn Two such algebras B and B prime areequivalent if B and B prime are isomorphic as algebras (but not necessarily as Frobenius alge-bras) Let Msimp(C) be the set of Morita classes of simple special symmetric Frobeniusalgebras in C Define the map z Msimp(C) rarr Cmax(C2plusmn) by z A rarr [Z(A)] whereA denotes the Morita class of A From Theorem 322 (i) and [KR Thm 11] we learn

Corollary 325 The map z Msimp(C) rarr Cmax(C2plusmn) is a bijection

A Appendix

A1 Proof of Lemma 27 We will show that if (F ψ F2 ψ

F0 ) is a colax tensor functor

from C1 to C2 then (G φG2 φ

G0 ) is a lax tensor functor from C2 to C1 Applying this

result to the opposed categories then gives the converse statementWe need to show that φG

0 and φG2 make the diagrams (21) and (22) commute We

first prove the commutativity of (21) Consider the following diagram

G(A)otimes (G(B)otimes G(C))

GψF2 δ

idG(A)otimesφG2 G(A)otimes G(B otimes C)

GψF2 δ

G(FG(A)otimes F(G(B)otimes G(C)))

G(F(idG(A))otimesF(φG2 ))

G(idFG(A)otimesψ F2 )

G(FG(A)otimes FG(B otimes C))

G(ρAotimesρBotimesC )

G(FG(A)otimes (FG(B)otimes FG(C)))

G(ρAotimes(ρBotimesρC )) G(A otimes (B otimes C))

(A1)

The top subdiagram is commutative because of the naturality of Gψ F2 δ The commu-

tativity of the bottom subdiagram follows from the following identities

(ρB otimes ρC ) ψ F2 = (ρB otimes ρC ) ψ F

2 ρF Fδ

= ρBotimesC FG(ρB otimes ρC ) FG(ψ F2 ) Fδ

= ρBotimesC F(φG2 ) (A2)

as a map F(G(B) otimes G(C)) rarr B otimes C The commutativity of (A1) implies that thecomposition of maps in the left column in (21) can be replaced by

G(ρA otimes (ρB otimes ρC )) G((idFG(A) otimes ψ F

2 ) ψ F2

) δ (A3)

906 L Kong I Runkel

Similarly we can show that the composition of maps in the right column in (21) can bereplaced by

G((ρA otimes ρB)otimes ρC ) G((ψ F

2 otimes idFG(C)) ψ F2

) δ (A4)

Using the commutativity of (23) it is easy to see that (21) with the left and right col-umns of (21) replaced by (A3) and (A4) respectively is commutative Hence (21) iscommutative

Now we prove the commutativity of the first diagram in (22)

φG2 (φG

0 otimes idG(A))

(1)= G(ρ12 otimes ρA) Gψ F2 δ

[(Gψ F

0 δ11)otimes idG(A)

]

(2)= G(ρ12 otimes ρA) Gψ F2 G F(Gψ0 otimes idG(A)) G F(δ11 otimes idG(A)) δ

(3)= G(ρ12 otimes ρA) G(FG(ψ F0 )otimes idFG(A)) Gψ F

2 G F(δ11 otimes idG(A)) δ(4)= G(id12 otimes ρA) G

([ψ F

0 ρF(11) (Fδ)11] otimes idFG(A)

) Gψ F

2 δ(5)= G(id12 otimes ρA) G(ψ F

0 otimes idFG(A)) Gψ F2 δ

(6)= G(id12 otimes ρA) G(lminus1FG(A)) G F(lG(A)) δ

(7)= G(lminus1A ) GρA δG lG(A)

(8)= G(lminus1A ) lG(A) (A5)

where in step (1) we substituted the definition of φG0 φ

G2 given in (210) in step (2)

we used the naturality of δ in step (3) we used the naturality of Gψ F2 in step (4) we

switched the position between Gρ12 and G FG(ψ F0 ) and the position between Gψ2 and

G F(δ12 otimes idG(A)) using the naturality of ρ and FψG2 respectively in step (5) we applied

the second identity in (28) in step (6) we used (24) in step (7) we used the naturalityof lminus1 and δ in step (8) we used the first identity in (28)

The proof of the commutativity of the second diagram in (22) is similar Thus wehave shown that G is a lax tensor functor

A2 Proof of Lemma 320 To prepare the proof recall that for a given object B isin Cthe modular group P SL(2Z) acts on the space oplusi HomC(B otimes Ui Ui ) see eg [BKSect 31] and [K3 Eq (455)] We will only need the action of S and Sminus1 Let f isinoplusi HomC(B otimes Ui Ui ) Then

S oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A6)

Cardy Algebras and Sewing Constraints I 907

Sminus1 oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A7)

By Lemma 32 to establish that (337) is S-invariant it is enough to prove the identity(34) when f is given by (337) Using (A6) and (A7) we can see that Eq (34) simplysays that oplusi j [RHS of (34)] is invariant under the action of S times S Consider the elementg of oplus jkisinIHomC2plusmn(R(A)otimes (Uor

j times Uk)Uorj times Uk) given by

g =oplusjkisinI

sumα

R(A)

R(A)

R(A)

R(A)

Uorj times Uk

Uorj times Uk

α

α

(A8)

By the above arguments proving S-invariance of (337) is equivalent to proving invari-ance of g under the action of S times S

For i isin I we denote by gi the component of g in(oplus jisinIHomC(A otimes Uor

i otimes Uorj U

orj )

)otimes (opluskisinIHomC(Ui otimes UkUk))

We view the second Hom-space in the above tensor product as a Hom-space in C+ insteadof Cminus It is enough to show that gi is invariant under the action of S times Sminus1 Note thatthe action of Sminus1 in C+ is equivalent to that of S in Cminus

The morphism gi can be canonically identified with a bilinear pairing

( middot middot )i ⎛⎝oplus

jisinIHomC(Uor

j A otimes Uori otimes Uor

j )

⎞⎠

times(oplus

kisinIHomC(UkUi otimes Uk)

)minusrarr C (A9)

as follows For h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) we set

(h1 h2)i = (dim U j dim Uk)minus1 trUor

j timesUk[gi (h1 times h2)] (A10)

908 L Kong I Runkel

When substituting the explicit form of the product m R(A) of R(A) = (A times 1) otimes R(1)after a short calculation one finds

(h1 h2)i =sumα

1

dim U j

A

A

A

A

A A

A α

α

h1

h2

U jUk

Uorj

Ui

Uk (A11)

Here the top morphism PlR(A) has been simplified with the help of the identity

PR(A) m R(A) (PR(A) otimes PR(A))

=(oplus

iisinI((m A A)otimes idUor

i)times idUi

) m R(A) (PR(A) otimes PR(A)) (A12)

which can be checked by direct calculation along the same lines as in the proof of [FrLem 310]

The action of the modular transformation S on oplusiisinI(B otimes Ui Ui ) for B isin C nat-urally induces an action on oplusiisinI(Ui B otimes Ui ) [K3 Prop 514] which we denote bySlowast In the present case we get an action of Slowast on oplus jisinIHomC(Uor

j A otimes Uori otimes Uor

j ) and

opluskisinIHomC(UkUi otimes Uk) Then to show gi is invariant under the action of S times Sminus1

amounts to showing that

(h1 h2)i = ((Sminus1)lowast h1 Slowast h2)i (A13)

Cardy Algebras and Sewing Constraints I 909

for all h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) We have

((Sminus1)lowast h1 Slowast h2)i =sum

mnα

dim Un

DimC

A

A

A

A

A A

A

Uk

α

α

h1

h2

Ui

U j

UmUn

(A14)

Now drag the upper vertex indexed by α in the above graph along its Uorm -leg until it

meets the lower vertex also indexed by α then sum over α and m This gives

((Sminus1)lowast h1 Slowast h2)i =sum

n

dim Un

DimC

A

A

A

A

A A

A

Uk

Un

h1

h2

Ui

U j

(A15)

910 L Kong I Runkel

If we just look at the neighbourhood of the Un-loop in the above graph we see thefollowing subgraphs

sumn

dim Un

DimC

A Aor

Uorj U j

Uk Uork

Un=

sumα

1

dim U j

A Aor Uk Uork

Uorj U j

αα

(A16)

where we have applied [BK Cor 3111] Substituting this subgraph back to the originalgraph in (A15) we obtain

((Sminus1)lowast h1 Slowast h2)i =sumα

1

dim U j

A

A

A

A

A A

A

A

A

Uorj

α

α

h1

h2m A

Ui

U jUk

Uk

(A17)

The graph in (A17) is equal to that in (A11) In order to see this we first drag theldquobubblerdquo (m A A) along A lines and through the m A vertex (because m A A is abimodule map) until it reaches the lower-left leg of the upper vertex indexed by α Thendrag the m A vertex along the (red) dotted line in the above graph Finally we applythe associativity of A (A12) and [Fr Lem 311] Then we see that the graph in (A17)exactly matches with the one in (A11)

Acknowledgements We would like to thank the organisers of the Oberwolfach Arbeitsgemeinschaft ldquoAlge-braic structures in conformal field theoriesrdquo (April 2007) where this work was started for an inspiring meetingWe would further like to thank the Hausdorff Institute for Mathematics in Bonn and the organisers of the stim-ulating meeting ldquoGeometry and Physicsrdquo (May 2008) We are indebted to Alexei Davydov Jens FjelstadJuumlrgen Fuchs Yi-Zhi Huang Alexei Kitaev Urs Schreiber Christoph Schweigert Stephan Stolz and PeterTeichner for helpful discussions andor comments on a draft of this paper The research of IR was partiallysupported by the EPSRC First Grant EPE0050471 the PPARC rolling grant PPC5071451 and the MarieCurie network lsquoSuperstring Theoryrsquo (MRTN-CT-2004-512194)

Cardy Algebras and Sewing Constraints I 911

References

[AN] Alexeevski A Natanzon SM Noncommutative two-dimensional topological field theories andhurwitz numbers for real algebraic curves Sel Math New Ser 12 307ndash377 (2006)

[Bi] Bichon J Cosovereign hopf algebras J Pure Appl Alg 157 121ndash133 (2001)[BK] Bakalov B Kirillov AA Lectures on Tensor Categories and Modular Functors Providence

RI Amer Math Soc 2001[C1] Cardy JL Operator content of two-dimensional conformal invariant theories Nucl Phys

B 270 186ndash204 (1986)[C2] Cardy JL Boundary conditions fusion rules and the verlinde formula Nucl Phys B 324

581ndash596 (1989)[CL] Cardy JL Lewellen DC Bulk and boundary operators in conformal field theory Phys Lett

B 259 274ndash278 (1991)[DMZ] Dong C-Y Mason G Zhu Y-C Discrete series of the Virasoro algebra and the moonshine

module In Algebraic Groups and Their Generalizations Quantum and infinite-dimensional Meth-ods Proc Symp Pure Math 56 Part 2 Providence RI Amer MathSoc 1994 pp 295ndash316

[DP] Day B Pastro C Note on frobenius monoidal functors New York J Math 14 733ndash742 (2008)[ENO1] Etingof PI Nikshych D Ostrik V On fusion categories Ann Math 162 581ndash642 (2005)[ENO2] Etingof PI Nikshych D Ostrik V An analogue of radfordrsquos s4 formula for finite tensor

categories Int Math Research Notices 54 2915ndash2933 (2004)[Fe] Felder G Froumlhlich J Fuchs J Schweigert C Correlation functions and boundary conditions in

rational conformal field theory and three-dimensional topology Comp Math 131 189ndash237 (2002)[FHL] Frenkel IB Huang Y-Z Lepowsky J On axiomatic approaches to vertex operator algebras

and modules Mem Amer Math Soc 104 (1993)[Fj] Fjelstad J Fuchs J Runkel I Schweigert C Uniqueness of openclosed rational cft with

given algebra of open states Adv Theor Math Phys 12 1283ndash1375 (2008)[FK] Froumlhlich J King C The chern-simons theory and knot polynomials Commun Math

Phys 126 167ndash199 (1989)[Fr] Froumlhlich J Fuchs J Runkel I Schweigert C Correspondences of ribbon categories Adv

Math 199 192ndash329 (2006)[FRS] Fuchs J Runkel I Schweigert C Tft construction of rcft correlators I partition functions Nucl

Phys B 646 353ndash497 (2002)[FS] Fuchs J Schweigert C Category theory for conformal boundary conditions Fields Inst Com-

mun 39 25ndash71 (2003)[H1] Huang Y-Z Two-dimensional conformal geometry and vertex operator algebras Progress in

Mathematics Vol 148 Boston Birkhaumluser 1997[H2] Huang Y-Z Rigidity and modularity of vertex tensor categories Comm Contemp Math 10

871ndash911 (2008)[HK1] Huang Y-Z Kong L Open-string vertex algebra category and operads Commun Math

Phys 250 433ndash471 (2004)[HK2] Huang Y-Z Kong L Full field algebras Commun Math Phys 272 345ndash396 (2007)[HK3] Huang Y-Z Kong L Modular invariance for conformal full field algebras httparxivorgabs

math0609570v2[mathQA] 2006[HL] Huang Y-Z Lepowsky J Tensor products of modules for a vertex operator algebra and vertex

tensor categories In Lie Theory and Geometry in honor of Bertram Kostant ed R BrylinskiJ-L Brylinski V Guillemin V Kac Boston Birkhaumluser 1994 pp 349ndash383

[JS] Joyal A Street R Braided tensor categories Adv Math 102 20ndash78 (1993)[K1] Kong L Full field algebras operads and tensor categories Adv Math 213 271ndash340 (2007)[K2] Kong L Open-closed field algebras Commun Math Phys 280 207ndash261 (2008)[K3] Kong L Cardy condition for open-closed field algebras Commun Math Phys 283 25ndash92 (2008)[Ki] Kitaev A Private communication[KO] Kirillov AA Ostrik V On q-analog of mckay correspondence and ade classification of sl(2)

conformal field theories Adv Math 171 183ndash227 (2002)[KR] Kong L Runkel I Morita classes of algebras in modular tensor categories Adv Math 219

1548ndash1576 (2008)[Ld] Lauda AD Frobenius algebras and ambidextrous adjunctions Theo Appl Cat 16 84ndash

122 (2006)[Li1] Li H-S Regular representations of vertex operator algebras Commun Contemp Math 4

639ndash683 (2002)[Li2] Li H-S Regular representations and huang-lepowsky tensor functors for vertex operator

algebras J Alge 255 423ndash462 (2002)

894 L Kong I Runkel

bα(i j) =

Acl

Acl

Acl

Ui times U j

α

b(i j)β = DimC

dim Ui dim U j

1

ζcl

Acl

Ui times U jAcl

Acl

β

(38)

where the nonzero factor in b(i j)β is included for convenience Notice that bα(i j) b(i j)

β isa left Acl-module map Since Acl is simple as a left module over itself we have

bα(i j) b(i j)β = λαβ idAcl (39)

for some λαβ isin C By computing tr(bα(i j) b(i j)β ) it is easy to verify that λαβ = δαβ

We will now prove the following identity

1

ζcl

Acl

Acl

Ui times U j

Ui times U j

Acl

=sumα

DimCdim Ui dim U j

1

ζcl

Acl

Ui times U j

Ui times U j

Acl

Acl

Acl

α

α

(310)

One checks that the left-hand side of this equation is an idempotent which we denote byζminus1

cl PlAcl(Ui timesU j ) cf [Fr Sect 31] By [Fr Prop 41] the image Im(ζminus1

cl PlAcl(Ui timesU j ))

is a local Acl-module and hence isomorphic to AoplusNcl for some N isin Zge0

All left-module morphisms from Acl to Aclotimes(Ui timesU j ) are linear combinations of the

b(i j)β Furthermore one verifies that ζminus1

cl PlAcl(Ui timesU j )b(i j)

β = b(i j)β Therefore the b(i j)

β

describe precisely the image of the idempotent ie ζminus1cl Pl

Acl(Ui timesU j ) = sum

α b(i j)α bα(i j)

which is nothing but (310) Composing (310) with ζcl middot εcl otimes idUi timesU j from the left (iefrom the top) produces (34)

In addition since Acl is commutative symmetric Frobenius it satisfies θAcl = idAcl

[Fr Prop 225] Altogether this shows that Acl is modular invariant Remark 35 As we were writing this paper we heard that the results in Theorem 34were obtained independently by Kitaev and Muumlger [Ki]

Remark 36 Setting i = j = 0 in (36) gives the identity dim Acl = Z00 Dim C [KRProp 23] Combining this with Theorem 34 (ii) one may wonder if a general modularinvariant commutative symmetric Frobenius algebra Acl in C2plusmn is isomorphic to a directsum of simple such algebras However this is not so For example one can take the com-mutative symmetric Frobenius algebra Acl = C[x]x2 in the category of vector spacesequipped with the non-degenerate trace ε(ax + b) = a In this case the modular invari-ance condition holds automatically but Acl is clearly not a direct sum of two algebras

Cardy Algebras and Sewing Constraints I 895

For a general modular tensor category C the algebra C[x]x2 R(1) understood as analgebra in C2plusmn via the braided monoidal isomorphism Vect f (C) C2plusmn rarr C2plusmn providesanother counter-example

32 Two definitions Define a morphism PlA A rarr A for a Frobenius algebra A in C

or C2plusmn as follows [Fr Sect 24]

PlA =

A

A

A

m

(311)

If A is also commutative and obeys m A A = ζA idA we have PlA = ζA idA In partic-

ular this holds if A is commutative and special Using the fact that the Frobenius algebraR(1) is commutative and normalised-special one can check that Pl

R(A) R(A) rarr R(A)takes the following form

PlR(A) =

oplusiisinI

A

A

Uori

Uori

A

m

times

Ui

Ui

(312)

With these ingredients we can now give the first definition of a Cardy C|C2plusmn-algebrawhich was introduced in [K3 Def 514] cf Remark 315 below

Definition 37 (CardyC|C2plusmn-algebra I) A CardyC|C2plusmn-algebra is a triple (Aop|Acl ιcl-op)where (Aclmcl ηclcl εcl) is a modular invariant commutative symmetric Frobe-nius C2plusmn-algebra (Aopmop ηopop εop) is a symmetric Frobenius C-algebra andιcl-op Acl rarr R(Aop) an algebra homomorphism such that the following conditionsare satisfied

(i) Centre condition

Acl R(Aop)

R(Aop)

R(Aop)

ιcl-op

m R(Aop)

=

Acl

ιcl-op

R(Aop)

R(Aop)

R(Aop)

m R(Aop)

(313)

896 L Kong I Runkel

(ii) Cardy condition

ιcl-op ιlowastcl-op =

R(Aop)

R(Aop)

R(Aop)

(314)

Remark 38

(i) The name ldquoCardy C|C2plusmn-algebrardquo in Definition 37 was chosen because many ofthe important ingredients were first studied by Cardy the modular invariance ofthe closed theory [C1] the consistency of the annulus amplitude [C2] and thebulk-boundary OPE [CL] On the other hand the boundary-boundary OPE andthe OPE analogue of the centre condition were first considered in [Lw]

(ii) One can easily see that in the special case that C is the category Vect f (C) of finite-dimensional C-vector spaces a Cardy C|C2plusmn-algebra gives exactly the algebraicformulation of two-dimensional open-closed topological field theory over C (cfRemark 614 in [K3]) see [Lz Sect 48] [Mo Thm 11] [AN Thm 45] [LPCor 43] [MS Sect 22] When passing to a general modular tensor category Cthere are two important differences to the two-dimensional topological field the-ory Firstly the algebras Acl and Aop now live in different categories which inparticular affects the formulation of the centre condition and the Cardy conditionSecondly the modular invariance condition has to be imposed on Acl In the caseC = Vect f (C) modular invariance holds automatically

Definition 39 A homomorphism of Cardy C|C2plusmn-algebras (A(1)op |A(1)cl ι(1)cl-op) rarr (A(2)op

|A(2)cl ι(2)cl-op) is a pair ( fop fcl) of Frobenius algebra homomorphisms fop A(1)op rarr A(2)op

and fcl A(1)cl rarr A(2)cl such that the diagram

A(1)cl

fcl

ι(1)cl-op

A(2)cl

ι(2)cl-op

R(A(1)op )

R( fop) R(A(2)op )

(315)

commutes

Remark 310 Since a homomorphism of Frobenius algebras is invertible (cf Lemma218 (iv)) a homomorphism of Cardy algebras is always an isomorphism

For a homomorphism ( fop fcl) of Cardy C|C2plusmn-algebras using the commutativity of(315) and the fact that fcl and fop are both algebra and coalgebra homomorphisms itis easy to show that (315) commutes iff

R(A(1)op )R( fop)

(ι(1)cl-op)

lowast

R(A(2)op )

(ι(2)cl-op)

lowast

A(1)cl

fcl A(2)cl

(316)

Cardy Algebras and Sewing Constraints I 897

commutesLet (Aop|Acl ιcl-op) be a Cardy C|C2plusmn-algebra Define the morphism

ιcl-op = χminus1(ιcl-op) T (Acl) minusrarr Aop (317)

Decompose Acl as Acl = oplusNn=1Cl

n times Crn such that Cl

1 times Cr1 = ηcl(1 times 1) We use ι(n)cl-op

to denote the restriction of ιcl-op to Cln times Cr

n and ι(n)cl-op to denote the restriction of ιcl-op

to Cln otimes Cr

n We introduce the following graphical notation

ι(n)cl-op =

Cln Cr

n

Aop

(318)

By (243) ιcl-op can be expressed in terms of ιcl-op as follows

ιcl-op =Noplus

n=1

oplusiisinI

sumα

Cln

Crn

Aop

α

Uori

times α

Crn

Ui

(319)

Lemma 311 The centre condition (313) is equivalent to the following condition in C

Cln Cr

n

Aop

Aop

Aop

=

Cln Cr

n

Aop

Aop

Aop

for n = 1 N (320)

Proof First insert (319) and the definition (217) of m R(Aop) into (313) Then apply thecommutativity of R(1) to the left hand side of (313) The equivalence between (313)and (320) follows immediately Remark 312 The centre condition (320) is very natural from the open-closed confor-mal field theory point of view Correlators on the upper half plane are expressed in termsof conformal blocks on the full complex plane The objects Cl

n and Crn are associated to

the field insertion at a point z in the upper half plane and at the complex conjugate pointz in the lower half plane respectively The object Aop corresponds to a field inserted ata point r on the real axis The centre condition (320) simply says that the correlationfunctions in the disjoint domains |z| gt r gt 0 and r gt |z| gt 0 are analytic continuationsof each other see [K2 Prop 118]

898 L Kong I Runkel

Recall that we define ιlowastcl-op Aop rarr T (Acl) as in (230) We introduce the graphicalnotation

ιlowastcl-op =Noplus

n=1

Cln Cr

n

Aop

=Noplus

n=1

Cln Cr

n

Aop

Aop T (minus1Acl) (321)

where the second equality follows from (215) (221) and (259)

Lemma 313 The Cardy condition (314) is equivalent to the following identity in C

Noplusn=1

DimCdim Ui

sumα

Cln

Crn

Crn

α

α

Aop

Aop Uori

Uori

=Aop

Aop

Aop

Uori

Uori

op

mop for all i isin I (322)

Proof By (244) (312) and (319) it is easy to see that (322) is equivalent to thefollowing identity

ιcl-op χ(ιlowastcl-op) = PlR(Aop)

(323)

Therefore it is enough to show that

χ(ιlowastcl-op) = ιlowastcl-op (324)

We have

χminus1(ιlowastcl-op)(1)= T (ιlowastcl-op) δ (2)= T (ιcl-op)

lowast δlowastlowast (3)= (ρ T (ιcl-op)

)lowast (4)= (ιcl-op

)lowast (325)

In step (1) we use the expression (253) for χminus1 step (2) follows from Lemma 218 (v)and Lemma 219 Step (3) is Lemma 218 (i) and Lemma 228 and finally step (4)amounts to substituting (253) and (317) Acting with χ on both sides of the aboveequality produces (324)

Combining Lemmas 311 and 313 and Proposition 216 we obtain the followingequivalent definition of Cardy C|C2plusmn-algebra (recall the graphical notation (318) forιcl-op and (321) for ιlowastcl-op)

Cardy Algebras and Sewing Constraints I 899

Definition 314 (CardyC|C2plusmn-algebra II) A CardyC-algebra is a triple (Aop|Acl ιcl-op)where Acl is a commutative symmetric Frobenius C2plusmn-algebra satisfying property (34)with f = mcl Aop is a symmetric Frobenius C-algebra and ιcl-op T (Acl) rarr Aop isan algebra homomorphism satisfying the conditions (320) and (322)

Remark 315 Up to a choice of normalisation Definition 314 is the same as the orig-inal one in [K3 Def 613] The difference between the two definitions is the factorDim C dim Ui on the left hand side of (322) which in [K3 Def 613] is given byradic

Dim C dim Ui The two definitions are related by rescaling the coproduct cl andcounit εcl of Acl by 1

radicDim C and

radicDim C respectively We chose the convention in

(322) to remove all dimension factors from the expression (314) for the Cardy condition

33 Uniqueness and existence theorems In this subsection we investigate the structureof Cardy algebras We start with the following proposition which when combined withthe results of part II provides an alternative proof of [Fj Prop 422]

Proposition 316 Let (Aop|Acl ιcl-op) be a Cardy C|C2plusmn-algebra If Acl is simple anddim Aop = 0 then Aop is simple and special

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis therefore haploid Restricting the Cardy condition (322) to the case Ui = 1 andcomposing both sides with εop from the left we see that εop kills all terms associated toU j times 1 isin Acl in the sum except for a single 1 times 1 term Thus we obtain the followingidentity

β εop = dAop (mop otimes idAorop) (idAop otimes bAop) (326)

where β isin C Composing with ηop from the right in turn implies that βεop ηop =dim Aop which is nonzero by assumption Thus also β = 0 and εop is a nonzero multipleof the morphism on the right hand side of (326) By [FRS Lem 311] Aop is special

Since Aop is a special Frobenius algebra Aop is semi-simple as an Aop-bimodule(apply [FS Prop 524] to Aop tensored with its opposite algebra) Suppose Aop is not

simple so that we can write Aop = A(1)op oplus A(2)op for nonzero Aop-bimodules A(1)op and A(2)op We denote the canonical embeddings and projections associated to this decompositionas ι12 and π12 We have the identities

mop (ι1 otimes ι2) = 0 εop ηop =2sum

i=1

εop ιi πi ηop (327)

The first identity follows since π1 mop (ι1 otimes ι2) = 0 (as mop gives the left action of

Aop on Aop and hence it preserves A(2)op ) and similarly π2 mop (ι1 otimes ι2) = 0 Thesecond identity is just the completeness of ι12 π12

Since εop ηop = 0 without losing generality we can assume εop ι1 π1 ηop = 0Using that π2 is a bimodule map we compute

π2 [LHS of (322)]Ui =1 ι1 π1 ηop = π2 [RHS of (322)]Ui =1 ι1 π1 ηop

= PlA(2)op

π2 ι1 π1 ηop = 0 (328)

900 L Kong I Runkel

On the other hand using that Acl is haploid that ιcl-op is an algebra map and thatι lowastcl-op is a coalgebra map one can check that the left-hand side of (328) is equal toλ(εop ι1 π1 ηop) π2 ηop for some λ = 0 This implies that π2 ηop = 0 Thusηop = sum

i ιi πi ηop = ι1 π1 ηop Hence we have

0 = π2 ι2 = π2 mop (ηop otimes ι2) = π2 mop ((ι1 π1 ηop)otimes ι2) (329)

However the right-hand side is zero by (327) This is a contradiction and hence Aopmust be simple

To formulate the next theorem we need the notion of the full centre of an algebra [FjDef 49] Recall that an algebra A in a braided tensor category has a left centre and aright centre [VZO] both of which are sub-algebras of A Of these two we will onlyneed the left centre The following definition is [Fr Def 231] which in our setting isequivalent to that of [VZO]

Definition 317 Let A be a symmetric special Frobenius algebra such that m A A =ζA idA

(i) The left centre Cl(A) of A is the image of the idempotent ζminus1A Pl

A(ii) The full centre Z(A) is Cl(R(A))

That ζminus1A Pl

A is an idempotent follows from [FRS Lem 52] when keeping track ofthe factors ζA ([FRS] assumes normalised-special ie ζA = 1) Note that Cl(A) is againan object of C while Z(A) is an object of C2plusmn Let el

A Cl(A) rarr A be the embeddingof Cl(A) into A The left centre is in fact the maximal subobject of A such that

m A cAA (elA otimes idA) = m A (330)

see [Fr Lem 232] This observation explains the name left centre and also makes theconnection to [O Def 15]

The full centre is by definition the image of the idempotent ζminus1A Pl

R(A) R(A) rarrR(A) Since C2plusmn is abelian the idempotent splits and we obtain the embedding andrestriction morphisms

e Z(A) rarr R(A) and r R(A) Z(A) (331)

which obey r e = idZ(A) and e r = ζminus1A Pl

R(A) It follows from Proposition 225 and

[Fr Prop 237] that Z(A) is a commutative symmetric Frobenius algebra in C2plusmn withstructure morphisms5

m Z(A) = r m R(A) (e otimes e) ηZ(A) = r ηR(A)

Z(A) = ζA middot (r otimes r) R(A) e εZ(A) = ζminus1A middot εR(A) e

(332)

Moreover if A is simple then Z(A) is simple and if A is simple and dim A = 0 thenZ(A) is simple and special The normalisation of the counit is such that

εZ(A) ηZ(A) = ζminus2A dim A DimC (333)

5 The normalisation of product and unit is the standard one The factors in the coproduct and counit haveto be included in order for (A|Z(A) e) to be a Cardy algebra see Theorem 318 below The normalisation ofthe counit enters the Cardy condition (314) through the definition of ( middot )lowast

Cardy Algebras and Sewing Constraints I 901

Theorem 318 Let A be a special symmetric Frobenius C-algebra Then (A|Z(A) e)is a Cardy C|C2plusmn-algebra

The proof of this theorem makes use the following two lemmas

Lemma 319 e Z(A) rarr R(A) is an algebra map and elowast = ζA middot r

Proof It follows from [Fr Lem 229] (or by direct calculation using in particularm R(A) R(A) = ζAidR(A)) that

m R(A) (e otimes e) = ζminus1A middot Pl

R(A) m R(A) (e otimes e) (334)

Substituting e r = ζminus1A Pl

R(A) shows that e is compatible with multiplication For theunit one finds

e ηZ(A) = e r ηR(A) = ζminus1A Pl

R(A) ηR(A) = ηR(A) (335)

Thus e is an algebra map For the second statement one computes

elowast (1)= ζA

R(A)

r

R(A)

R(A)

Z(A)

R(A)

Z(A)

r

e

e

(2)=

R(A)

r

R(A)

R(A)

R(A)

Z(A)

(3)=

R(A)

r

R(A)

Z(A)

(4)= ζA middot r (336)

where in (1) the definitions (230) and (332) have been substituted step (2) is e r = ζminus1

A PlR(A) step (3) uses that R(A) is symmetric Frobenius and step (4) is again

e r = ζminus1A Pl

R(A) Lemma 320 Let A be a symmetric Frobenius algebra in C The morphism

PlR(A) m R(A)

(Pl

R(A) otimes PlR(A)

) R(A)otimes R(A) minusrarr R(A) (337)

is S-invariant

The proof of this lemma is a slightly lengthy explicit calculation and has been deferredto Appendix A2

902 L Kong I Runkel

Proof of Theorem 318 That e is an algebra map was proved in Lemma 319 The centrecondition (313) holds by property (330) of the left centre The Cardy condition (314)also is an immediate consequence of Lemma 319

ιcl-op ιlowastcl-op = e (ζA r) = PlR(A) (338)

The full centre Z(A) is a commutative symmetric Frobenius algebra It remains to provemodular invariance That θZ(A) = idZ(A) is implied by commutativity and symmetryof Z(A) [Fr Prop 225] The S-invariance condition (33) follows from Lemma 320In (337) substitute Pl

R(A) = ζA e r and then put the resulting morphism into (33)Compose the resulting equation with e otimes idW from the right (ie from the bottom) andsubstitute the definition (332) of m Z(A) This results in the statement that m Z(A) isS-invariant

The following theorem is analogous to [LR Prop 29] and [Fj Thm 426] whichroughly speaking answer the question under which circumstances the restriction of atwo-dimensional conformal field theory to the boundary already determines the entireconformal field theory The first work is set in Minkowski space and uses operator alge-bras and subfactors while the second work is set in Euclidean space and uses modulartensor categories

Theorem 321 Let (A|Acl ιcl-op) be a Cardy C|C2plusmn-algebra such that dim A = 0 andAcl is simple Then A is special and (A|Acl ιcl-op) sim= (A|Z(A) e) as Cardy algebras

Proof By Proposition 316 A is simple and special Since Acl is simple the algebramap ιcl-op Acl rarr R(A) is either zero or a monomorphism But ιcl-op ηcl = ηR(A)and so ιcl-op = 0 Thus ιcl-op is monic By Lemma 218 (ii) ζminus1

A ιlowastcl-op is epi The Cardycondition (314) implies

ιcl-op ζminus1A ιlowastcl-op = ζminus1

A PlR(A) = e r (339)

Composing this with e r from the left yields e r ιcl-op ζminus1A ιlowastcl-op = e r =

ιcl-op ζminus1A ιlowastcl-op Since ζminus1

A ιlowastcl-op is epi we have

e r ιcl-op = ιcl-op (340)

Actually (340) also follows from (313) and specialness of R(A) We will prove that

( fop fcl) (A|Acl ιcl-op) minusrarr (A|Z(A) e) where fop = idA fcl = r ιcl-op

(341)

is an isomorphism of Cardy algebrasfcl is an algebra map Compatibility with the units follows since ιcl-op is an algebramap

fcl ηcl = r ιcl-op ηcl = r ηR(A) = ηZ(A) (342)

Compatibility with the multiplication also follows since ιcl-op is an algebra map

m Z(A) ( fcl otimes fcl) = r m R(A) (e otimes e) (r otimes r) (ιcl-op otimes ιcl-op)

= r m R(A) (ιcl-op otimes ιcl-op) = r ιcl-op mcl = fcl mcl (343)

where in the second step we used (340)

Cardy Algebras and Sewing Constraints I 903

fcl is an isomorphism As above since fcl is an algebra map and since Acl is simple fcl

has to be monic By Lemma 319 rlowast = ζminus1A e Thus f lowast

cl = ιlowastcl-op rlowast = ζminus1A ιlowastcl-op e and

fcl f lowastcl = r ιcl-op ζminus1

A ιlowastcl-op e = r e r e = idZ(A) (344)

and so fcl is also epi and hence isofcl is a coalgebra map Since fcl is an algebra map so is f minus1

cl By (344) f minus1cl = f lowast

cland by Lemma 218 (iii) this implies that fcl is a also coalgebra mapThe diagram (315) commutes Commutativity of (315) is equivalent to e fcl = ιcl-opwhich holds by (340)

Let A be a special symmetric Frobenius algebra So far we have seen that (A|Z(A) e)is a Cardy algebra and that all Cardy algebras with Aop = A and simple Acl are of thisform It is now natural to ask if every simple Acl does occur as part of a Cardy algebraThe following theorem provides an affirmative answer Recall that for an A-left moduleM the object Mor otimesA M is an algebra (see eg [KR Lem 42])

Theorem 322 If Acl is a simple modular invariant commutative symmetric FrobeniusC2plusmn-algebra then there exist a simple special symmetric Frobenius C-algebra A and amorphism ιcl-op Acl rarr R(A) such that

(i) Acl sim= Z(A) as Frobenius algebras(ii) (A|Acl ιcl-op) is a Cardy C|C2plusmn-algebra

(iii) T (Acl) sim= oplusκisinJ Morκ otimesA Mκ as algebras where MκκisinJ is a set of representa-

tives of the isomorphism classes of simple A-left modules

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis haploid It then follows from Theorem 34 that Acl is special By Proposition 224T (Acl) is a special symmetric Frobenius algebra in C Thus T (Acl) = oplusi Ai where theAi are simple symmetric Frobenius algebras We will show that at least one of the Aiis special Since T (Acl) is special we have mT (Acl) T (Acl) = ζ idT (Acl) for someζ isin C

times Restricting this to the summand Ai shows mi i = ζ idAi FurthermoreεT (Acl) ηT (Acl) = ξ id1 for some ξ isin C

times But εT (Acl) ηT (Acl) = sumi εi ηi and so

at least one of the εi ηi has to be nonzero Therefore at least one of the Ai is speciallet A equiv Ai be this summand We denote the embedding A rarr T (Acl) by e0 and therestriction T (Acl) A by r0 Notice that r0 is an algebra homomorphism Define

ιcl-op = χ (r0) Acl minusrarr R(A) (345)

By Proposition 216 ιcl-op is an algebra homomorphism Next we verify the centre con-dition (313) or rather its equivalent form (320) By substituting the definitions onecan convince oneself that the commutativity mcl cAclAcl = mcl of Acl in C2plusmn implies thecondition mT (Acl) = mT (Acl) in C see [K2 Prop 36] Here T (Acl)otimes T (Acl) rarrT (Acl)otimes T (Acl) is given by

=oplusmn

(idCl

notimes cCl

m Crnotimes idCr

m

)

(cCl

m Clnotimes cminus1

Crn C

rm

)

(idCl

motimes cminus1

Cln C

rm

otimes idCrn

)

(346)

and we decomposed Acl as Acl = oplusnCln times Cr

n As a consequence we obtain the identity

r0 mT (Acl) (idAcl otimes e0) = r0 mT (Acl) (idAcl otimes e0) (347)

904 L Kong I Runkel

Using that r0 is an algebra map and that by definition ιcl-op = r0 we obtain (320)In order to show that (A|Acl ιcl-op) is a Cardy algebra it remains to show that the

Cardy condition (314) is satisfied We will demonstrate this via a detour by first provingthat Acl sim= Z(A) as Frobenius algebras

Recall the notations e and r given in (331) Using the centre condition (320) onecan check that Pl

R(A) ιcl-op = m R(A) R(A) ιcl-op By specialness of A we have

m A A = ζA idA and so together with e r = ζminus1A Pl

R(A) we get

e r ιcl-op = ιcl-op (348)

Next consider the morphism

fcl = r ιcl-op Acl minusrarr Z(A) (349)

By the same derivation as in (342) and (343) one sees that fcl is an algebra map Inparticular fcl ηcl = ηZ(A) = 0 and so fcl = 0 Since Acl is simple fcl has to be amonomorphism

By the same argument as used in the proof of Theorem 34 (ii) up to isomorphismAcl is the unique simple local Acl-(left-)module The algebra monomorphism fcl turnsZ(A) into an Acl-module Since Z(A) is commutative it is local as an Acl-module andso Z(A) sim= AoplusN

cl for some N ge 1 By construction A is a simple special symmetricFrobenius algebra Proposition 225 and Corollary 227 show that R(A) inherits all theseproperties and thus Z(A) is simple (see the comment below Eq (332)) By Theorem318 Z(A) is modular invariant and then by Theorem 34 (i) dim Z(A) = Dim C Thisimplies that N = 1 in Z(A) sim= AoplusN

cl and so fcl is in fact an isomorphismSince Acl and Z(A) are both haploid we have εZ(A) fcl = ξ εcl for some ξ isin C

timesThe counit uniquely determines the Frobenius structure on Acl and Z(A) (see eg [FRSLemma 37]) so that fcl is a coalgebra isomorphism iff ξ = 1 To compute ξ we composethe above identity with ηcl from the right Defining ζcl via εcl ηcl = ζminus1

cl Dim C middot id1 andusing (333) gives ξ = dim A ζclζ

2A By rescaling the comultiplication and the counit

of A and consequently changing ζA we can always achieve ξ = 1 This proves part (i)of the theorem

Equation (348) implies that ιcl-op = e fcl Since fcl is an isomorphism of Frobeniusalgebras by Lemmas 218 and 319 we have

ιcl-op ιlowastcl-op = e fcl f lowastcl elowast = ζA e r = Pl

R(A) (350)

Thus (A|Acl ιcl-op) is a Cardy algebra This proves part (ii) of the theoremPart (iii) can be seen as follows By [KR Prop 43] T Z(A) sim= oplusκisinJ Mor

κ otimesA Mκ asalgebras Together with the observation that T ( fcl) T (Acl) rarr T Z(A) is an isomor-phism of algebras this proves part (iii) Remark 323 Part (i) of Theorem 322 was announced by Muumlger [Muuml2] We provide anindependent proof in the setting of Cardy algebras

The above theorem together with Lemma 211 and Theorem 34 shows that a sim-ple commutative symmetric Frobenius C2plusmn-algebra Acl with dim Acl = Dim C is alwayspart of a Cardy algebra (Aop|Acl ιcl-op) for some simple special symmetric Frobeniusalgebra Aop in C However the above proof also illustrates that Aop is not unique Thisraises the question how two Cardy algebras with a given Acl can differ This question isanswered by [KR Thm 11] which in the present framework can be restated as follows

Cardy Algebras and Sewing Constraints I 905

Theorem 324 If (A(i)op |A(i)cl ι(i)cl-op) i = 1 2 are two Cardy C|C2plusmn-algebras such that

A(i)cl is simple and dim A(i)op = 0 for i = 1 2 then A(1)clsim= A(2)cl as algebras if and only

if A(1)op and A(2)op are Morita equivalent

Proof Theorem 11 in [KR] is stated for A(i)op being non-degenerate algebras and A(i)cl =Z(A(i)op ) for i = 1 2 By Proposition 316 A(i)op are simple and special for i = 1 2

Then by [KR Lem 21] A(i)op are non-degenerate algebras By Theorem 321 we have

A(i)clsim= Z(A(i)op ) as Frobenius algebras Finally by [KR Thm 11] Z(A(1)op ) sim= Z(A(2)op )

as algebras iff A(1)op and A(2)op are Morita equivalent Let Cmax(C2plusmn) be the set of equivalence classes [B] of simple modular invariant

commutative symmetric Frobenius algebras B in C2plusmn Two such algebras B and B prime areequivalent if B and B prime are isomorphic as algebras (but not necessarily as Frobenius alge-bras) Let Msimp(C) be the set of Morita classes of simple special symmetric Frobeniusalgebras in C Define the map z Msimp(C) rarr Cmax(C2plusmn) by z A rarr [Z(A)] whereA denotes the Morita class of A From Theorem 322 (i) and [KR Thm 11] we learn

Corollary 325 The map z Msimp(C) rarr Cmax(C2plusmn) is a bijection

A Appendix

A1 Proof of Lemma 27 We will show that if (F ψ F2 ψ

F0 ) is a colax tensor functor

from C1 to C2 then (G φG2 φ

G0 ) is a lax tensor functor from C2 to C1 Applying this

result to the opposed categories then gives the converse statementWe need to show that φG

0 and φG2 make the diagrams (21) and (22) commute We

first prove the commutativity of (21) Consider the following diagram

G(A)otimes (G(B)otimes G(C))

GψF2 δ

idG(A)otimesφG2 G(A)otimes G(B otimes C)

GψF2 δ

G(FG(A)otimes F(G(B)otimes G(C)))

G(F(idG(A))otimesF(φG2 ))

G(idFG(A)otimesψ F2 )

G(FG(A)otimes FG(B otimes C))

G(ρAotimesρBotimesC )

G(FG(A)otimes (FG(B)otimes FG(C)))

G(ρAotimes(ρBotimesρC )) G(A otimes (B otimes C))

(A1)

The top subdiagram is commutative because of the naturality of Gψ F2 δ The commu-

tativity of the bottom subdiagram follows from the following identities

(ρB otimes ρC ) ψ F2 = (ρB otimes ρC ) ψ F

2 ρF Fδ

= ρBotimesC FG(ρB otimes ρC ) FG(ψ F2 ) Fδ

= ρBotimesC F(φG2 ) (A2)

as a map F(G(B) otimes G(C)) rarr B otimes C The commutativity of (A1) implies that thecomposition of maps in the left column in (21) can be replaced by

G(ρA otimes (ρB otimes ρC )) G((idFG(A) otimes ψ F

2 ) ψ F2

) δ (A3)

906 L Kong I Runkel

Similarly we can show that the composition of maps in the right column in (21) can bereplaced by

G((ρA otimes ρB)otimes ρC ) G((ψ F

2 otimes idFG(C)) ψ F2

) δ (A4)

Using the commutativity of (23) it is easy to see that (21) with the left and right col-umns of (21) replaced by (A3) and (A4) respectively is commutative Hence (21) iscommutative

Now we prove the commutativity of the first diagram in (22)

φG2 (φG

0 otimes idG(A))

(1)= G(ρ12 otimes ρA) Gψ F2 δ

[(Gψ F

0 δ11)otimes idG(A)

]

(2)= G(ρ12 otimes ρA) Gψ F2 G F(Gψ0 otimes idG(A)) G F(δ11 otimes idG(A)) δ

(3)= G(ρ12 otimes ρA) G(FG(ψ F0 )otimes idFG(A)) Gψ F

2 G F(δ11 otimes idG(A)) δ(4)= G(id12 otimes ρA) G

([ψ F

0 ρF(11) (Fδ)11] otimes idFG(A)

) Gψ F

2 δ(5)= G(id12 otimes ρA) G(ψ F

0 otimes idFG(A)) Gψ F2 δ

(6)= G(id12 otimes ρA) G(lminus1FG(A)) G F(lG(A)) δ

(7)= G(lminus1A ) GρA δG lG(A)

(8)= G(lminus1A ) lG(A) (A5)

where in step (1) we substituted the definition of φG0 φ

G2 given in (210) in step (2)

we used the naturality of δ in step (3) we used the naturality of Gψ F2 in step (4) we

switched the position between Gρ12 and G FG(ψ F0 ) and the position between Gψ2 and

G F(δ12 otimes idG(A)) using the naturality of ρ and FψG2 respectively in step (5) we applied

the second identity in (28) in step (6) we used (24) in step (7) we used the naturalityof lminus1 and δ in step (8) we used the first identity in (28)

The proof of the commutativity of the second diagram in (22) is similar Thus wehave shown that G is a lax tensor functor

A2 Proof of Lemma 320 To prepare the proof recall that for a given object B isin Cthe modular group P SL(2Z) acts on the space oplusi HomC(B otimes Ui Ui ) see eg [BKSect 31] and [K3 Eq (455)] We will only need the action of S and Sminus1 Let f isinoplusi HomC(B otimes Ui Ui ) Then

S oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A6)

Cardy Algebras and Sewing Constraints I 907

Sminus1 oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A7)

By Lemma 32 to establish that (337) is S-invariant it is enough to prove the identity(34) when f is given by (337) Using (A6) and (A7) we can see that Eq (34) simplysays that oplusi j [RHS of (34)] is invariant under the action of S times S Consider the elementg of oplus jkisinIHomC2plusmn(R(A)otimes (Uor

j times Uk)Uorj times Uk) given by

g =oplusjkisinI

sumα

R(A)

R(A)

R(A)

R(A)

Uorj times Uk

Uorj times Uk

α

α

(A8)

By the above arguments proving S-invariance of (337) is equivalent to proving invari-ance of g under the action of S times S

For i isin I we denote by gi the component of g in(oplus jisinIHomC(A otimes Uor

i otimes Uorj U

orj )

)otimes (opluskisinIHomC(Ui otimes UkUk))

We view the second Hom-space in the above tensor product as a Hom-space in C+ insteadof Cminus It is enough to show that gi is invariant under the action of S times Sminus1 Note thatthe action of Sminus1 in C+ is equivalent to that of S in Cminus

The morphism gi can be canonically identified with a bilinear pairing

( middot middot )i ⎛⎝oplus

jisinIHomC(Uor

j A otimes Uori otimes Uor

j )

⎞⎠

times(oplus

kisinIHomC(UkUi otimes Uk)

)minusrarr C (A9)

as follows For h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) we set

(h1 h2)i = (dim U j dim Uk)minus1 trUor

j timesUk[gi (h1 times h2)] (A10)

908 L Kong I Runkel

When substituting the explicit form of the product m R(A) of R(A) = (A times 1) otimes R(1)after a short calculation one finds

(h1 h2)i =sumα

1

dim U j

A

A

A

A

A A

A α

α

h1

h2

U jUk

Uorj

Ui

Uk (A11)

Here the top morphism PlR(A) has been simplified with the help of the identity

PR(A) m R(A) (PR(A) otimes PR(A))

=(oplus

iisinI((m A A)otimes idUor

i)times idUi

) m R(A) (PR(A) otimes PR(A)) (A12)

which can be checked by direct calculation along the same lines as in the proof of [FrLem 310]

The action of the modular transformation S on oplusiisinI(B otimes Ui Ui ) for B isin C nat-urally induces an action on oplusiisinI(Ui B otimes Ui ) [K3 Prop 514] which we denote bySlowast In the present case we get an action of Slowast on oplus jisinIHomC(Uor

j A otimes Uori otimes Uor

j ) and

opluskisinIHomC(UkUi otimes Uk) Then to show gi is invariant under the action of S times Sminus1

amounts to showing that

(h1 h2)i = ((Sminus1)lowast h1 Slowast h2)i (A13)

Cardy Algebras and Sewing Constraints I 909

for all h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) We have

((Sminus1)lowast h1 Slowast h2)i =sum

mnα

dim Un

DimC

A

A

A

A

A A

A

Uk

α

α

h1

h2

Ui

U j

UmUn

(A14)

Now drag the upper vertex indexed by α in the above graph along its Uorm -leg until it

meets the lower vertex also indexed by α then sum over α and m This gives

((Sminus1)lowast h1 Slowast h2)i =sum

n

dim Un

DimC

A

A

A

A

A A

A

Uk

Un

h1

h2

Ui

U j

(A15)

910 L Kong I Runkel

If we just look at the neighbourhood of the Un-loop in the above graph we see thefollowing subgraphs

sumn

dim Un

DimC

A Aor

Uorj U j

Uk Uork

Un=

sumα

1

dim U j

A Aor Uk Uork

Uorj U j

αα

(A16)

where we have applied [BK Cor 3111] Substituting this subgraph back to the originalgraph in (A15) we obtain

((Sminus1)lowast h1 Slowast h2)i =sumα

1

dim U j

A

A

A

A

A A

A

A

A

Uorj

α

α

h1

h2m A

Ui

U jUk

Uk

(A17)

The graph in (A17) is equal to that in (A11) In order to see this we first drag theldquobubblerdquo (m A A) along A lines and through the m A vertex (because m A A is abimodule map) until it reaches the lower-left leg of the upper vertex indexed by α Thendrag the m A vertex along the (red) dotted line in the above graph Finally we applythe associativity of A (A12) and [Fr Lem 311] Then we see that the graph in (A17)exactly matches with the one in (A11)

Acknowledgements We would like to thank the organisers of the Oberwolfach Arbeitsgemeinschaft ldquoAlge-braic structures in conformal field theoriesrdquo (April 2007) where this work was started for an inspiring meetingWe would further like to thank the Hausdorff Institute for Mathematics in Bonn and the organisers of the stim-ulating meeting ldquoGeometry and Physicsrdquo (May 2008) We are indebted to Alexei Davydov Jens FjelstadJuumlrgen Fuchs Yi-Zhi Huang Alexei Kitaev Urs Schreiber Christoph Schweigert Stephan Stolz and PeterTeichner for helpful discussions andor comments on a draft of this paper The research of IR was partiallysupported by the EPSRC First Grant EPE0050471 the PPARC rolling grant PPC5071451 and the MarieCurie network lsquoSuperstring Theoryrsquo (MRTN-CT-2004-512194)

Cardy Algebras and Sewing Constraints I 911

References

[AN] Alexeevski A Natanzon SM Noncommutative two-dimensional topological field theories andhurwitz numbers for real algebraic curves Sel Math New Ser 12 307ndash377 (2006)

[Bi] Bichon J Cosovereign hopf algebras J Pure Appl Alg 157 121ndash133 (2001)[BK] Bakalov B Kirillov AA Lectures on Tensor Categories and Modular Functors Providence

RI Amer Math Soc 2001[C1] Cardy JL Operator content of two-dimensional conformal invariant theories Nucl Phys

B 270 186ndash204 (1986)[C2] Cardy JL Boundary conditions fusion rules and the verlinde formula Nucl Phys B 324

581ndash596 (1989)[CL] Cardy JL Lewellen DC Bulk and boundary operators in conformal field theory Phys Lett

B 259 274ndash278 (1991)[DMZ] Dong C-Y Mason G Zhu Y-C Discrete series of the Virasoro algebra and the moonshine

module In Algebraic Groups and Their Generalizations Quantum and infinite-dimensional Meth-ods Proc Symp Pure Math 56 Part 2 Providence RI Amer MathSoc 1994 pp 295ndash316

[DP] Day B Pastro C Note on frobenius monoidal functors New York J Math 14 733ndash742 (2008)[ENO1] Etingof PI Nikshych D Ostrik V On fusion categories Ann Math 162 581ndash642 (2005)[ENO2] Etingof PI Nikshych D Ostrik V An analogue of radfordrsquos s4 formula for finite tensor

categories Int Math Research Notices 54 2915ndash2933 (2004)[Fe] Felder G Froumlhlich J Fuchs J Schweigert C Correlation functions and boundary conditions in

rational conformal field theory and three-dimensional topology Comp Math 131 189ndash237 (2002)[FHL] Frenkel IB Huang Y-Z Lepowsky J On axiomatic approaches to vertex operator algebras

and modules Mem Amer Math Soc 104 (1993)[Fj] Fjelstad J Fuchs J Runkel I Schweigert C Uniqueness of openclosed rational cft with

given algebra of open states Adv Theor Math Phys 12 1283ndash1375 (2008)[FK] Froumlhlich J King C The chern-simons theory and knot polynomials Commun Math

Phys 126 167ndash199 (1989)[Fr] Froumlhlich J Fuchs J Runkel I Schweigert C Correspondences of ribbon categories Adv

Math 199 192ndash329 (2006)[FRS] Fuchs J Runkel I Schweigert C Tft construction of rcft correlators I partition functions Nucl

Phys B 646 353ndash497 (2002)[FS] Fuchs J Schweigert C Category theory for conformal boundary conditions Fields Inst Com-

mun 39 25ndash71 (2003)[H1] Huang Y-Z Two-dimensional conformal geometry and vertex operator algebras Progress in

Mathematics Vol 148 Boston Birkhaumluser 1997[H2] Huang Y-Z Rigidity and modularity of vertex tensor categories Comm Contemp Math 10

871ndash911 (2008)[HK1] Huang Y-Z Kong L Open-string vertex algebra category and operads Commun Math

Phys 250 433ndash471 (2004)[HK2] Huang Y-Z Kong L Full field algebras Commun Math Phys 272 345ndash396 (2007)[HK3] Huang Y-Z Kong L Modular invariance for conformal full field algebras httparxivorgabs

math0609570v2[mathQA] 2006[HL] Huang Y-Z Lepowsky J Tensor products of modules for a vertex operator algebra and vertex

tensor categories In Lie Theory and Geometry in honor of Bertram Kostant ed R BrylinskiJ-L Brylinski V Guillemin V Kac Boston Birkhaumluser 1994 pp 349ndash383

[JS] Joyal A Street R Braided tensor categories Adv Math 102 20ndash78 (1993)[K1] Kong L Full field algebras operads and tensor categories Adv Math 213 271ndash340 (2007)[K2] Kong L Open-closed field algebras Commun Math Phys 280 207ndash261 (2008)[K3] Kong L Cardy condition for open-closed field algebras Commun Math Phys 283 25ndash92 (2008)[Ki] Kitaev A Private communication[KO] Kirillov AA Ostrik V On q-analog of mckay correspondence and ade classification of sl(2)

conformal field theories Adv Math 171 183ndash227 (2002)[KR] Kong L Runkel I Morita classes of algebras in modular tensor categories Adv Math 219

1548ndash1576 (2008)[Ld] Lauda AD Frobenius algebras and ambidextrous adjunctions Theo Appl Cat 16 84ndash

122 (2006)[Li1] Li H-S Regular representations of vertex operator algebras Commun Contemp Math 4

639ndash683 (2002)[Li2] Li H-S Regular representations and huang-lepowsky tensor functors for vertex operator

algebras J Alge 255 423ndash462 (2002)

Cardy Algebras and Sewing Constraints I 895

For a general modular tensor category C the algebra C[x]x2 R(1) understood as analgebra in C2plusmn via the braided monoidal isomorphism Vect f (C) C2plusmn rarr C2plusmn providesanother counter-example

32 Two definitions Define a morphism PlA A rarr A for a Frobenius algebra A in C

or C2plusmn as follows [Fr Sect 24]

PlA =

A

A

A

m

(311)

If A is also commutative and obeys m A A = ζA idA we have PlA = ζA idA In partic-

ular this holds if A is commutative and special Using the fact that the Frobenius algebraR(1) is commutative and normalised-special one can check that Pl

R(A) R(A) rarr R(A)takes the following form

PlR(A) =

oplusiisinI

A

A

Uori

Uori

A

m

times

Ui

Ui

(312)

With these ingredients we can now give the first definition of a Cardy C|C2plusmn-algebrawhich was introduced in [K3 Def 514] cf Remark 315 below

Definition 37 (CardyC|C2plusmn-algebra I) A CardyC|C2plusmn-algebra is a triple (Aop|Acl ιcl-op)where (Aclmcl ηclcl εcl) is a modular invariant commutative symmetric Frobe-nius C2plusmn-algebra (Aopmop ηopop εop) is a symmetric Frobenius C-algebra andιcl-op Acl rarr R(Aop) an algebra homomorphism such that the following conditionsare satisfied

(i) Centre condition

Acl R(Aop)

R(Aop)

R(Aop)

ιcl-op

m R(Aop)

=

Acl

ιcl-op

R(Aop)

R(Aop)

R(Aop)

m R(Aop)

(313)

896 L Kong I Runkel

(ii) Cardy condition

ιcl-op ιlowastcl-op =

R(Aop)

R(Aop)

R(Aop)

(314)

Remark 38

(i) The name ldquoCardy C|C2plusmn-algebrardquo in Definition 37 was chosen because many ofthe important ingredients were first studied by Cardy the modular invariance ofthe closed theory [C1] the consistency of the annulus amplitude [C2] and thebulk-boundary OPE [CL] On the other hand the boundary-boundary OPE andthe OPE analogue of the centre condition were first considered in [Lw]

(ii) One can easily see that in the special case that C is the category Vect f (C) of finite-dimensional C-vector spaces a Cardy C|C2plusmn-algebra gives exactly the algebraicformulation of two-dimensional open-closed topological field theory over C (cfRemark 614 in [K3]) see [Lz Sect 48] [Mo Thm 11] [AN Thm 45] [LPCor 43] [MS Sect 22] When passing to a general modular tensor category Cthere are two important differences to the two-dimensional topological field the-ory Firstly the algebras Acl and Aop now live in different categories which inparticular affects the formulation of the centre condition and the Cardy conditionSecondly the modular invariance condition has to be imposed on Acl In the caseC = Vect f (C) modular invariance holds automatically

Definition 39 A homomorphism of Cardy C|C2plusmn-algebras (A(1)op |A(1)cl ι(1)cl-op) rarr (A(2)op

|A(2)cl ι(2)cl-op) is a pair ( fop fcl) of Frobenius algebra homomorphisms fop A(1)op rarr A(2)op

and fcl A(1)cl rarr A(2)cl such that the diagram

A(1)cl

fcl

ι(1)cl-op

A(2)cl

ι(2)cl-op

R(A(1)op )

R( fop) R(A(2)op )

(315)

commutes

Remark 310 Since a homomorphism of Frobenius algebras is invertible (cf Lemma218 (iv)) a homomorphism of Cardy algebras is always an isomorphism

For a homomorphism ( fop fcl) of Cardy C|C2plusmn-algebras using the commutativity of(315) and the fact that fcl and fop are both algebra and coalgebra homomorphisms itis easy to show that (315) commutes iff

R(A(1)op )R( fop)

(ι(1)cl-op)

lowast

R(A(2)op )

(ι(2)cl-op)

lowast

A(1)cl

fcl A(2)cl

(316)

Cardy Algebras and Sewing Constraints I 897

commutesLet (Aop|Acl ιcl-op) be a Cardy C|C2plusmn-algebra Define the morphism

ιcl-op = χminus1(ιcl-op) T (Acl) minusrarr Aop (317)

Decompose Acl as Acl = oplusNn=1Cl

n times Crn such that Cl

1 times Cr1 = ηcl(1 times 1) We use ι(n)cl-op

to denote the restriction of ιcl-op to Cln times Cr

n and ι(n)cl-op to denote the restriction of ιcl-op

to Cln otimes Cr

n We introduce the following graphical notation

ι(n)cl-op =

Cln Cr

n

Aop

(318)

By (243) ιcl-op can be expressed in terms of ιcl-op as follows

ιcl-op =Noplus

n=1

oplusiisinI

sumα

Cln

Crn

Aop

α

Uori

times α

Crn

Ui

(319)

Lemma 311 The centre condition (313) is equivalent to the following condition in C

Cln Cr

n

Aop

Aop

Aop

=

Cln Cr

n

Aop

Aop

Aop

for n = 1 N (320)

Proof First insert (319) and the definition (217) of m R(Aop) into (313) Then apply thecommutativity of R(1) to the left hand side of (313) The equivalence between (313)and (320) follows immediately Remark 312 The centre condition (320) is very natural from the open-closed confor-mal field theory point of view Correlators on the upper half plane are expressed in termsof conformal blocks on the full complex plane The objects Cl

n and Crn are associated to

the field insertion at a point z in the upper half plane and at the complex conjugate pointz in the lower half plane respectively The object Aop corresponds to a field inserted ata point r on the real axis The centre condition (320) simply says that the correlationfunctions in the disjoint domains |z| gt r gt 0 and r gt |z| gt 0 are analytic continuationsof each other see [K2 Prop 118]

898 L Kong I Runkel

Recall that we define ιlowastcl-op Aop rarr T (Acl) as in (230) We introduce the graphicalnotation

ιlowastcl-op =Noplus

n=1

Cln Cr

n

Aop

=Noplus

n=1

Cln Cr

n

Aop

Aop T (minus1Acl) (321)

where the second equality follows from (215) (221) and (259)

Lemma 313 The Cardy condition (314) is equivalent to the following identity in C

Noplusn=1

DimCdim Ui

sumα

Cln

Crn

Crn

α

α

Aop

Aop Uori

Uori

=Aop

Aop

Aop

Uori

Uori

op

mop for all i isin I (322)

Proof By (244) (312) and (319) it is easy to see that (322) is equivalent to thefollowing identity

ιcl-op χ(ιlowastcl-op) = PlR(Aop)

(323)

Therefore it is enough to show that

χ(ιlowastcl-op) = ιlowastcl-op (324)

We have

χminus1(ιlowastcl-op)(1)= T (ιlowastcl-op) δ (2)= T (ιcl-op)

lowast δlowastlowast (3)= (ρ T (ιcl-op)

)lowast (4)= (ιcl-op

)lowast (325)

In step (1) we use the expression (253) for χminus1 step (2) follows from Lemma 218 (v)and Lemma 219 Step (3) is Lemma 218 (i) and Lemma 228 and finally step (4)amounts to substituting (253) and (317) Acting with χ on both sides of the aboveequality produces (324)

Combining Lemmas 311 and 313 and Proposition 216 we obtain the followingequivalent definition of Cardy C|C2plusmn-algebra (recall the graphical notation (318) forιcl-op and (321) for ιlowastcl-op)

Cardy Algebras and Sewing Constraints I 899

Definition 314 (CardyC|C2plusmn-algebra II) A CardyC-algebra is a triple (Aop|Acl ιcl-op)where Acl is a commutative symmetric Frobenius C2plusmn-algebra satisfying property (34)with f = mcl Aop is a symmetric Frobenius C-algebra and ιcl-op T (Acl) rarr Aop isan algebra homomorphism satisfying the conditions (320) and (322)

Remark 315 Up to a choice of normalisation Definition 314 is the same as the orig-inal one in [K3 Def 613] The difference between the two definitions is the factorDim C dim Ui on the left hand side of (322) which in [K3 Def 613] is given byradic

Dim C dim Ui The two definitions are related by rescaling the coproduct cl andcounit εcl of Acl by 1

radicDim C and

radicDim C respectively We chose the convention in

(322) to remove all dimension factors from the expression (314) for the Cardy condition

33 Uniqueness and existence theorems In this subsection we investigate the structureof Cardy algebras We start with the following proposition which when combined withthe results of part II provides an alternative proof of [Fj Prop 422]

Proposition 316 Let (Aop|Acl ιcl-op) be a Cardy C|C2plusmn-algebra If Acl is simple anddim Aop = 0 then Aop is simple and special

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis therefore haploid Restricting the Cardy condition (322) to the case Ui = 1 andcomposing both sides with εop from the left we see that εop kills all terms associated toU j times 1 isin Acl in the sum except for a single 1 times 1 term Thus we obtain the followingidentity

β εop = dAop (mop otimes idAorop) (idAop otimes bAop) (326)

where β isin C Composing with ηop from the right in turn implies that βεop ηop =dim Aop which is nonzero by assumption Thus also β = 0 and εop is a nonzero multipleof the morphism on the right hand side of (326) By [FRS Lem 311] Aop is special

Since Aop is a special Frobenius algebra Aop is semi-simple as an Aop-bimodule(apply [FS Prop 524] to Aop tensored with its opposite algebra) Suppose Aop is not

simple so that we can write Aop = A(1)op oplus A(2)op for nonzero Aop-bimodules A(1)op and A(2)op We denote the canonical embeddings and projections associated to this decompositionas ι12 and π12 We have the identities

mop (ι1 otimes ι2) = 0 εop ηop =2sum

i=1

εop ιi πi ηop (327)

The first identity follows since π1 mop (ι1 otimes ι2) = 0 (as mop gives the left action of

Aop on Aop and hence it preserves A(2)op ) and similarly π2 mop (ι1 otimes ι2) = 0 Thesecond identity is just the completeness of ι12 π12

Since εop ηop = 0 without losing generality we can assume εop ι1 π1 ηop = 0Using that π2 is a bimodule map we compute

π2 [LHS of (322)]Ui =1 ι1 π1 ηop = π2 [RHS of (322)]Ui =1 ι1 π1 ηop

= PlA(2)op

π2 ι1 π1 ηop = 0 (328)

900 L Kong I Runkel

On the other hand using that Acl is haploid that ιcl-op is an algebra map and thatι lowastcl-op is a coalgebra map one can check that the left-hand side of (328) is equal toλ(εop ι1 π1 ηop) π2 ηop for some λ = 0 This implies that π2 ηop = 0 Thusηop = sum

i ιi πi ηop = ι1 π1 ηop Hence we have

0 = π2 ι2 = π2 mop (ηop otimes ι2) = π2 mop ((ι1 π1 ηop)otimes ι2) (329)

However the right-hand side is zero by (327) This is a contradiction and hence Aopmust be simple

To formulate the next theorem we need the notion of the full centre of an algebra [FjDef 49] Recall that an algebra A in a braided tensor category has a left centre and aright centre [VZO] both of which are sub-algebras of A Of these two we will onlyneed the left centre The following definition is [Fr Def 231] which in our setting isequivalent to that of [VZO]

Definition 317 Let A be a symmetric special Frobenius algebra such that m A A =ζA idA

(i) The left centre Cl(A) of A is the image of the idempotent ζminus1A Pl

A(ii) The full centre Z(A) is Cl(R(A))

That ζminus1A Pl

A is an idempotent follows from [FRS Lem 52] when keeping track ofthe factors ζA ([FRS] assumes normalised-special ie ζA = 1) Note that Cl(A) is againan object of C while Z(A) is an object of C2plusmn Let el

A Cl(A) rarr A be the embeddingof Cl(A) into A The left centre is in fact the maximal subobject of A such that

m A cAA (elA otimes idA) = m A (330)

see [Fr Lem 232] This observation explains the name left centre and also makes theconnection to [O Def 15]

The full centre is by definition the image of the idempotent ζminus1A Pl

R(A) R(A) rarrR(A) Since C2plusmn is abelian the idempotent splits and we obtain the embedding andrestriction morphisms

e Z(A) rarr R(A) and r R(A) Z(A) (331)

which obey r e = idZ(A) and e r = ζminus1A Pl

R(A) It follows from Proposition 225 and

[Fr Prop 237] that Z(A) is a commutative symmetric Frobenius algebra in C2plusmn withstructure morphisms5

m Z(A) = r m R(A) (e otimes e) ηZ(A) = r ηR(A)

Z(A) = ζA middot (r otimes r) R(A) e εZ(A) = ζminus1A middot εR(A) e

(332)

Moreover if A is simple then Z(A) is simple and if A is simple and dim A = 0 thenZ(A) is simple and special The normalisation of the counit is such that

εZ(A) ηZ(A) = ζminus2A dim A DimC (333)

5 The normalisation of product and unit is the standard one The factors in the coproduct and counit haveto be included in order for (A|Z(A) e) to be a Cardy algebra see Theorem 318 below The normalisation ofthe counit enters the Cardy condition (314) through the definition of ( middot )lowast

Cardy Algebras and Sewing Constraints I 901

Theorem 318 Let A be a special symmetric Frobenius C-algebra Then (A|Z(A) e)is a Cardy C|C2plusmn-algebra

The proof of this theorem makes use the following two lemmas

Lemma 319 e Z(A) rarr R(A) is an algebra map and elowast = ζA middot r

Proof It follows from [Fr Lem 229] (or by direct calculation using in particularm R(A) R(A) = ζAidR(A)) that

m R(A) (e otimes e) = ζminus1A middot Pl

R(A) m R(A) (e otimes e) (334)

Substituting e r = ζminus1A Pl

R(A) shows that e is compatible with multiplication For theunit one finds

e ηZ(A) = e r ηR(A) = ζminus1A Pl

R(A) ηR(A) = ηR(A) (335)

Thus e is an algebra map For the second statement one computes

elowast (1)= ζA

R(A)

r

R(A)

R(A)

Z(A)

R(A)

Z(A)

r

e

e

(2)=

R(A)

r

R(A)

R(A)

R(A)

Z(A)

(3)=

R(A)

r

R(A)

Z(A)

(4)= ζA middot r (336)

where in (1) the definitions (230) and (332) have been substituted step (2) is e r = ζminus1

A PlR(A) step (3) uses that R(A) is symmetric Frobenius and step (4) is again

e r = ζminus1A Pl

R(A) Lemma 320 Let A be a symmetric Frobenius algebra in C The morphism

PlR(A) m R(A)

(Pl

R(A) otimes PlR(A)

) R(A)otimes R(A) minusrarr R(A) (337)

is S-invariant

The proof of this lemma is a slightly lengthy explicit calculation and has been deferredto Appendix A2

902 L Kong I Runkel

Proof of Theorem 318 That e is an algebra map was proved in Lemma 319 The centrecondition (313) holds by property (330) of the left centre The Cardy condition (314)also is an immediate consequence of Lemma 319

ιcl-op ιlowastcl-op = e (ζA r) = PlR(A) (338)

The full centre Z(A) is a commutative symmetric Frobenius algebra It remains to provemodular invariance That θZ(A) = idZ(A) is implied by commutativity and symmetryof Z(A) [Fr Prop 225] The S-invariance condition (33) follows from Lemma 320In (337) substitute Pl

R(A) = ζA e r and then put the resulting morphism into (33)Compose the resulting equation with e otimes idW from the right (ie from the bottom) andsubstitute the definition (332) of m Z(A) This results in the statement that m Z(A) isS-invariant

The following theorem is analogous to [LR Prop 29] and [Fj Thm 426] whichroughly speaking answer the question under which circumstances the restriction of atwo-dimensional conformal field theory to the boundary already determines the entireconformal field theory The first work is set in Minkowski space and uses operator alge-bras and subfactors while the second work is set in Euclidean space and uses modulartensor categories

Theorem 321 Let (A|Acl ιcl-op) be a Cardy C|C2plusmn-algebra such that dim A = 0 andAcl is simple Then A is special and (A|Acl ιcl-op) sim= (A|Z(A) e) as Cardy algebras

Proof By Proposition 316 A is simple and special Since Acl is simple the algebramap ιcl-op Acl rarr R(A) is either zero or a monomorphism But ιcl-op ηcl = ηR(A)and so ιcl-op = 0 Thus ιcl-op is monic By Lemma 218 (ii) ζminus1

A ιlowastcl-op is epi The Cardycondition (314) implies

ιcl-op ζminus1A ιlowastcl-op = ζminus1

A PlR(A) = e r (339)

Composing this with e r from the left yields e r ιcl-op ζminus1A ιlowastcl-op = e r =

ιcl-op ζminus1A ιlowastcl-op Since ζminus1

A ιlowastcl-op is epi we have

e r ιcl-op = ιcl-op (340)

Actually (340) also follows from (313) and specialness of R(A) We will prove that

( fop fcl) (A|Acl ιcl-op) minusrarr (A|Z(A) e) where fop = idA fcl = r ιcl-op

(341)

is an isomorphism of Cardy algebrasfcl is an algebra map Compatibility with the units follows since ιcl-op is an algebramap

fcl ηcl = r ιcl-op ηcl = r ηR(A) = ηZ(A) (342)

Compatibility with the multiplication also follows since ιcl-op is an algebra map

m Z(A) ( fcl otimes fcl) = r m R(A) (e otimes e) (r otimes r) (ιcl-op otimes ιcl-op)

= r m R(A) (ιcl-op otimes ιcl-op) = r ιcl-op mcl = fcl mcl (343)

where in the second step we used (340)

Cardy Algebras and Sewing Constraints I 903

fcl is an isomorphism As above since fcl is an algebra map and since Acl is simple fcl

has to be monic By Lemma 319 rlowast = ζminus1A e Thus f lowast

cl = ιlowastcl-op rlowast = ζminus1A ιlowastcl-op e and

fcl f lowastcl = r ιcl-op ζminus1

A ιlowastcl-op e = r e r e = idZ(A) (344)

and so fcl is also epi and hence isofcl is a coalgebra map Since fcl is an algebra map so is f minus1

cl By (344) f minus1cl = f lowast

cland by Lemma 218 (iii) this implies that fcl is a also coalgebra mapThe diagram (315) commutes Commutativity of (315) is equivalent to e fcl = ιcl-opwhich holds by (340)

Let A be a special symmetric Frobenius algebra So far we have seen that (A|Z(A) e)is a Cardy algebra and that all Cardy algebras with Aop = A and simple Acl are of thisform It is now natural to ask if every simple Acl does occur as part of a Cardy algebraThe following theorem provides an affirmative answer Recall that for an A-left moduleM the object Mor otimesA M is an algebra (see eg [KR Lem 42])

Theorem 322 If Acl is a simple modular invariant commutative symmetric FrobeniusC2plusmn-algebra then there exist a simple special symmetric Frobenius C-algebra A and amorphism ιcl-op Acl rarr R(A) such that

(i) Acl sim= Z(A) as Frobenius algebras(ii) (A|Acl ιcl-op) is a Cardy C|C2plusmn-algebra

(iii) T (Acl) sim= oplusκisinJ Morκ otimesA Mκ as algebras where MκκisinJ is a set of representa-

tives of the isomorphism classes of simple A-left modules

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis haploid It then follows from Theorem 34 that Acl is special By Proposition 224T (Acl) is a special symmetric Frobenius algebra in C Thus T (Acl) = oplusi Ai where theAi are simple symmetric Frobenius algebras We will show that at least one of the Aiis special Since T (Acl) is special we have mT (Acl) T (Acl) = ζ idT (Acl) for someζ isin C

times Restricting this to the summand Ai shows mi i = ζ idAi FurthermoreεT (Acl) ηT (Acl) = ξ id1 for some ξ isin C

times But εT (Acl) ηT (Acl) = sumi εi ηi and so

at least one of the εi ηi has to be nonzero Therefore at least one of the Ai is speciallet A equiv Ai be this summand We denote the embedding A rarr T (Acl) by e0 and therestriction T (Acl) A by r0 Notice that r0 is an algebra homomorphism Define

ιcl-op = χ (r0) Acl minusrarr R(A) (345)

By Proposition 216 ιcl-op is an algebra homomorphism Next we verify the centre con-dition (313) or rather its equivalent form (320) By substituting the definitions onecan convince oneself that the commutativity mcl cAclAcl = mcl of Acl in C2plusmn implies thecondition mT (Acl) = mT (Acl) in C see [K2 Prop 36] Here T (Acl)otimes T (Acl) rarrT (Acl)otimes T (Acl) is given by

=oplusmn

(idCl

notimes cCl

m Crnotimes idCr

m

)

(cCl

m Clnotimes cminus1

Crn C

rm

)

(idCl

motimes cminus1

Cln C

rm

otimes idCrn

)

(346)

and we decomposed Acl as Acl = oplusnCln times Cr

n As a consequence we obtain the identity

r0 mT (Acl) (idAcl otimes e0) = r0 mT (Acl) (idAcl otimes e0) (347)

904 L Kong I Runkel

Using that r0 is an algebra map and that by definition ιcl-op = r0 we obtain (320)In order to show that (A|Acl ιcl-op) is a Cardy algebra it remains to show that the

Cardy condition (314) is satisfied We will demonstrate this via a detour by first provingthat Acl sim= Z(A) as Frobenius algebras

Recall the notations e and r given in (331) Using the centre condition (320) onecan check that Pl

R(A) ιcl-op = m R(A) R(A) ιcl-op By specialness of A we have

m A A = ζA idA and so together with e r = ζminus1A Pl

R(A) we get

e r ιcl-op = ιcl-op (348)

Next consider the morphism

fcl = r ιcl-op Acl minusrarr Z(A) (349)

By the same derivation as in (342) and (343) one sees that fcl is an algebra map Inparticular fcl ηcl = ηZ(A) = 0 and so fcl = 0 Since Acl is simple fcl has to be amonomorphism

By the same argument as used in the proof of Theorem 34 (ii) up to isomorphismAcl is the unique simple local Acl-(left-)module The algebra monomorphism fcl turnsZ(A) into an Acl-module Since Z(A) is commutative it is local as an Acl-module andso Z(A) sim= AoplusN

cl for some N ge 1 By construction A is a simple special symmetricFrobenius algebra Proposition 225 and Corollary 227 show that R(A) inherits all theseproperties and thus Z(A) is simple (see the comment below Eq (332)) By Theorem318 Z(A) is modular invariant and then by Theorem 34 (i) dim Z(A) = Dim C Thisimplies that N = 1 in Z(A) sim= AoplusN

cl and so fcl is in fact an isomorphismSince Acl and Z(A) are both haploid we have εZ(A) fcl = ξ εcl for some ξ isin C

timesThe counit uniquely determines the Frobenius structure on Acl and Z(A) (see eg [FRSLemma 37]) so that fcl is a coalgebra isomorphism iff ξ = 1 To compute ξ we composethe above identity with ηcl from the right Defining ζcl via εcl ηcl = ζminus1

cl Dim C middot id1 andusing (333) gives ξ = dim A ζclζ

2A By rescaling the comultiplication and the counit

of A and consequently changing ζA we can always achieve ξ = 1 This proves part (i)of the theorem

Equation (348) implies that ιcl-op = e fcl Since fcl is an isomorphism of Frobeniusalgebras by Lemmas 218 and 319 we have

ιcl-op ιlowastcl-op = e fcl f lowastcl elowast = ζA e r = Pl

R(A) (350)

Thus (A|Acl ιcl-op) is a Cardy algebra This proves part (ii) of the theoremPart (iii) can be seen as follows By [KR Prop 43] T Z(A) sim= oplusκisinJ Mor

κ otimesA Mκ asalgebras Together with the observation that T ( fcl) T (Acl) rarr T Z(A) is an isomor-phism of algebras this proves part (iii) Remark 323 Part (i) of Theorem 322 was announced by Muumlger [Muuml2] We provide anindependent proof in the setting of Cardy algebras

The above theorem together with Lemma 211 and Theorem 34 shows that a sim-ple commutative symmetric Frobenius C2plusmn-algebra Acl with dim Acl = Dim C is alwayspart of a Cardy algebra (Aop|Acl ιcl-op) for some simple special symmetric Frobeniusalgebra Aop in C However the above proof also illustrates that Aop is not unique Thisraises the question how two Cardy algebras with a given Acl can differ This question isanswered by [KR Thm 11] which in the present framework can be restated as follows

Cardy Algebras and Sewing Constraints I 905

Theorem 324 If (A(i)op |A(i)cl ι(i)cl-op) i = 1 2 are two Cardy C|C2plusmn-algebras such that

A(i)cl is simple and dim A(i)op = 0 for i = 1 2 then A(1)clsim= A(2)cl as algebras if and only

if A(1)op and A(2)op are Morita equivalent

Proof Theorem 11 in [KR] is stated for A(i)op being non-degenerate algebras and A(i)cl =Z(A(i)op ) for i = 1 2 By Proposition 316 A(i)op are simple and special for i = 1 2

Then by [KR Lem 21] A(i)op are non-degenerate algebras By Theorem 321 we have

A(i)clsim= Z(A(i)op ) as Frobenius algebras Finally by [KR Thm 11] Z(A(1)op ) sim= Z(A(2)op )

as algebras iff A(1)op and A(2)op are Morita equivalent Let Cmax(C2plusmn) be the set of equivalence classes [B] of simple modular invariant

commutative symmetric Frobenius algebras B in C2plusmn Two such algebras B and B prime areequivalent if B and B prime are isomorphic as algebras (but not necessarily as Frobenius alge-bras) Let Msimp(C) be the set of Morita classes of simple special symmetric Frobeniusalgebras in C Define the map z Msimp(C) rarr Cmax(C2plusmn) by z A rarr [Z(A)] whereA denotes the Morita class of A From Theorem 322 (i) and [KR Thm 11] we learn

Corollary 325 The map z Msimp(C) rarr Cmax(C2plusmn) is a bijection

A Appendix

A1 Proof of Lemma 27 We will show that if (F ψ F2 ψ

F0 ) is a colax tensor functor

from C1 to C2 then (G φG2 φ

G0 ) is a lax tensor functor from C2 to C1 Applying this

result to the opposed categories then gives the converse statementWe need to show that φG

0 and φG2 make the diagrams (21) and (22) commute We

first prove the commutativity of (21) Consider the following diagram

G(A)otimes (G(B)otimes G(C))

GψF2 δ

idG(A)otimesφG2 G(A)otimes G(B otimes C)

GψF2 δ

G(FG(A)otimes F(G(B)otimes G(C)))

G(F(idG(A))otimesF(φG2 ))

G(idFG(A)otimesψ F2 )

G(FG(A)otimes FG(B otimes C))

G(ρAotimesρBotimesC )

G(FG(A)otimes (FG(B)otimes FG(C)))

G(ρAotimes(ρBotimesρC )) G(A otimes (B otimes C))

(A1)

The top subdiagram is commutative because of the naturality of Gψ F2 δ The commu-

tativity of the bottom subdiagram follows from the following identities

(ρB otimes ρC ) ψ F2 = (ρB otimes ρC ) ψ F

2 ρF Fδ

= ρBotimesC FG(ρB otimes ρC ) FG(ψ F2 ) Fδ

= ρBotimesC F(φG2 ) (A2)

as a map F(G(B) otimes G(C)) rarr B otimes C The commutativity of (A1) implies that thecomposition of maps in the left column in (21) can be replaced by

G(ρA otimes (ρB otimes ρC )) G((idFG(A) otimes ψ F

2 ) ψ F2

) δ (A3)

906 L Kong I Runkel

Similarly we can show that the composition of maps in the right column in (21) can bereplaced by

G((ρA otimes ρB)otimes ρC ) G((ψ F

2 otimes idFG(C)) ψ F2

) δ (A4)

Using the commutativity of (23) it is easy to see that (21) with the left and right col-umns of (21) replaced by (A3) and (A4) respectively is commutative Hence (21) iscommutative

Now we prove the commutativity of the first diagram in (22)

φG2 (φG

0 otimes idG(A))

(1)= G(ρ12 otimes ρA) Gψ F2 δ

[(Gψ F

0 δ11)otimes idG(A)

]

(2)= G(ρ12 otimes ρA) Gψ F2 G F(Gψ0 otimes idG(A)) G F(δ11 otimes idG(A)) δ

(3)= G(ρ12 otimes ρA) G(FG(ψ F0 )otimes idFG(A)) Gψ F

2 G F(δ11 otimes idG(A)) δ(4)= G(id12 otimes ρA) G

([ψ F

0 ρF(11) (Fδ)11] otimes idFG(A)

) Gψ F

2 δ(5)= G(id12 otimes ρA) G(ψ F

0 otimes idFG(A)) Gψ F2 δ

(6)= G(id12 otimes ρA) G(lminus1FG(A)) G F(lG(A)) δ

(7)= G(lminus1A ) GρA δG lG(A)

(8)= G(lminus1A ) lG(A) (A5)

where in step (1) we substituted the definition of φG0 φ

G2 given in (210) in step (2)

we used the naturality of δ in step (3) we used the naturality of Gψ F2 in step (4) we

switched the position between Gρ12 and G FG(ψ F0 ) and the position between Gψ2 and

G F(δ12 otimes idG(A)) using the naturality of ρ and FψG2 respectively in step (5) we applied

the second identity in (28) in step (6) we used (24) in step (7) we used the naturalityof lminus1 and δ in step (8) we used the first identity in (28)

The proof of the commutativity of the second diagram in (22) is similar Thus wehave shown that G is a lax tensor functor

A2 Proof of Lemma 320 To prepare the proof recall that for a given object B isin Cthe modular group P SL(2Z) acts on the space oplusi HomC(B otimes Ui Ui ) see eg [BKSect 31] and [K3 Eq (455)] We will only need the action of S and Sminus1 Let f isinoplusi HomC(B otimes Ui Ui ) Then

S oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A6)

Cardy Algebras and Sewing Constraints I 907

Sminus1 oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A7)

By Lemma 32 to establish that (337) is S-invariant it is enough to prove the identity(34) when f is given by (337) Using (A6) and (A7) we can see that Eq (34) simplysays that oplusi j [RHS of (34)] is invariant under the action of S times S Consider the elementg of oplus jkisinIHomC2plusmn(R(A)otimes (Uor

j times Uk)Uorj times Uk) given by

g =oplusjkisinI

sumα

R(A)

R(A)

R(A)

R(A)

Uorj times Uk

Uorj times Uk

α

α

(A8)

By the above arguments proving S-invariance of (337) is equivalent to proving invari-ance of g under the action of S times S

For i isin I we denote by gi the component of g in(oplus jisinIHomC(A otimes Uor

i otimes Uorj U

orj )

)otimes (opluskisinIHomC(Ui otimes UkUk))

We view the second Hom-space in the above tensor product as a Hom-space in C+ insteadof Cminus It is enough to show that gi is invariant under the action of S times Sminus1 Note thatthe action of Sminus1 in C+ is equivalent to that of S in Cminus

The morphism gi can be canonically identified with a bilinear pairing

( middot middot )i ⎛⎝oplus

jisinIHomC(Uor

j A otimes Uori otimes Uor

j )

⎞⎠

times(oplus

kisinIHomC(UkUi otimes Uk)

)minusrarr C (A9)

as follows For h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) we set

(h1 h2)i = (dim U j dim Uk)minus1 trUor

j timesUk[gi (h1 times h2)] (A10)

908 L Kong I Runkel

When substituting the explicit form of the product m R(A) of R(A) = (A times 1) otimes R(1)after a short calculation one finds

(h1 h2)i =sumα

1

dim U j

A

A

A

A

A A

A α

α

h1

h2

U jUk

Uorj

Ui

Uk (A11)

Here the top morphism PlR(A) has been simplified with the help of the identity

PR(A) m R(A) (PR(A) otimes PR(A))

=(oplus

iisinI((m A A)otimes idUor

i)times idUi

) m R(A) (PR(A) otimes PR(A)) (A12)

which can be checked by direct calculation along the same lines as in the proof of [FrLem 310]

The action of the modular transformation S on oplusiisinI(B otimes Ui Ui ) for B isin C nat-urally induces an action on oplusiisinI(Ui B otimes Ui ) [K3 Prop 514] which we denote bySlowast In the present case we get an action of Slowast on oplus jisinIHomC(Uor

j A otimes Uori otimes Uor

j ) and

opluskisinIHomC(UkUi otimes Uk) Then to show gi is invariant under the action of S times Sminus1

amounts to showing that

(h1 h2)i = ((Sminus1)lowast h1 Slowast h2)i (A13)

Cardy Algebras and Sewing Constraints I 909

for all h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) We have

((Sminus1)lowast h1 Slowast h2)i =sum

mnα

dim Un

DimC

A

A

A

A

A A

A

Uk

α

α

h1

h2

Ui

U j

UmUn

(A14)

Now drag the upper vertex indexed by α in the above graph along its Uorm -leg until it

meets the lower vertex also indexed by α then sum over α and m This gives

((Sminus1)lowast h1 Slowast h2)i =sum

n

dim Un

DimC

A

A

A

A

A A

A

Uk

Un

h1

h2

Ui

U j

(A15)

910 L Kong I Runkel

If we just look at the neighbourhood of the Un-loop in the above graph we see thefollowing subgraphs

sumn

dim Un

DimC

A Aor

Uorj U j

Uk Uork

Un=

sumα

1

dim U j

A Aor Uk Uork

Uorj U j

αα

(A16)

where we have applied [BK Cor 3111] Substituting this subgraph back to the originalgraph in (A15) we obtain

((Sminus1)lowast h1 Slowast h2)i =sumα

1

dim U j

A

A

A

A

A A

A

A

A

Uorj

α

α

h1

h2m A

Ui

U jUk

Uk

(A17)

The graph in (A17) is equal to that in (A11) In order to see this we first drag theldquobubblerdquo (m A A) along A lines and through the m A vertex (because m A A is abimodule map) until it reaches the lower-left leg of the upper vertex indexed by α Thendrag the m A vertex along the (red) dotted line in the above graph Finally we applythe associativity of A (A12) and [Fr Lem 311] Then we see that the graph in (A17)exactly matches with the one in (A11)

Acknowledgements We would like to thank the organisers of the Oberwolfach Arbeitsgemeinschaft ldquoAlge-braic structures in conformal field theoriesrdquo (April 2007) where this work was started for an inspiring meetingWe would further like to thank the Hausdorff Institute for Mathematics in Bonn and the organisers of the stim-ulating meeting ldquoGeometry and Physicsrdquo (May 2008) We are indebted to Alexei Davydov Jens FjelstadJuumlrgen Fuchs Yi-Zhi Huang Alexei Kitaev Urs Schreiber Christoph Schweigert Stephan Stolz and PeterTeichner for helpful discussions andor comments on a draft of this paper The research of IR was partiallysupported by the EPSRC First Grant EPE0050471 the PPARC rolling grant PPC5071451 and the MarieCurie network lsquoSuperstring Theoryrsquo (MRTN-CT-2004-512194)

Cardy Algebras and Sewing Constraints I 911

References

[AN] Alexeevski A Natanzon SM Noncommutative two-dimensional topological field theories andhurwitz numbers for real algebraic curves Sel Math New Ser 12 307ndash377 (2006)

[Bi] Bichon J Cosovereign hopf algebras J Pure Appl Alg 157 121ndash133 (2001)[BK] Bakalov B Kirillov AA Lectures on Tensor Categories and Modular Functors Providence

RI Amer Math Soc 2001[C1] Cardy JL Operator content of two-dimensional conformal invariant theories Nucl Phys

B 270 186ndash204 (1986)[C2] Cardy JL Boundary conditions fusion rules and the verlinde formula Nucl Phys B 324

581ndash596 (1989)[CL] Cardy JL Lewellen DC Bulk and boundary operators in conformal field theory Phys Lett

B 259 274ndash278 (1991)[DMZ] Dong C-Y Mason G Zhu Y-C Discrete series of the Virasoro algebra and the moonshine

module In Algebraic Groups and Their Generalizations Quantum and infinite-dimensional Meth-ods Proc Symp Pure Math 56 Part 2 Providence RI Amer MathSoc 1994 pp 295ndash316

[DP] Day B Pastro C Note on frobenius monoidal functors New York J Math 14 733ndash742 (2008)[ENO1] Etingof PI Nikshych D Ostrik V On fusion categories Ann Math 162 581ndash642 (2005)[ENO2] Etingof PI Nikshych D Ostrik V An analogue of radfordrsquos s4 formula for finite tensor

categories Int Math Research Notices 54 2915ndash2933 (2004)[Fe] Felder G Froumlhlich J Fuchs J Schweigert C Correlation functions and boundary conditions in

rational conformal field theory and three-dimensional topology Comp Math 131 189ndash237 (2002)[FHL] Frenkel IB Huang Y-Z Lepowsky J On axiomatic approaches to vertex operator algebras

and modules Mem Amer Math Soc 104 (1993)[Fj] Fjelstad J Fuchs J Runkel I Schweigert C Uniqueness of openclosed rational cft with

given algebra of open states Adv Theor Math Phys 12 1283ndash1375 (2008)[FK] Froumlhlich J King C The chern-simons theory and knot polynomials Commun Math

Phys 126 167ndash199 (1989)[Fr] Froumlhlich J Fuchs J Runkel I Schweigert C Correspondences of ribbon categories Adv

Math 199 192ndash329 (2006)[FRS] Fuchs J Runkel I Schweigert C Tft construction of rcft correlators I partition functions Nucl

Phys B 646 353ndash497 (2002)[FS] Fuchs J Schweigert C Category theory for conformal boundary conditions Fields Inst Com-

mun 39 25ndash71 (2003)[H1] Huang Y-Z Two-dimensional conformal geometry and vertex operator algebras Progress in

Mathematics Vol 148 Boston Birkhaumluser 1997[H2] Huang Y-Z Rigidity and modularity of vertex tensor categories Comm Contemp Math 10

871ndash911 (2008)[HK1] Huang Y-Z Kong L Open-string vertex algebra category and operads Commun Math

Phys 250 433ndash471 (2004)[HK2] Huang Y-Z Kong L Full field algebras Commun Math Phys 272 345ndash396 (2007)[HK3] Huang Y-Z Kong L Modular invariance for conformal full field algebras httparxivorgabs

math0609570v2[mathQA] 2006[HL] Huang Y-Z Lepowsky J Tensor products of modules for a vertex operator algebra and vertex

tensor categories In Lie Theory and Geometry in honor of Bertram Kostant ed R BrylinskiJ-L Brylinski V Guillemin V Kac Boston Birkhaumluser 1994 pp 349ndash383

[JS] Joyal A Street R Braided tensor categories Adv Math 102 20ndash78 (1993)[K1] Kong L Full field algebras operads and tensor categories Adv Math 213 271ndash340 (2007)[K2] Kong L Open-closed field algebras Commun Math Phys 280 207ndash261 (2008)[K3] Kong L Cardy condition for open-closed field algebras Commun Math Phys 283 25ndash92 (2008)[Ki] Kitaev A Private communication[KO] Kirillov AA Ostrik V On q-analog of mckay correspondence and ade classification of sl(2)

conformal field theories Adv Math 171 183ndash227 (2002)[KR] Kong L Runkel I Morita classes of algebras in modular tensor categories Adv Math 219

1548ndash1576 (2008)[Ld] Lauda AD Frobenius algebras and ambidextrous adjunctions Theo Appl Cat 16 84ndash

122 (2006)[Li1] Li H-S Regular representations of vertex operator algebras Commun Contemp Math 4

639ndash683 (2002)[Li2] Li H-S Regular representations and huang-lepowsky tensor functors for vertex operator

algebras J Alge 255 423ndash462 (2002)

896 L Kong I Runkel

(ii) Cardy condition

ιcl-op ιlowastcl-op =

R(Aop)

R(Aop)

R(Aop)

(314)

Remark 38

(i) The name ldquoCardy C|C2plusmn-algebrardquo in Definition 37 was chosen because many ofthe important ingredients were first studied by Cardy the modular invariance ofthe closed theory [C1] the consistency of the annulus amplitude [C2] and thebulk-boundary OPE [CL] On the other hand the boundary-boundary OPE andthe OPE analogue of the centre condition were first considered in [Lw]

(ii) One can easily see that in the special case that C is the category Vect f (C) of finite-dimensional C-vector spaces a Cardy C|C2plusmn-algebra gives exactly the algebraicformulation of two-dimensional open-closed topological field theory over C (cfRemark 614 in [K3]) see [Lz Sect 48] [Mo Thm 11] [AN Thm 45] [LPCor 43] [MS Sect 22] When passing to a general modular tensor category Cthere are two important differences to the two-dimensional topological field the-ory Firstly the algebras Acl and Aop now live in different categories which inparticular affects the formulation of the centre condition and the Cardy conditionSecondly the modular invariance condition has to be imposed on Acl In the caseC = Vect f (C) modular invariance holds automatically

Definition 39 A homomorphism of Cardy C|C2plusmn-algebras (A(1)op |A(1)cl ι(1)cl-op) rarr (A(2)op

|A(2)cl ι(2)cl-op) is a pair ( fop fcl) of Frobenius algebra homomorphisms fop A(1)op rarr A(2)op

and fcl A(1)cl rarr A(2)cl such that the diagram

A(1)cl

fcl

ι(1)cl-op

A(2)cl

ι(2)cl-op

R(A(1)op )

R( fop) R(A(2)op )

(315)

commutes

Remark 310 Since a homomorphism of Frobenius algebras is invertible (cf Lemma218 (iv)) a homomorphism of Cardy algebras is always an isomorphism

For a homomorphism ( fop fcl) of Cardy C|C2plusmn-algebras using the commutativity of(315) and the fact that fcl and fop are both algebra and coalgebra homomorphisms itis easy to show that (315) commutes iff

R(A(1)op )R( fop)

(ι(1)cl-op)

lowast

R(A(2)op )

(ι(2)cl-op)

lowast

A(1)cl

fcl A(2)cl

(316)

Cardy Algebras and Sewing Constraints I 897

commutesLet (Aop|Acl ιcl-op) be a Cardy C|C2plusmn-algebra Define the morphism

ιcl-op = χminus1(ιcl-op) T (Acl) minusrarr Aop (317)

Decompose Acl as Acl = oplusNn=1Cl

n times Crn such that Cl

1 times Cr1 = ηcl(1 times 1) We use ι(n)cl-op

to denote the restriction of ιcl-op to Cln times Cr

n and ι(n)cl-op to denote the restriction of ιcl-op

to Cln otimes Cr

n We introduce the following graphical notation

ι(n)cl-op =

Cln Cr

n

Aop

(318)

By (243) ιcl-op can be expressed in terms of ιcl-op as follows

ιcl-op =Noplus

n=1

oplusiisinI

sumα

Cln

Crn

Aop

α

Uori

times α

Crn

Ui

(319)

Lemma 311 The centre condition (313) is equivalent to the following condition in C

Cln Cr

n

Aop

Aop

Aop

=

Cln Cr

n

Aop

Aop

Aop

for n = 1 N (320)

Proof First insert (319) and the definition (217) of m R(Aop) into (313) Then apply thecommutativity of R(1) to the left hand side of (313) The equivalence between (313)and (320) follows immediately Remark 312 The centre condition (320) is very natural from the open-closed confor-mal field theory point of view Correlators on the upper half plane are expressed in termsof conformal blocks on the full complex plane The objects Cl

n and Crn are associated to

the field insertion at a point z in the upper half plane and at the complex conjugate pointz in the lower half plane respectively The object Aop corresponds to a field inserted ata point r on the real axis The centre condition (320) simply says that the correlationfunctions in the disjoint domains |z| gt r gt 0 and r gt |z| gt 0 are analytic continuationsof each other see [K2 Prop 118]

898 L Kong I Runkel

Recall that we define ιlowastcl-op Aop rarr T (Acl) as in (230) We introduce the graphicalnotation

ιlowastcl-op =Noplus

n=1

Cln Cr

n

Aop

=Noplus

n=1

Cln Cr

n

Aop

Aop T (minus1Acl) (321)

where the second equality follows from (215) (221) and (259)

Lemma 313 The Cardy condition (314) is equivalent to the following identity in C

Noplusn=1

DimCdim Ui

sumα

Cln

Crn

Crn

α

α

Aop

Aop Uori

Uori

=Aop

Aop

Aop

Uori

Uori

op

mop for all i isin I (322)

Proof By (244) (312) and (319) it is easy to see that (322) is equivalent to thefollowing identity

ιcl-op χ(ιlowastcl-op) = PlR(Aop)

(323)

Therefore it is enough to show that

χ(ιlowastcl-op) = ιlowastcl-op (324)

We have

χminus1(ιlowastcl-op)(1)= T (ιlowastcl-op) δ (2)= T (ιcl-op)

lowast δlowastlowast (3)= (ρ T (ιcl-op)

)lowast (4)= (ιcl-op

)lowast (325)

In step (1) we use the expression (253) for χminus1 step (2) follows from Lemma 218 (v)and Lemma 219 Step (3) is Lemma 218 (i) and Lemma 228 and finally step (4)amounts to substituting (253) and (317) Acting with χ on both sides of the aboveequality produces (324)

Combining Lemmas 311 and 313 and Proposition 216 we obtain the followingequivalent definition of Cardy C|C2plusmn-algebra (recall the graphical notation (318) forιcl-op and (321) for ιlowastcl-op)

Cardy Algebras and Sewing Constraints I 899

Definition 314 (CardyC|C2plusmn-algebra II) A CardyC-algebra is a triple (Aop|Acl ιcl-op)where Acl is a commutative symmetric Frobenius C2plusmn-algebra satisfying property (34)with f = mcl Aop is a symmetric Frobenius C-algebra and ιcl-op T (Acl) rarr Aop isan algebra homomorphism satisfying the conditions (320) and (322)

Remark 315 Up to a choice of normalisation Definition 314 is the same as the orig-inal one in [K3 Def 613] The difference between the two definitions is the factorDim C dim Ui on the left hand side of (322) which in [K3 Def 613] is given byradic

Dim C dim Ui The two definitions are related by rescaling the coproduct cl andcounit εcl of Acl by 1

radicDim C and

radicDim C respectively We chose the convention in

(322) to remove all dimension factors from the expression (314) for the Cardy condition

33 Uniqueness and existence theorems In this subsection we investigate the structureof Cardy algebras We start with the following proposition which when combined withthe results of part II provides an alternative proof of [Fj Prop 422]

Proposition 316 Let (Aop|Acl ιcl-op) be a Cardy C|C2plusmn-algebra If Acl is simple anddim Aop = 0 then Aop is simple and special

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis therefore haploid Restricting the Cardy condition (322) to the case Ui = 1 andcomposing both sides with εop from the left we see that εop kills all terms associated toU j times 1 isin Acl in the sum except for a single 1 times 1 term Thus we obtain the followingidentity

β εop = dAop (mop otimes idAorop) (idAop otimes bAop) (326)

where β isin C Composing with ηop from the right in turn implies that βεop ηop =dim Aop which is nonzero by assumption Thus also β = 0 and εop is a nonzero multipleof the morphism on the right hand side of (326) By [FRS Lem 311] Aop is special

Since Aop is a special Frobenius algebra Aop is semi-simple as an Aop-bimodule(apply [FS Prop 524] to Aop tensored with its opposite algebra) Suppose Aop is not

simple so that we can write Aop = A(1)op oplus A(2)op for nonzero Aop-bimodules A(1)op and A(2)op We denote the canonical embeddings and projections associated to this decompositionas ι12 and π12 We have the identities

mop (ι1 otimes ι2) = 0 εop ηop =2sum

i=1

εop ιi πi ηop (327)

The first identity follows since π1 mop (ι1 otimes ι2) = 0 (as mop gives the left action of

Aop on Aop and hence it preserves A(2)op ) and similarly π2 mop (ι1 otimes ι2) = 0 Thesecond identity is just the completeness of ι12 π12

Since εop ηop = 0 without losing generality we can assume εop ι1 π1 ηop = 0Using that π2 is a bimodule map we compute

π2 [LHS of (322)]Ui =1 ι1 π1 ηop = π2 [RHS of (322)]Ui =1 ι1 π1 ηop

= PlA(2)op

π2 ι1 π1 ηop = 0 (328)

900 L Kong I Runkel

On the other hand using that Acl is haploid that ιcl-op is an algebra map and thatι lowastcl-op is a coalgebra map one can check that the left-hand side of (328) is equal toλ(εop ι1 π1 ηop) π2 ηop for some λ = 0 This implies that π2 ηop = 0 Thusηop = sum

i ιi πi ηop = ι1 π1 ηop Hence we have

0 = π2 ι2 = π2 mop (ηop otimes ι2) = π2 mop ((ι1 π1 ηop)otimes ι2) (329)

However the right-hand side is zero by (327) This is a contradiction and hence Aopmust be simple

To formulate the next theorem we need the notion of the full centre of an algebra [FjDef 49] Recall that an algebra A in a braided tensor category has a left centre and aright centre [VZO] both of which are sub-algebras of A Of these two we will onlyneed the left centre The following definition is [Fr Def 231] which in our setting isequivalent to that of [VZO]

Definition 317 Let A be a symmetric special Frobenius algebra such that m A A =ζA idA

(i) The left centre Cl(A) of A is the image of the idempotent ζminus1A Pl

A(ii) The full centre Z(A) is Cl(R(A))

That ζminus1A Pl

A is an idempotent follows from [FRS Lem 52] when keeping track ofthe factors ζA ([FRS] assumes normalised-special ie ζA = 1) Note that Cl(A) is againan object of C while Z(A) is an object of C2plusmn Let el

A Cl(A) rarr A be the embeddingof Cl(A) into A The left centre is in fact the maximal subobject of A such that

m A cAA (elA otimes idA) = m A (330)

see [Fr Lem 232] This observation explains the name left centre and also makes theconnection to [O Def 15]

The full centre is by definition the image of the idempotent ζminus1A Pl

R(A) R(A) rarrR(A) Since C2plusmn is abelian the idempotent splits and we obtain the embedding andrestriction morphisms

e Z(A) rarr R(A) and r R(A) Z(A) (331)

which obey r e = idZ(A) and e r = ζminus1A Pl

R(A) It follows from Proposition 225 and

[Fr Prop 237] that Z(A) is a commutative symmetric Frobenius algebra in C2plusmn withstructure morphisms5

m Z(A) = r m R(A) (e otimes e) ηZ(A) = r ηR(A)

Z(A) = ζA middot (r otimes r) R(A) e εZ(A) = ζminus1A middot εR(A) e

(332)

Moreover if A is simple then Z(A) is simple and if A is simple and dim A = 0 thenZ(A) is simple and special The normalisation of the counit is such that

εZ(A) ηZ(A) = ζminus2A dim A DimC (333)

5 The normalisation of product and unit is the standard one The factors in the coproduct and counit haveto be included in order for (A|Z(A) e) to be a Cardy algebra see Theorem 318 below The normalisation ofthe counit enters the Cardy condition (314) through the definition of ( middot )lowast

Cardy Algebras and Sewing Constraints I 901

Theorem 318 Let A be a special symmetric Frobenius C-algebra Then (A|Z(A) e)is a Cardy C|C2plusmn-algebra

The proof of this theorem makes use the following two lemmas

Lemma 319 e Z(A) rarr R(A) is an algebra map and elowast = ζA middot r

Proof It follows from [Fr Lem 229] (or by direct calculation using in particularm R(A) R(A) = ζAidR(A)) that

m R(A) (e otimes e) = ζminus1A middot Pl

R(A) m R(A) (e otimes e) (334)

Substituting e r = ζminus1A Pl

R(A) shows that e is compatible with multiplication For theunit one finds

e ηZ(A) = e r ηR(A) = ζminus1A Pl

R(A) ηR(A) = ηR(A) (335)

Thus e is an algebra map For the second statement one computes

elowast (1)= ζA

R(A)

r

R(A)

R(A)

Z(A)

R(A)

Z(A)

r

e

e

(2)=

R(A)

r

R(A)

R(A)

R(A)

Z(A)

(3)=

R(A)

r

R(A)

Z(A)

(4)= ζA middot r (336)

where in (1) the definitions (230) and (332) have been substituted step (2) is e r = ζminus1

A PlR(A) step (3) uses that R(A) is symmetric Frobenius and step (4) is again

e r = ζminus1A Pl

R(A) Lemma 320 Let A be a symmetric Frobenius algebra in C The morphism

PlR(A) m R(A)

(Pl

R(A) otimes PlR(A)

) R(A)otimes R(A) minusrarr R(A) (337)

is S-invariant

The proof of this lemma is a slightly lengthy explicit calculation and has been deferredto Appendix A2

902 L Kong I Runkel

Proof of Theorem 318 That e is an algebra map was proved in Lemma 319 The centrecondition (313) holds by property (330) of the left centre The Cardy condition (314)also is an immediate consequence of Lemma 319

ιcl-op ιlowastcl-op = e (ζA r) = PlR(A) (338)

The full centre Z(A) is a commutative symmetric Frobenius algebra It remains to provemodular invariance That θZ(A) = idZ(A) is implied by commutativity and symmetryof Z(A) [Fr Prop 225] The S-invariance condition (33) follows from Lemma 320In (337) substitute Pl

R(A) = ζA e r and then put the resulting morphism into (33)Compose the resulting equation with e otimes idW from the right (ie from the bottom) andsubstitute the definition (332) of m Z(A) This results in the statement that m Z(A) isS-invariant

The following theorem is analogous to [LR Prop 29] and [Fj Thm 426] whichroughly speaking answer the question under which circumstances the restriction of atwo-dimensional conformal field theory to the boundary already determines the entireconformal field theory The first work is set in Minkowski space and uses operator alge-bras and subfactors while the second work is set in Euclidean space and uses modulartensor categories

Theorem 321 Let (A|Acl ιcl-op) be a Cardy C|C2plusmn-algebra such that dim A = 0 andAcl is simple Then A is special and (A|Acl ιcl-op) sim= (A|Z(A) e) as Cardy algebras

Proof By Proposition 316 A is simple and special Since Acl is simple the algebramap ιcl-op Acl rarr R(A) is either zero or a monomorphism But ιcl-op ηcl = ηR(A)and so ιcl-op = 0 Thus ιcl-op is monic By Lemma 218 (ii) ζminus1

A ιlowastcl-op is epi The Cardycondition (314) implies

ιcl-op ζminus1A ιlowastcl-op = ζminus1

A PlR(A) = e r (339)

Composing this with e r from the left yields e r ιcl-op ζminus1A ιlowastcl-op = e r =

ιcl-op ζminus1A ιlowastcl-op Since ζminus1

A ιlowastcl-op is epi we have

e r ιcl-op = ιcl-op (340)

Actually (340) also follows from (313) and specialness of R(A) We will prove that

( fop fcl) (A|Acl ιcl-op) minusrarr (A|Z(A) e) where fop = idA fcl = r ιcl-op

(341)

is an isomorphism of Cardy algebrasfcl is an algebra map Compatibility with the units follows since ιcl-op is an algebramap

fcl ηcl = r ιcl-op ηcl = r ηR(A) = ηZ(A) (342)

Compatibility with the multiplication also follows since ιcl-op is an algebra map

m Z(A) ( fcl otimes fcl) = r m R(A) (e otimes e) (r otimes r) (ιcl-op otimes ιcl-op)

= r m R(A) (ιcl-op otimes ιcl-op) = r ιcl-op mcl = fcl mcl (343)

where in the second step we used (340)

Cardy Algebras and Sewing Constraints I 903

fcl is an isomorphism As above since fcl is an algebra map and since Acl is simple fcl

has to be monic By Lemma 319 rlowast = ζminus1A e Thus f lowast

cl = ιlowastcl-op rlowast = ζminus1A ιlowastcl-op e and

fcl f lowastcl = r ιcl-op ζminus1

A ιlowastcl-op e = r e r e = idZ(A) (344)

and so fcl is also epi and hence isofcl is a coalgebra map Since fcl is an algebra map so is f minus1

cl By (344) f minus1cl = f lowast

cland by Lemma 218 (iii) this implies that fcl is a also coalgebra mapThe diagram (315) commutes Commutativity of (315) is equivalent to e fcl = ιcl-opwhich holds by (340)

Let A be a special symmetric Frobenius algebra So far we have seen that (A|Z(A) e)is a Cardy algebra and that all Cardy algebras with Aop = A and simple Acl are of thisform It is now natural to ask if every simple Acl does occur as part of a Cardy algebraThe following theorem provides an affirmative answer Recall that for an A-left moduleM the object Mor otimesA M is an algebra (see eg [KR Lem 42])

Theorem 322 If Acl is a simple modular invariant commutative symmetric FrobeniusC2plusmn-algebra then there exist a simple special symmetric Frobenius C-algebra A and amorphism ιcl-op Acl rarr R(A) such that

(i) Acl sim= Z(A) as Frobenius algebras(ii) (A|Acl ιcl-op) is a Cardy C|C2plusmn-algebra

(iii) T (Acl) sim= oplusκisinJ Morκ otimesA Mκ as algebras where MκκisinJ is a set of representa-

tives of the isomorphism classes of simple A-left modules

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis haploid It then follows from Theorem 34 that Acl is special By Proposition 224T (Acl) is a special symmetric Frobenius algebra in C Thus T (Acl) = oplusi Ai where theAi are simple symmetric Frobenius algebras We will show that at least one of the Aiis special Since T (Acl) is special we have mT (Acl) T (Acl) = ζ idT (Acl) for someζ isin C

times Restricting this to the summand Ai shows mi i = ζ idAi FurthermoreεT (Acl) ηT (Acl) = ξ id1 for some ξ isin C

times But εT (Acl) ηT (Acl) = sumi εi ηi and so

at least one of the εi ηi has to be nonzero Therefore at least one of the Ai is speciallet A equiv Ai be this summand We denote the embedding A rarr T (Acl) by e0 and therestriction T (Acl) A by r0 Notice that r0 is an algebra homomorphism Define

ιcl-op = χ (r0) Acl minusrarr R(A) (345)

By Proposition 216 ιcl-op is an algebra homomorphism Next we verify the centre con-dition (313) or rather its equivalent form (320) By substituting the definitions onecan convince oneself that the commutativity mcl cAclAcl = mcl of Acl in C2plusmn implies thecondition mT (Acl) = mT (Acl) in C see [K2 Prop 36] Here T (Acl)otimes T (Acl) rarrT (Acl)otimes T (Acl) is given by

=oplusmn

(idCl

notimes cCl

m Crnotimes idCr

m

)

(cCl

m Clnotimes cminus1

Crn C

rm

)

(idCl

motimes cminus1

Cln C

rm

otimes idCrn

)

(346)

and we decomposed Acl as Acl = oplusnCln times Cr

n As a consequence we obtain the identity

r0 mT (Acl) (idAcl otimes e0) = r0 mT (Acl) (idAcl otimes e0) (347)

904 L Kong I Runkel

Using that r0 is an algebra map and that by definition ιcl-op = r0 we obtain (320)In order to show that (A|Acl ιcl-op) is a Cardy algebra it remains to show that the

Cardy condition (314) is satisfied We will demonstrate this via a detour by first provingthat Acl sim= Z(A) as Frobenius algebras

Recall the notations e and r given in (331) Using the centre condition (320) onecan check that Pl

R(A) ιcl-op = m R(A) R(A) ιcl-op By specialness of A we have

m A A = ζA idA and so together with e r = ζminus1A Pl

R(A) we get

e r ιcl-op = ιcl-op (348)

Next consider the morphism

fcl = r ιcl-op Acl minusrarr Z(A) (349)

By the same derivation as in (342) and (343) one sees that fcl is an algebra map Inparticular fcl ηcl = ηZ(A) = 0 and so fcl = 0 Since Acl is simple fcl has to be amonomorphism

By the same argument as used in the proof of Theorem 34 (ii) up to isomorphismAcl is the unique simple local Acl-(left-)module The algebra monomorphism fcl turnsZ(A) into an Acl-module Since Z(A) is commutative it is local as an Acl-module andso Z(A) sim= AoplusN

cl for some N ge 1 By construction A is a simple special symmetricFrobenius algebra Proposition 225 and Corollary 227 show that R(A) inherits all theseproperties and thus Z(A) is simple (see the comment below Eq (332)) By Theorem318 Z(A) is modular invariant and then by Theorem 34 (i) dim Z(A) = Dim C Thisimplies that N = 1 in Z(A) sim= AoplusN

cl and so fcl is in fact an isomorphismSince Acl and Z(A) are both haploid we have εZ(A) fcl = ξ εcl for some ξ isin C

timesThe counit uniquely determines the Frobenius structure on Acl and Z(A) (see eg [FRSLemma 37]) so that fcl is a coalgebra isomorphism iff ξ = 1 To compute ξ we composethe above identity with ηcl from the right Defining ζcl via εcl ηcl = ζminus1

cl Dim C middot id1 andusing (333) gives ξ = dim A ζclζ

2A By rescaling the comultiplication and the counit

of A and consequently changing ζA we can always achieve ξ = 1 This proves part (i)of the theorem

Equation (348) implies that ιcl-op = e fcl Since fcl is an isomorphism of Frobeniusalgebras by Lemmas 218 and 319 we have

ιcl-op ιlowastcl-op = e fcl f lowastcl elowast = ζA e r = Pl

R(A) (350)

Thus (A|Acl ιcl-op) is a Cardy algebra This proves part (ii) of the theoremPart (iii) can be seen as follows By [KR Prop 43] T Z(A) sim= oplusκisinJ Mor

κ otimesA Mκ asalgebras Together with the observation that T ( fcl) T (Acl) rarr T Z(A) is an isomor-phism of algebras this proves part (iii) Remark 323 Part (i) of Theorem 322 was announced by Muumlger [Muuml2] We provide anindependent proof in the setting of Cardy algebras

The above theorem together with Lemma 211 and Theorem 34 shows that a sim-ple commutative symmetric Frobenius C2plusmn-algebra Acl with dim Acl = Dim C is alwayspart of a Cardy algebra (Aop|Acl ιcl-op) for some simple special symmetric Frobeniusalgebra Aop in C However the above proof also illustrates that Aop is not unique Thisraises the question how two Cardy algebras with a given Acl can differ This question isanswered by [KR Thm 11] which in the present framework can be restated as follows

Cardy Algebras and Sewing Constraints I 905

Theorem 324 If (A(i)op |A(i)cl ι(i)cl-op) i = 1 2 are two Cardy C|C2plusmn-algebras such that

A(i)cl is simple and dim A(i)op = 0 for i = 1 2 then A(1)clsim= A(2)cl as algebras if and only

if A(1)op and A(2)op are Morita equivalent

Proof Theorem 11 in [KR] is stated for A(i)op being non-degenerate algebras and A(i)cl =Z(A(i)op ) for i = 1 2 By Proposition 316 A(i)op are simple and special for i = 1 2

Then by [KR Lem 21] A(i)op are non-degenerate algebras By Theorem 321 we have

A(i)clsim= Z(A(i)op ) as Frobenius algebras Finally by [KR Thm 11] Z(A(1)op ) sim= Z(A(2)op )

as algebras iff A(1)op and A(2)op are Morita equivalent Let Cmax(C2plusmn) be the set of equivalence classes [B] of simple modular invariant

commutative symmetric Frobenius algebras B in C2plusmn Two such algebras B and B prime areequivalent if B and B prime are isomorphic as algebras (but not necessarily as Frobenius alge-bras) Let Msimp(C) be the set of Morita classes of simple special symmetric Frobeniusalgebras in C Define the map z Msimp(C) rarr Cmax(C2plusmn) by z A rarr [Z(A)] whereA denotes the Morita class of A From Theorem 322 (i) and [KR Thm 11] we learn

Corollary 325 The map z Msimp(C) rarr Cmax(C2plusmn) is a bijection

A Appendix

A1 Proof of Lemma 27 We will show that if (F ψ F2 ψ

F0 ) is a colax tensor functor

from C1 to C2 then (G φG2 φ

G0 ) is a lax tensor functor from C2 to C1 Applying this

result to the opposed categories then gives the converse statementWe need to show that φG

0 and φG2 make the diagrams (21) and (22) commute We

first prove the commutativity of (21) Consider the following diagram

G(A)otimes (G(B)otimes G(C))

GψF2 δ

idG(A)otimesφG2 G(A)otimes G(B otimes C)

GψF2 δ

G(FG(A)otimes F(G(B)otimes G(C)))

G(F(idG(A))otimesF(φG2 ))

G(idFG(A)otimesψ F2 )

G(FG(A)otimes FG(B otimes C))

G(ρAotimesρBotimesC )

G(FG(A)otimes (FG(B)otimes FG(C)))

G(ρAotimes(ρBotimesρC )) G(A otimes (B otimes C))

(A1)

The top subdiagram is commutative because of the naturality of Gψ F2 δ The commu-

tativity of the bottom subdiagram follows from the following identities

(ρB otimes ρC ) ψ F2 = (ρB otimes ρC ) ψ F

2 ρF Fδ

= ρBotimesC FG(ρB otimes ρC ) FG(ψ F2 ) Fδ

= ρBotimesC F(φG2 ) (A2)

as a map F(G(B) otimes G(C)) rarr B otimes C The commutativity of (A1) implies that thecomposition of maps in the left column in (21) can be replaced by

G(ρA otimes (ρB otimes ρC )) G((idFG(A) otimes ψ F

2 ) ψ F2

) δ (A3)

906 L Kong I Runkel

Similarly we can show that the composition of maps in the right column in (21) can bereplaced by

G((ρA otimes ρB)otimes ρC ) G((ψ F

2 otimes idFG(C)) ψ F2

) δ (A4)

Using the commutativity of (23) it is easy to see that (21) with the left and right col-umns of (21) replaced by (A3) and (A4) respectively is commutative Hence (21) iscommutative

Now we prove the commutativity of the first diagram in (22)

φG2 (φG

0 otimes idG(A))

(1)= G(ρ12 otimes ρA) Gψ F2 δ

[(Gψ F

0 δ11)otimes idG(A)

]

(2)= G(ρ12 otimes ρA) Gψ F2 G F(Gψ0 otimes idG(A)) G F(δ11 otimes idG(A)) δ

(3)= G(ρ12 otimes ρA) G(FG(ψ F0 )otimes idFG(A)) Gψ F

2 G F(δ11 otimes idG(A)) δ(4)= G(id12 otimes ρA) G

([ψ F

0 ρF(11) (Fδ)11] otimes idFG(A)

) Gψ F

2 δ(5)= G(id12 otimes ρA) G(ψ F

0 otimes idFG(A)) Gψ F2 δ

(6)= G(id12 otimes ρA) G(lminus1FG(A)) G F(lG(A)) δ

(7)= G(lminus1A ) GρA δG lG(A)

(8)= G(lminus1A ) lG(A) (A5)

where in step (1) we substituted the definition of φG0 φ

G2 given in (210) in step (2)

we used the naturality of δ in step (3) we used the naturality of Gψ F2 in step (4) we

switched the position between Gρ12 and G FG(ψ F0 ) and the position between Gψ2 and

G F(δ12 otimes idG(A)) using the naturality of ρ and FψG2 respectively in step (5) we applied

the second identity in (28) in step (6) we used (24) in step (7) we used the naturalityof lminus1 and δ in step (8) we used the first identity in (28)

The proof of the commutativity of the second diagram in (22) is similar Thus wehave shown that G is a lax tensor functor

A2 Proof of Lemma 320 To prepare the proof recall that for a given object B isin Cthe modular group P SL(2Z) acts on the space oplusi HomC(B otimes Ui Ui ) see eg [BKSect 31] and [K3 Eq (455)] We will only need the action of S and Sminus1 Let f isinoplusi HomC(B otimes Ui Ui ) Then

S oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A6)

Cardy Algebras and Sewing Constraints I 907

Sminus1 oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A7)

By Lemma 32 to establish that (337) is S-invariant it is enough to prove the identity(34) when f is given by (337) Using (A6) and (A7) we can see that Eq (34) simplysays that oplusi j [RHS of (34)] is invariant under the action of S times S Consider the elementg of oplus jkisinIHomC2plusmn(R(A)otimes (Uor

j times Uk)Uorj times Uk) given by

g =oplusjkisinI

sumα

R(A)

R(A)

R(A)

R(A)

Uorj times Uk

Uorj times Uk

α

α

(A8)

By the above arguments proving S-invariance of (337) is equivalent to proving invari-ance of g under the action of S times S

For i isin I we denote by gi the component of g in(oplus jisinIHomC(A otimes Uor

i otimes Uorj U

orj )

)otimes (opluskisinIHomC(Ui otimes UkUk))

We view the second Hom-space in the above tensor product as a Hom-space in C+ insteadof Cminus It is enough to show that gi is invariant under the action of S times Sminus1 Note thatthe action of Sminus1 in C+ is equivalent to that of S in Cminus

The morphism gi can be canonically identified with a bilinear pairing

( middot middot )i ⎛⎝oplus

jisinIHomC(Uor

j A otimes Uori otimes Uor

j )

⎞⎠

times(oplus

kisinIHomC(UkUi otimes Uk)

)minusrarr C (A9)

as follows For h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) we set

(h1 h2)i = (dim U j dim Uk)minus1 trUor

j timesUk[gi (h1 times h2)] (A10)

908 L Kong I Runkel

When substituting the explicit form of the product m R(A) of R(A) = (A times 1) otimes R(1)after a short calculation one finds

(h1 h2)i =sumα

1

dim U j

A

A

A

A

A A

A α

α

h1

h2

U jUk

Uorj

Ui

Uk (A11)

Here the top morphism PlR(A) has been simplified with the help of the identity

PR(A) m R(A) (PR(A) otimes PR(A))

=(oplus

iisinI((m A A)otimes idUor

i)times idUi

) m R(A) (PR(A) otimes PR(A)) (A12)

which can be checked by direct calculation along the same lines as in the proof of [FrLem 310]

The action of the modular transformation S on oplusiisinI(B otimes Ui Ui ) for B isin C nat-urally induces an action on oplusiisinI(Ui B otimes Ui ) [K3 Prop 514] which we denote bySlowast In the present case we get an action of Slowast on oplus jisinIHomC(Uor

j A otimes Uori otimes Uor

j ) and

opluskisinIHomC(UkUi otimes Uk) Then to show gi is invariant under the action of S times Sminus1

amounts to showing that

(h1 h2)i = ((Sminus1)lowast h1 Slowast h2)i (A13)

Cardy Algebras and Sewing Constraints I 909

for all h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) We have

((Sminus1)lowast h1 Slowast h2)i =sum

mnα

dim Un

DimC

A

A

A

A

A A

A

Uk

α

α

h1

h2

Ui

U j

UmUn

(A14)

Now drag the upper vertex indexed by α in the above graph along its Uorm -leg until it

meets the lower vertex also indexed by α then sum over α and m This gives

((Sminus1)lowast h1 Slowast h2)i =sum

n

dim Un

DimC

A

A

A

A

A A

A

Uk

Un

h1

h2

Ui

U j

(A15)

910 L Kong I Runkel

If we just look at the neighbourhood of the Un-loop in the above graph we see thefollowing subgraphs

sumn

dim Un

DimC

A Aor

Uorj U j

Uk Uork

Un=

sumα

1

dim U j

A Aor Uk Uork

Uorj U j

αα

(A16)

where we have applied [BK Cor 3111] Substituting this subgraph back to the originalgraph in (A15) we obtain

((Sminus1)lowast h1 Slowast h2)i =sumα

1

dim U j

A

A

A

A

A A

A

A

A

Uorj

α

α

h1

h2m A

Ui

U jUk

Uk

(A17)

The graph in (A17) is equal to that in (A11) In order to see this we first drag theldquobubblerdquo (m A A) along A lines and through the m A vertex (because m A A is abimodule map) until it reaches the lower-left leg of the upper vertex indexed by α Thendrag the m A vertex along the (red) dotted line in the above graph Finally we applythe associativity of A (A12) and [Fr Lem 311] Then we see that the graph in (A17)exactly matches with the one in (A11)

Acknowledgements We would like to thank the organisers of the Oberwolfach Arbeitsgemeinschaft ldquoAlge-braic structures in conformal field theoriesrdquo (April 2007) where this work was started for an inspiring meetingWe would further like to thank the Hausdorff Institute for Mathematics in Bonn and the organisers of the stim-ulating meeting ldquoGeometry and Physicsrdquo (May 2008) We are indebted to Alexei Davydov Jens FjelstadJuumlrgen Fuchs Yi-Zhi Huang Alexei Kitaev Urs Schreiber Christoph Schweigert Stephan Stolz and PeterTeichner for helpful discussions andor comments on a draft of this paper The research of IR was partiallysupported by the EPSRC First Grant EPE0050471 the PPARC rolling grant PPC5071451 and the MarieCurie network lsquoSuperstring Theoryrsquo (MRTN-CT-2004-512194)

Cardy Algebras and Sewing Constraints I 911

References

[AN] Alexeevski A Natanzon SM Noncommutative two-dimensional topological field theories andhurwitz numbers for real algebraic curves Sel Math New Ser 12 307ndash377 (2006)

[Bi] Bichon J Cosovereign hopf algebras J Pure Appl Alg 157 121ndash133 (2001)[BK] Bakalov B Kirillov AA Lectures on Tensor Categories and Modular Functors Providence

RI Amer Math Soc 2001[C1] Cardy JL Operator content of two-dimensional conformal invariant theories Nucl Phys

B 270 186ndash204 (1986)[C2] Cardy JL Boundary conditions fusion rules and the verlinde formula Nucl Phys B 324

581ndash596 (1989)[CL] Cardy JL Lewellen DC Bulk and boundary operators in conformal field theory Phys Lett

B 259 274ndash278 (1991)[DMZ] Dong C-Y Mason G Zhu Y-C Discrete series of the Virasoro algebra and the moonshine

module In Algebraic Groups and Their Generalizations Quantum and infinite-dimensional Meth-ods Proc Symp Pure Math 56 Part 2 Providence RI Amer MathSoc 1994 pp 295ndash316

[DP] Day B Pastro C Note on frobenius monoidal functors New York J Math 14 733ndash742 (2008)[ENO1] Etingof PI Nikshych D Ostrik V On fusion categories Ann Math 162 581ndash642 (2005)[ENO2] Etingof PI Nikshych D Ostrik V An analogue of radfordrsquos s4 formula for finite tensor

categories Int Math Research Notices 54 2915ndash2933 (2004)[Fe] Felder G Froumlhlich J Fuchs J Schweigert C Correlation functions and boundary conditions in

rational conformal field theory and three-dimensional topology Comp Math 131 189ndash237 (2002)[FHL] Frenkel IB Huang Y-Z Lepowsky J On axiomatic approaches to vertex operator algebras

and modules Mem Amer Math Soc 104 (1993)[Fj] Fjelstad J Fuchs J Runkel I Schweigert C Uniqueness of openclosed rational cft with

given algebra of open states Adv Theor Math Phys 12 1283ndash1375 (2008)[FK] Froumlhlich J King C The chern-simons theory and knot polynomials Commun Math

Phys 126 167ndash199 (1989)[Fr] Froumlhlich J Fuchs J Runkel I Schweigert C Correspondences of ribbon categories Adv

Math 199 192ndash329 (2006)[FRS] Fuchs J Runkel I Schweigert C Tft construction of rcft correlators I partition functions Nucl

Phys B 646 353ndash497 (2002)[FS] Fuchs J Schweigert C Category theory for conformal boundary conditions Fields Inst Com-

mun 39 25ndash71 (2003)[H1] Huang Y-Z Two-dimensional conformal geometry and vertex operator algebras Progress in

Mathematics Vol 148 Boston Birkhaumluser 1997[H2] Huang Y-Z Rigidity and modularity of vertex tensor categories Comm Contemp Math 10

871ndash911 (2008)[HK1] Huang Y-Z Kong L Open-string vertex algebra category and operads Commun Math

Phys 250 433ndash471 (2004)[HK2] Huang Y-Z Kong L Full field algebras Commun Math Phys 272 345ndash396 (2007)[HK3] Huang Y-Z Kong L Modular invariance for conformal full field algebras httparxivorgabs

math0609570v2[mathQA] 2006[HL] Huang Y-Z Lepowsky J Tensor products of modules for a vertex operator algebra and vertex

tensor categories In Lie Theory and Geometry in honor of Bertram Kostant ed R BrylinskiJ-L Brylinski V Guillemin V Kac Boston Birkhaumluser 1994 pp 349ndash383

[JS] Joyal A Street R Braided tensor categories Adv Math 102 20ndash78 (1993)[K1] Kong L Full field algebras operads and tensor categories Adv Math 213 271ndash340 (2007)[K2] Kong L Open-closed field algebras Commun Math Phys 280 207ndash261 (2008)[K3] Kong L Cardy condition for open-closed field algebras Commun Math Phys 283 25ndash92 (2008)[Ki] Kitaev A Private communication[KO] Kirillov AA Ostrik V On q-analog of mckay correspondence and ade classification of sl(2)

conformal field theories Adv Math 171 183ndash227 (2002)[KR] Kong L Runkel I Morita classes of algebras in modular tensor categories Adv Math 219

1548ndash1576 (2008)[Ld] Lauda AD Frobenius algebras and ambidextrous adjunctions Theo Appl Cat 16 84ndash

122 (2006)[Li1] Li H-S Regular representations of vertex operator algebras Commun Contemp Math 4

639ndash683 (2002)[Li2] Li H-S Regular representations and huang-lepowsky tensor functors for vertex operator

algebras J Alge 255 423ndash462 (2002)

Cardy Algebras and Sewing Constraints I 897

commutesLet (Aop|Acl ιcl-op) be a Cardy C|C2plusmn-algebra Define the morphism

ιcl-op = χminus1(ιcl-op) T (Acl) minusrarr Aop (317)

Decompose Acl as Acl = oplusNn=1Cl

n times Crn such that Cl

1 times Cr1 = ηcl(1 times 1) We use ι(n)cl-op

to denote the restriction of ιcl-op to Cln times Cr

n and ι(n)cl-op to denote the restriction of ιcl-op

to Cln otimes Cr

n We introduce the following graphical notation

ι(n)cl-op =

Cln Cr

n

Aop

(318)

By (243) ιcl-op can be expressed in terms of ιcl-op as follows

ιcl-op =Noplus

n=1

oplusiisinI

sumα

Cln

Crn

Aop

α

Uori

times α

Crn

Ui

(319)

Lemma 311 The centre condition (313) is equivalent to the following condition in C

Cln Cr

n

Aop

Aop

Aop

=

Cln Cr

n

Aop

Aop

Aop

for n = 1 N (320)

Proof First insert (319) and the definition (217) of m R(Aop) into (313) Then apply thecommutativity of R(1) to the left hand side of (313) The equivalence between (313)and (320) follows immediately Remark 312 The centre condition (320) is very natural from the open-closed confor-mal field theory point of view Correlators on the upper half plane are expressed in termsof conformal blocks on the full complex plane The objects Cl

n and Crn are associated to

the field insertion at a point z in the upper half plane and at the complex conjugate pointz in the lower half plane respectively The object Aop corresponds to a field inserted ata point r on the real axis The centre condition (320) simply says that the correlationfunctions in the disjoint domains |z| gt r gt 0 and r gt |z| gt 0 are analytic continuationsof each other see [K2 Prop 118]

898 L Kong I Runkel

Recall that we define ιlowastcl-op Aop rarr T (Acl) as in (230) We introduce the graphicalnotation

ιlowastcl-op =Noplus

n=1

Cln Cr

n

Aop

=Noplus

n=1

Cln Cr

n

Aop

Aop T (minus1Acl) (321)

where the second equality follows from (215) (221) and (259)

Lemma 313 The Cardy condition (314) is equivalent to the following identity in C

Noplusn=1

DimCdim Ui

sumα

Cln

Crn

Crn

α

α

Aop

Aop Uori

Uori

=Aop

Aop

Aop

Uori

Uori

op

mop for all i isin I (322)

Proof By (244) (312) and (319) it is easy to see that (322) is equivalent to thefollowing identity

ιcl-op χ(ιlowastcl-op) = PlR(Aop)

(323)

Therefore it is enough to show that

χ(ιlowastcl-op) = ιlowastcl-op (324)

We have

χminus1(ιlowastcl-op)(1)= T (ιlowastcl-op) δ (2)= T (ιcl-op)

lowast δlowastlowast (3)= (ρ T (ιcl-op)

)lowast (4)= (ιcl-op

)lowast (325)

In step (1) we use the expression (253) for χminus1 step (2) follows from Lemma 218 (v)and Lemma 219 Step (3) is Lemma 218 (i) and Lemma 228 and finally step (4)amounts to substituting (253) and (317) Acting with χ on both sides of the aboveequality produces (324)

Combining Lemmas 311 and 313 and Proposition 216 we obtain the followingequivalent definition of Cardy C|C2plusmn-algebra (recall the graphical notation (318) forιcl-op and (321) for ιlowastcl-op)

Cardy Algebras and Sewing Constraints I 899

Definition 314 (CardyC|C2plusmn-algebra II) A CardyC-algebra is a triple (Aop|Acl ιcl-op)where Acl is a commutative symmetric Frobenius C2plusmn-algebra satisfying property (34)with f = mcl Aop is a symmetric Frobenius C-algebra and ιcl-op T (Acl) rarr Aop isan algebra homomorphism satisfying the conditions (320) and (322)

Remark 315 Up to a choice of normalisation Definition 314 is the same as the orig-inal one in [K3 Def 613] The difference between the two definitions is the factorDim C dim Ui on the left hand side of (322) which in [K3 Def 613] is given byradic

Dim C dim Ui The two definitions are related by rescaling the coproduct cl andcounit εcl of Acl by 1

radicDim C and

radicDim C respectively We chose the convention in

(322) to remove all dimension factors from the expression (314) for the Cardy condition

33 Uniqueness and existence theorems In this subsection we investigate the structureof Cardy algebras We start with the following proposition which when combined withthe results of part II provides an alternative proof of [Fj Prop 422]

Proposition 316 Let (Aop|Acl ιcl-op) be a Cardy C|C2plusmn-algebra If Acl is simple anddim Aop = 0 then Aop is simple and special

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis therefore haploid Restricting the Cardy condition (322) to the case Ui = 1 andcomposing both sides with εop from the left we see that εop kills all terms associated toU j times 1 isin Acl in the sum except for a single 1 times 1 term Thus we obtain the followingidentity

β εop = dAop (mop otimes idAorop) (idAop otimes bAop) (326)

where β isin C Composing with ηop from the right in turn implies that βεop ηop =dim Aop which is nonzero by assumption Thus also β = 0 and εop is a nonzero multipleof the morphism on the right hand side of (326) By [FRS Lem 311] Aop is special

Since Aop is a special Frobenius algebra Aop is semi-simple as an Aop-bimodule(apply [FS Prop 524] to Aop tensored with its opposite algebra) Suppose Aop is not

simple so that we can write Aop = A(1)op oplus A(2)op for nonzero Aop-bimodules A(1)op and A(2)op We denote the canonical embeddings and projections associated to this decompositionas ι12 and π12 We have the identities

mop (ι1 otimes ι2) = 0 εop ηop =2sum

i=1

εop ιi πi ηop (327)

The first identity follows since π1 mop (ι1 otimes ι2) = 0 (as mop gives the left action of

Aop on Aop and hence it preserves A(2)op ) and similarly π2 mop (ι1 otimes ι2) = 0 Thesecond identity is just the completeness of ι12 π12

Since εop ηop = 0 without losing generality we can assume εop ι1 π1 ηop = 0Using that π2 is a bimodule map we compute

π2 [LHS of (322)]Ui =1 ι1 π1 ηop = π2 [RHS of (322)]Ui =1 ι1 π1 ηop

= PlA(2)op

π2 ι1 π1 ηop = 0 (328)

900 L Kong I Runkel

On the other hand using that Acl is haploid that ιcl-op is an algebra map and thatι lowastcl-op is a coalgebra map one can check that the left-hand side of (328) is equal toλ(εop ι1 π1 ηop) π2 ηop for some λ = 0 This implies that π2 ηop = 0 Thusηop = sum

i ιi πi ηop = ι1 π1 ηop Hence we have

0 = π2 ι2 = π2 mop (ηop otimes ι2) = π2 mop ((ι1 π1 ηop)otimes ι2) (329)

However the right-hand side is zero by (327) This is a contradiction and hence Aopmust be simple

To formulate the next theorem we need the notion of the full centre of an algebra [FjDef 49] Recall that an algebra A in a braided tensor category has a left centre and aright centre [VZO] both of which are sub-algebras of A Of these two we will onlyneed the left centre The following definition is [Fr Def 231] which in our setting isequivalent to that of [VZO]

Definition 317 Let A be a symmetric special Frobenius algebra such that m A A =ζA idA

(i) The left centre Cl(A) of A is the image of the idempotent ζminus1A Pl

A(ii) The full centre Z(A) is Cl(R(A))

That ζminus1A Pl

A is an idempotent follows from [FRS Lem 52] when keeping track ofthe factors ζA ([FRS] assumes normalised-special ie ζA = 1) Note that Cl(A) is againan object of C while Z(A) is an object of C2plusmn Let el

A Cl(A) rarr A be the embeddingof Cl(A) into A The left centre is in fact the maximal subobject of A such that

m A cAA (elA otimes idA) = m A (330)

see [Fr Lem 232] This observation explains the name left centre and also makes theconnection to [O Def 15]

The full centre is by definition the image of the idempotent ζminus1A Pl

R(A) R(A) rarrR(A) Since C2plusmn is abelian the idempotent splits and we obtain the embedding andrestriction morphisms

e Z(A) rarr R(A) and r R(A) Z(A) (331)

which obey r e = idZ(A) and e r = ζminus1A Pl

R(A) It follows from Proposition 225 and

[Fr Prop 237] that Z(A) is a commutative symmetric Frobenius algebra in C2plusmn withstructure morphisms5

m Z(A) = r m R(A) (e otimes e) ηZ(A) = r ηR(A)

Z(A) = ζA middot (r otimes r) R(A) e εZ(A) = ζminus1A middot εR(A) e

(332)

Moreover if A is simple then Z(A) is simple and if A is simple and dim A = 0 thenZ(A) is simple and special The normalisation of the counit is such that

εZ(A) ηZ(A) = ζminus2A dim A DimC (333)

5 The normalisation of product and unit is the standard one The factors in the coproduct and counit haveto be included in order for (A|Z(A) e) to be a Cardy algebra see Theorem 318 below The normalisation ofthe counit enters the Cardy condition (314) through the definition of ( middot )lowast

Cardy Algebras and Sewing Constraints I 901

Theorem 318 Let A be a special symmetric Frobenius C-algebra Then (A|Z(A) e)is a Cardy C|C2plusmn-algebra

The proof of this theorem makes use the following two lemmas

Lemma 319 e Z(A) rarr R(A) is an algebra map and elowast = ζA middot r

Proof It follows from [Fr Lem 229] (or by direct calculation using in particularm R(A) R(A) = ζAidR(A)) that

m R(A) (e otimes e) = ζminus1A middot Pl

R(A) m R(A) (e otimes e) (334)

Substituting e r = ζminus1A Pl

R(A) shows that e is compatible with multiplication For theunit one finds

e ηZ(A) = e r ηR(A) = ζminus1A Pl

R(A) ηR(A) = ηR(A) (335)

Thus e is an algebra map For the second statement one computes

elowast (1)= ζA

R(A)

r

R(A)

R(A)

Z(A)

R(A)

Z(A)

r

e

e

(2)=

R(A)

r

R(A)

R(A)

R(A)

Z(A)

(3)=

R(A)

r

R(A)

Z(A)

(4)= ζA middot r (336)

where in (1) the definitions (230) and (332) have been substituted step (2) is e r = ζminus1

A PlR(A) step (3) uses that R(A) is symmetric Frobenius and step (4) is again

e r = ζminus1A Pl

R(A) Lemma 320 Let A be a symmetric Frobenius algebra in C The morphism

PlR(A) m R(A)

(Pl

R(A) otimes PlR(A)

) R(A)otimes R(A) minusrarr R(A) (337)

is S-invariant

The proof of this lemma is a slightly lengthy explicit calculation and has been deferredto Appendix A2

902 L Kong I Runkel

Proof of Theorem 318 That e is an algebra map was proved in Lemma 319 The centrecondition (313) holds by property (330) of the left centre The Cardy condition (314)also is an immediate consequence of Lemma 319

ιcl-op ιlowastcl-op = e (ζA r) = PlR(A) (338)

The full centre Z(A) is a commutative symmetric Frobenius algebra It remains to provemodular invariance That θZ(A) = idZ(A) is implied by commutativity and symmetryof Z(A) [Fr Prop 225] The S-invariance condition (33) follows from Lemma 320In (337) substitute Pl

R(A) = ζA e r and then put the resulting morphism into (33)Compose the resulting equation with e otimes idW from the right (ie from the bottom) andsubstitute the definition (332) of m Z(A) This results in the statement that m Z(A) isS-invariant

The following theorem is analogous to [LR Prop 29] and [Fj Thm 426] whichroughly speaking answer the question under which circumstances the restriction of atwo-dimensional conformal field theory to the boundary already determines the entireconformal field theory The first work is set in Minkowski space and uses operator alge-bras and subfactors while the second work is set in Euclidean space and uses modulartensor categories

Theorem 321 Let (A|Acl ιcl-op) be a Cardy C|C2plusmn-algebra such that dim A = 0 andAcl is simple Then A is special and (A|Acl ιcl-op) sim= (A|Z(A) e) as Cardy algebras

Proof By Proposition 316 A is simple and special Since Acl is simple the algebramap ιcl-op Acl rarr R(A) is either zero or a monomorphism But ιcl-op ηcl = ηR(A)and so ιcl-op = 0 Thus ιcl-op is monic By Lemma 218 (ii) ζminus1

A ιlowastcl-op is epi The Cardycondition (314) implies

ιcl-op ζminus1A ιlowastcl-op = ζminus1

A PlR(A) = e r (339)

Composing this with e r from the left yields e r ιcl-op ζminus1A ιlowastcl-op = e r =

ιcl-op ζminus1A ιlowastcl-op Since ζminus1

A ιlowastcl-op is epi we have

e r ιcl-op = ιcl-op (340)

Actually (340) also follows from (313) and specialness of R(A) We will prove that

( fop fcl) (A|Acl ιcl-op) minusrarr (A|Z(A) e) where fop = idA fcl = r ιcl-op

(341)

is an isomorphism of Cardy algebrasfcl is an algebra map Compatibility with the units follows since ιcl-op is an algebramap

fcl ηcl = r ιcl-op ηcl = r ηR(A) = ηZ(A) (342)

Compatibility with the multiplication also follows since ιcl-op is an algebra map

m Z(A) ( fcl otimes fcl) = r m R(A) (e otimes e) (r otimes r) (ιcl-op otimes ιcl-op)

= r m R(A) (ιcl-op otimes ιcl-op) = r ιcl-op mcl = fcl mcl (343)

where in the second step we used (340)

Cardy Algebras and Sewing Constraints I 903

fcl is an isomorphism As above since fcl is an algebra map and since Acl is simple fcl

has to be monic By Lemma 319 rlowast = ζminus1A e Thus f lowast

cl = ιlowastcl-op rlowast = ζminus1A ιlowastcl-op e and

fcl f lowastcl = r ιcl-op ζminus1

A ιlowastcl-op e = r e r e = idZ(A) (344)

and so fcl is also epi and hence isofcl is a coalgebra map Since fcl is an algebra map so is f minus1

cl By (344) f minus1cl = f lowast

cland by Lemma 218 (iii) this implies that fcl is a also coalgebra mapThe diagram (315) commutes Commutativity of (315) is equivalent to e fcl = ιcl-opwhich holds by (340)

Let A be a special symmetric Frobenius algebra So far we have seen that (A|Z(A) e)is a Cardy algebra and that all Cardy algebras with Aop = A and simple Acl are of thisform It is now natural to ask if every simple Acl does occur as part of a Cardy algebraThe following theorem provides an affirmative answer Recall that for an A-left moduleM the object Mor otimesA M is an algebra (see eg [KR Lem 42])

Theorem 322 If Acl is a simple modular invariant commutative symmetric FrobeniusC2plusmn-algebra then there exist a simple special symmetric Frobenius C-algebra A and amorphism ιcl-op Acl rarr R(A) such that

(i) Acl sim= Z(A) as Frobenius algebras(ii) (A|Acl ιcl-op) is a Cardy C|C2plusmn-algebra

(iii) T (Acl) sim= oplusκisinJ Morκ otimesA Mκ as algebras where MκκisinJ is a set of representa-

tives of the isomorphism classes of simple A-left modules

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis haploid It then follows from Theorem 34 that Acl is special By Proposition 224T (Acl) is a special symmetric Frobenius algebra in C Thus T (Acl) = oplusi Ai where theAi are simple symmetric Frobenius algebras We will show that at least one of the Aiis special Since T (Acl) is special we have mT (Acl) T (Acl) = ζ idT (Acl) for someζ isin C

times Restricting this to the summand Ai shows mi i = ζ idAi FurthermoreεT (Acl) ηT (Acl) = ξ id1 for some ξ isin C

times But εT (Acl) ηT (Acl) = sumi εi ηi and so

at least one of the εi ηi has to be nonzero Therefore at least one of the Ai is speciallet A equiv Ai be this summand We denote the embedding A rarr T (Acl) by e0 and therestriction T (Acl) A by r0 Notice that r0 is an algebra homomorphism Define

ιcl-op = χ (r0) Acl minusrarr R(A) (345)

By Proposition 216 ιcl-op is an algebra homomorphism Next we verify the centre con-dition (313) or rather its equivalent form (320) By substituting the definitions onecan convince oneself that the commutativity mcl cAclAcl = mcl of Acl in C2plusmn implies thecondition mT (Acl) = mT (Acl) in C see [K2 Prop 36] Here T (Acl)otimes T (Acl) rarrT (Acl)otimes T (Acl) is given by

=oplusmn

(idCl

notimes cCl

m Crnotimes idCr

m

)

(cCl

m Clnotimes cminus1

Crn C

rm

)

(idCl

motimes cminus1

Cln C

rm

otimes idCrn

)

(346)

and we decomposed Acl as Acl = oplusnCln times Cr

n As a consequence we obtain the identity

r0 mT (Acl) (idAcl otimes e0) = r0 mT (Acl) (idAcl otimes e0) (347)

904 L Kong I Runkel

Using that r0 is an algebra map and that by definition ιcl-op = r0 we obtain (320)In order to show that (A|Acl ιcl-op) is a Cardy algebra it remains to show that the

Cardy condition (314) is satisfied We will demonstrate this via a detour by first provingthat Acl sim= Z(A) as Frobenius algebras

Recall the notations e and r given in (331) Using the centre condition (320) onecan check that Pl

R(A) ιcl-op = m R(A) R(A) ιcl-op By specialness of A we have

m A A = ζA idA and so together with e r = ζminus1A Pl

R(A) we get

e r ιcl-op = ιcl-op (348)

Next consider the morphism

fcl = r ιcl-op Acl minusrarr Z(A) (349)

By the same derivation as in (342) and (343) one sees that fcl is an algebra map Inparticular fcl ηcl = ηZ(A) = 0 and so fcl = 0 Since Acl is simple fcl has to be amonomorphism

By the same argument as used in the proof of Theorem 34 (ii) up to isomorphismAcl is the unique simple local Acl-(left-)module The algebra monomorphism fcl turnsZ(A) into an Acl-module Since Z(A) is commutative it is local as an Acl-module andso Z(A) sim= AoplusN

cl for some N ge 1 By construction A is a simple special symmetricFrobenius algebra Proposition 225 and Corollary 227 show that R(A) inherits all theseproperties and thus Z(A) is simple (see the comment below Eq (332)) By Theorem318 Z(A) is modular invariant and then by Theorem 34 (i) dim Z(A) = Dim C Thisimplies that N = 1 in Z(A) sim= AoplusN

cl and so fcl is in fact an isomorphismSince Acl and Z(A) are both haploid we have εZ(A) fcl = ξ εcl for some ξ isin C

timesThe counit uniquely determines the Frobenius structure on Acl and Z(A) (see eg [FRSLemma 37]) so that fcl is a coalgebra isomorphism iff ξ = 1 To compute ξ we composethe above identity with ηcl from the right Defining ζcl via εcl ηcl = ζminus1

cl Dim C middot id1 andusing (333) gives ξ = dim A ζclζ

2A By rescaling the comultiplication and the counit

of A and consequently changing ζA we can always achieve ξ = 1 This proves part (i)of the theorem

Equation (348) implies that ιcl-op = e fcl Since fcl is an isomorphism of Frobeniusalgebras by Lemmas 218 and 319 we have

ιcl-op ιlowastcl-op = e fcl f lowastcl elowast = ζA e r = Pl

R(A) (350)

Thus (A|Acl ιcl-op) is a Cardy algebra This proves part (ii) of the theoremPart (iii) can be seen as follows By [KR Prop 43] T Z(A) sim= oplusκisinJ Mor

κ otimesA Mκ asalgebras Together with the observation that T ( fcl) T (Acl) rarr T Z(A) is an isomor-phism of algebras this proves part (iii) Remark 323 Part (i) of Theorem 322 was announced by Muumlger [Muuml2] We provide anindependent proof in the setting of Cardy algebras

The above theorem together with Lemma 211 and Theorem 34 shows that a sim-ple commutative symmetric Frobenius C2plusmn-algebra Acl with dim Acl = Dim C is alwayspart of a Cardy algebra (Aop|Acl ιcl-op) for some simple special symmetric Frobeniusalgebra Aop in C However the above proof also illustrates that Aop is not unique Thisraises the question how two Cardy algebras with a given Acl can differ This question isanswered by [KR Thm 11] which in the present framework can be restated as follows

Cardy Algebras and Sewing Constraints I 905

Theorem 324 If (A(i)op |A(i)cl ι(i)cl-op) i = 1 2 are two Cardy C|C2plusmn-algebras such that

A(i)cl is simple and dim A(i)op = 0 for i = 1 2 then A(1)clsim= A(2)cl as algebras if and only

if A(1)op and A(2)op are Morita equivalent

Proof Theorem 11 in [KR] is stated for A(i)op being non-degenerate algebras and A(i)cl =Z(A(i)op ) for i = 1 2 By Proposition 316 A(i)op are simple and special for i = 1 2

Then by [KR Lem 21] A(i)op are non-degenerate algebras By Theorem 321 we have

A(i)clsim= Z(A(i)op ) as Frobenius algebras Finally by [KR Thm 11] Z(A(1)op ) sim= Z(A(2)op )

as algebras iff A(1)op and A(2)op are Morita equivalent Let Cmax(C2plusmn) be the set of equivalence classes [B] of simple modular invariant

commutative symmetric Frobenius algebras B in C2plusmn Two such algebras B and B prime areequivalent if B and B prime are isomorphic as algebras (but not necessarily as Frobenius alge-bras) Let Msimp(C) be the set of Morita classes of simple special symmetric Frobeniusalgebras in C Define the map z Msimp(C) rarr Cmax(C2plusmn) by z A rarr [Z(A)] whereA denotes the Morita class of A From Theorem 322 (i) and [KR Thm 11] we learn

Corollary 325 The map z Msimp(C) rarr Cmax(C2plusmn) is a bijection

A Appendix

A1 Proof of Lemma 27 We will show that if (F ψ F2 ψ

F0 ) is a colax tensor functor

from C1 to C2 then (G φG2 φ

G0 ) is a lax tensor functor from C2 to C1 Applying this

result to the opposed categories then gives the converse statementWe need to show that φG

0 and φG2 make the diagrams (21) and (22) commute We

first prove the commutativity of (21) Consider the following diagram

G(A)otimes (G(B)otimes G(C))

GψF2 δ

idG(A)otimesφG2 G(A)otimes G(B otimes C)

GψF2 δ

G(FG(A)otimes F(G(B)otimes G(C)))

G(F(idG(A))otimesF(φG2 ))

G(idFG(A)otimesψ F2 )

G(FG(A)otimes FG(B otimes C))

G(ρAotimesρBotimesC )

G(FG(A)otimes (FG(B)otimes FG(C)))

G(ρAotimes(ρBotimesρC )) G(A otimes (B otimes C))

(A1)

The top subdiagram is commutative because of the naturality of Gψ F2 δ The commu-

tativity of the bottom subdiagram follows from the following identities

(ρB otimes ρC ) ψ F2 = (ρB otimes ρC ) ψ F

2 ρF Fδ

= ρBotimesC FG(ρB otimes ρC ) FG(ψ F2 ) Fδ

= ρBotimesC F(φG2 ) (A2)

as a map F(G(B) otimes G(C)) rarr B otimes C The commutativity of (A1) implies that thecomposition of maps in the left column in (21) can be replaced by

G(ρA otimes (ρB otimes ρC )) G((idFG(A) otimes ψ F

2 ) ψ F2

) δ (A3)

906 L Kong I Runkel

Similarly we can show that the composition of maps in the right column in (21) can bereplaced by

G((ρA otimes ρB)otimes ρC ) G((ψ F

2 otimes idFG(C)) ψ F2

) δ (A4)

Using the commutativity of (23) it is easy to see that (21) with the left and right col-umns of (21) replaced by (A3) and (A4) respectively is commutative Hence (21) iscommutative

Now we prove the commutativity of the first diagram in (22)

φG2 (φG

0 otimes idG(A))

(1)= G(ρ12 otimes ρA) Gψ F2 δ

[(Gψ F

0 δ11)otimes idG(A)

]

(2)= G(ρ12 otimes ρA) Gψ F2 G F(Gψ0 otimes idG(A)) G F(δ11 otimes idG(A)) δ

(3)= G(ρ12 otimes ρA) G(FG(ψ F0 )otimes idFG(A)) Gψ F

2 G F(δ11 otimes idG(A)) δ(4)= G(id12 otimes ρA) G

([ψ F

0 ρF(11) (Fδ)11] otimes idFG(A)

) Gψ F

2 δ(5)= G(id12 otimes ρA) G(ψ F

0 otimes idFG(A)) Gψ F2 δ

(6)= G(id12 otimes ρA) G(lminus1FG(A)) G F(lG(A)) δ

(7)= G(lminus1A ) GρA δG lG(A)

(8)= G(lminus1A ) lG(A) (A5)

where in step (1) we substituted the definition of φG0 φ

G2 given in (210) in step (2)

we used the naturality of δ in step (3) we used the naturality of Gψ F2 in step (4) we

switched the position between Gρ12 and G FG(ψ F0 ) and the position between Gψ2 and

G F(δ12 otimes idG(A)) using the naturality of ρ and FψG2 respectively in step (5) we applied

the second identity in (28) in step (6) we used (24) in step (7) we used the naturalityof lminus1 and δ in step (8) we used the first identity in (28)

The proof of the commutativity of the second diagram in (22) is similar Thus wehave shown that G is a lax tensor functor

A2 Proof of Lemma 320 To prepare the proof recall that for a given object B isin Cthe modular group P SL(2Z) acts on the space oplusi HomC(B otimes Ui Ui ) see eg [BKSect 31] and [K3 Eq (455)] We will only need the action of S and Sminus1 Let f isinoplusi HomC(B otimes Ui Ui ) Then

S oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A6)

Cardy Algebras and Sewing Constraints I 907

Sminus1 oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A7)

By Lemma 32 to establish that (337) is S-invariant it is enough to prove the identity(34) when f is given by (337) Using (A6) and (A7) we can see that Eq (34) simplysays that oplusi j [RHS of (34)] is invariant under the action of S times S Consider the elementg of oplus jkisinIHomC2plusmn(R(A)otimes (Uor

j times Uk)Uorj times Uk) given by

g =oplusjkisinI

sumα

R(A)

R(A)

R(A)

R(A)

Uorj times Uk

Uorj times Uk

α

α

(A8)

By the above arguments proving S-invariance of (337) is equivalent to proving invari-ance of g under the action of S times S

For i isin I we denote by gi the component of g in(oplus jisinIHomC(A otimes Uor

i otimes Uorj U

orj )

)otimes (opluskisinIHomC(Ui otimes UkUk))

We view the second Hom-space in the above tensor product as a Hom-space in C+ insteadof Cminus It is enough to show that gi is invariant under the action of S times Sminus1 Note thatthe action of Sminus1 in C+ is equivalent to that of S in Cminus

The morphism gi can be canonically identified with a bilinear pairing

( middot middot )i ⎛⎝oplus

jisinIHomC(Uor

j A otimes Uori otimes Uor

j )

⎞⎠

times(oplus

kisinIHomC(UkUi otimes Uk)

)minusrarr C (A9)

as follows For h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) we set

(h1 h2)i = (dim U j dim Uk)minus1 trUor

j timesUk[gi (h1 times h2)] (A10)

908 L Kong I Runkel

When substituting the explicit form of the product m R(A) of R(A) = (A times 1) otimes R(1)after a short calculation one finds

(h1 h2)i =sumα

1

dim U j

A

A

A

A

A A

A α

α

h1

h2

U jUk

Uorj

Ui

Uk (A11)

Here the top morphism PlR(A) has been simplified with the help of the identity

PR(A) m R(A) (PR(A) otimes PR(A))

=(oplus

iisinI((m A A)otimes idUor

i)times idUi

) m R(A) (PR(A) otimes PR(A)) (A12)

which can be checked by direct calculation along the same lines as in the proof of [FrLem 310]

The action of the modular transformation S on oplusiisinI(B otimes Ui Ui ) for B isin C nat-urally induces an action on oplusiisinI(Ui B otimes Ui ) [K3 Prop 514] which we denote bySlowast In the present case we get an action of Slowast on oplus jisinIHomC(Uor

j A otimes Uori otimes Uor

j ) and

opluskisinIHomC(UkUi otimes Uk) Then to show gi is invariant under the action of S times Sminus1

amounts to showing that

(h1 h2)i = ((Sminus1)lowast h1 Slowast h2)i (A13)

Cardy Algebras and Sewing Constraints I 909

for all h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) We have

((Sminus1)lowast h1 Slowast h2)i =sum

mnα

dim Un

DimC

A

A

A

A

A A

A

Uk

α

α

h1

h2

Ui

U j

UmUn

(A14)

Now drag the upper vertex indexed by α in the above graph along its Uorm -leg until it

meets the lower vertex also indexed by α then sum over α and m This gives

((Sminus1)lowast h1 Slowast h2)i =sum

n

dim Un

DimC

A

A

A

A

A A

A

Uk

Un

h1

h2

Ui

U j

(A15)

910 L Kong I Runkel

If we just look at the neighbourhood of the Un-loop in the above graph we see thefollowing subgraphs

sumn

dim Un

DimC

A Aor

Uorj U j

Uk Uork

Un=

sumα

1

dim U j

A Aor Uk Uork

Uorj U j

αα

(A16)

where we have applied [BK Cor 3111] Substituting this subgraph back to the originalgraph in (A15) we obtain

((Sminus1)lowast h1 Slowast h2)i =sumα

1

dim U j

A

A

A

A

A A

A

A

A

Uorj

α

α

h1

h2m A

Ui

U jUk

Uk

(A17)

The graph in (A17) is equal to that in (A11) In order to see this we first drag theldquobubblerdquo (m A A) along A lines and through the m A vertex (because m A A is abimodule map) until it reaches the lower-left leg of the upper vertex indexed by α Thendrag the m A vertex along the (red) dotted line in the above graph Finally we applythe associativity of A (A12) and [Fr Lem 311] Then we see that the graph in (A17)exactly matches with the one in (A11)

Acknowledgements We would like to thank the organisers of the Oberwolfach Arbeitsgemeinschaft ldquoAlge-braic structures in conformal field theoriesrdquo (April 2007) where this work was started for an inspiring meetingWe would further like to thank the Hausdorff Institute for Mathematics in Bonn and the organisers of the stim-ulating meeting ldquoGeometry and Physicsrdquo (May 2008) We are indebted to Alexei Davydov Jens FjelstadJuumlrgen Fuchs Yi-Zhi Huang Alexei Kitaev Urs Schreiber Christoph Schweigert Stephan Stolz and PeterTeichner for helpful discussions andor comments on a draft of this paper The research of IR was partiallysupported by the EPSRC First Grant EPE0050471 the PPARC rolling grant PPC5071451 and the MarieCurie network lsquoSuperstring Theoryrsquo (MRTN-CT-2004-512194)

Cardy Algebras and Sewing Constraints I 911

References

[AN] Alexeevski A Natanzon SM Noncommutative two-dimensional topological field theories andhurwitz numbers for real algebraic curves Sel Math New Ser 12 307ndash377 (2006)

[Bi] Bichon J Cosovereign hopf algebras J Pure Appl Alg 157 121ndash133 (2001)[BK] Bakalov B Kirillov AA Lectures on Tensor Categories and Modular Functors Providence

RI Amer Math Soc 2001[C1] Cardy JL Operator content of two-dimensional conformal invariant theories Nucl Phys

B 270 186ndash204 (1986)[C2] Cardy JL Boundary conditions fusion rules and the verlinde formula Nucl Phys B 324

581ndash596 (1989)[CL] Cardy JL Lewellen DC Bulk and boundary operators in conformal field theory Phys Lett

B 259 274ndash278 (1991)[DMZ] Dong C-Y Mason G Zhu Y-C Discrete series of the Virasoro algebra and the moonshine

module In Algebraic Groups and Their Generalizations Quantum and infinite-dimensional Meth-ods Proc Symp Pure Math 56 Part 2 Providence RI Amer MathSoc 1994 pp 295ndash316

[DP] Day B Pastro C Note on frobenius monoidal functors New York J Math 14 733ndash742 (2008)[ENO1] Etingof PI Nikshych D Ostrik V On fusion categories Ann Math 162 581ndash642 (2005)[ENO2] Etingof PI Nikshych D Ostrik V An analogue of radfordrsquos s4 formula for finite tensor

categories Int Math Research Notices 54 2915ndash2933 (2004)[Fe] Felder G Froumlhlich J Fuchs J Schweigert C Correlation functions and boundary conditions in

rational conformal field theory and three-dimensional topology Comp Math 131 189ndash237 (2002)[FHL] Frenkel IB Huang Y-Z Lepowsky J On axiomatic approaches to vertex operator algebras

and modules Mem Amer Math Soc 104 (1993)[Fj] Fjelstad J Fuchs J Runkel I Schweigert C Uniqueness of openclosed rational cft with

given algebra of open states Adv Theor Math Phys 12 1283ndash1375 (2008)[FK] Froumlhlich J King C The chern-simons theory and knot polynomials Commun Math

Phys 126 167ndash199 (1989)[Fr] Froumlhlich J Fuchs J Runkel I Schweigert C Correspondences of ribbon categories Adv

Math 199 192ndash329 (2006)[FRS] Fuchs J Runkel I Schweigert C Tft construction of rcft correlators I partition functions Nucl

Phys B 646 353ndash497 (2002)[FS] Fuchs J Schweigert C Category theory for conformal boundary conditions Fields Inst Com-

mun 39 25ndash71 (2003)[H1] Huang Y-Z Two-dimensional conformal geometry and vertex operator algebras Progress in

Mathematics Vol 148 Boston Birkhaumluser 1997[H2] Huang Y-Z Rigidity and modularity of vertex tensor categories Comm Contemp Math 10

871ndash911 (2008)[HK1] Huang Y-Z Kong L Open-string vertex algebra category and operads Commun Math

Phys 250 433ndash471 (2004)[HK2] Huang Y-Z Kong L Full field algebras Commun Math Phys 272 345ndash396 (2007)[HK3] Huang Y-Z Kong L Modular invariance for conformal full field algebras httparxivorgabs

math0609570v2[mathQA] 2006[HL] Huang Y-Z Lepowsky J Tensor products of modules for a vertex operator algebra and vertex

tensor categories In Lie Theory and Geometry in honor of Bertram Kostant ed R BrylinskiJ-L Brylinski V Guillemin V Kac Boston Birkhaumluser 1994 pp 349ndash383

[JS] Joyal A Street R Braided tensor categories Adv Math 102 20ndash78 (1993)[K1] Kong L Full field algebras operads and tensor categories Adv Math 213 271ndash340 (2007)[K2] Kong L Open-closed field algebras Commun Math Phys 280 207ndash261 (2008)[K3] Kong L Cardy condition for open-closed field algebras Commun Math Phys 283 25ndash92 (2008)[Ki] Kitaev A Private communication[KO] Kirillov AA Ostrik V On q-analog of mckay correspondence and ade classification of sl(2)

conformal field theories Adv Math 171 183ndash227 (2002)[KR] Kong L Runkel I Morita classes of algebras in modular tensor categories Adv Math 219

1548ndash1576 (2008)[Ld] Lauda AD Frobenius algebras and ambidextrous adjunctions Theo Appl Cat 16 84ndash

122 (2006)[Li1] Li H-S Regular representations of vertex operator algebras Commun Contemp Math 4

639ndash683 (2002)[Li2] Li H-S Regular representations and huang-lepowsky tensor functors for vertex operator

algebras J Alge 255 423ndash462 (2002)

898 L Kong I Runkel

Recall that we define ιlowastcl-op Aop rarr T (Acl) as in (230) We introduce the graphicalnotation

ιlowastcl-op =Noplus

n=1

Cln Cr

n

Aop

=Noplus

n=1

Cln Cr

n

Aop

Aop T (minus1Acl) (321)

where the second equality follows from (215) (221) and (259)

Lemma 313 The Cardy condition (314) is equivalent to the following identity in C

Noplusn=1

DimCdim Ui

sumα

Cln

Crn

Crn

α

α

Aop

Aop Uori

Uori

=Aop

Aop

Aop

Uori

Uori

op

mop for all i isin I (322)

Proof By (244) (312) and (319) it is easy to see that (322) is equivalent to thefollowing identity

ιcl-op χ(ιlowastcl-op) = PlR(Aop)

(323)

Therefore it is enough to show that

χ(ιlowastcl-op) = ιlowastcl-op (324)

We have

χminus1(ιlowastcl-op)(1)= T (ιlowastcl-op) δ (2)= T (ιcl-op)

lowast δlowastlowast (3)= (ρ T (ιcl-op)

)lowast (4)= (ιcl-op

)lowast (325)

In step (1) we use the expression (253) for χminus1 step (2) follows from Lemma 218 (v)and Lemma 219 Step (3) is Lemma 218 (i) and Lemma 228 and finally step (4)amounts to substituting (253) and (317) Acting with χ on both sides of the aboveequality produces (324)

Combining Lemmas 311 and 313 and Proposition 216 we obtain the followingequivalent definition of Cardy C|C2plusmn-algebra (recall the graphical notation (318) forιcl-op and (321) for ιlowastcl-op)

Cardy Algebras and Sewing Constraints I 899

Definition 314 (CardyC|C2plusmn-algebra II) A CardyC-algebra is a triple (Aop|Acl ιcl-op)where Acl is a commutative symmetric Frobenius C2plusmn-algebra satisfying property (34)with f = mcl Aop is a symmetric Frobenius C-algebra and ιcl-op T (Acl) rarr Aop isan algebra homomorphism satisfying the conditions (320) and (322)

Remark 315 Up to a choice of normalisation Definition 314 is the same as the orig-inal one in [K3 Def 613] The difference between the two definitions is the factorDim C dim Ui on the left hand side of (322) which in [K3 Def 613] is given byradic

Dim C dim Ui The two definitions are related by rescaling the coproduct cl andcounit εcl of Acl by 1

radicDim C and

radicDim C respectively We chose the convention in

(322) to remove all dimension factors from the expression (314) for the Cardy condition

33 Uniqueness and existence theorems In this subsection we investigate the structureof Cardy algebras We start with the following proposition which when combined withthe results of part II provides an alternative proof of [Fj Prop 422]

Proposition 316 Let (Aop|Acl ιcl-op) be a Cardy C|C2plusmn-algebra If Acl is simple anddim Aop = 0 then Aop is simple and special

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis therefore haploid Restricting the Cardy condition (322) to the case Ui = 1 andcomposing both sides with εop from the left we see that εop kills all terms associated toU j times 1 isin Acl in the sum except for a single 1 times 1 term Thus we obtain the followingidentity

β εop = dAop (mop otimes idAorop) (idAop otimes bAop) (326)

where β isin C Composing with ηop from the right in turn implies that βεop ηop =dim Aop which is nonzero by assumption Thus also β = 0 and εop is a nonzero multipleof the morphism on the right hand side of (326) By [FRS Lem 311] Aop is special

Since Aop is a special Frobenius algebra Aop is semi-simple as an Aop-bimodule(apply [FS Prop 524] to Aop tensored with its opposite algebra) Suppose Aop is not

simple so that we can write Aop = A(1)op oplus A(2)op for nonzero Aop-bimodules A(1)op and A(2)op We denote the canonical embeddings and projections associated to this decompositionas ι12 and π12 We have the identities

mop (ι1 otimes ι2) = 0 εop ηop =2sum

i=1

εop ιi πi ηop (327)

The first identity follows since π1 mop (ι1 otimes ι2) = 0 (as mop gives the left action of

Aop on Aop and hence it preserves A(2)op ) and similarly π2 mop (ι1 otimes ι2) = 0 Thesecond identity is just the completeness of ι12 π12

Since εop ηop = 0 without losing generality we can assume εop ι1 π1 ηop = 0Using that π2 is a bimodule map we compute

π2 [LHS of (322)]Ui =1 ι1 π1 ηop = π2 [RHS of (322)]Ui =1 ι1 π1 ηop

= PlA(2)op

π2 ι1 π1 ηop = 0 (328)

900 L Kong I Runkel

On the other hand using that Acl is haploid that ιcl-op is an algebra map and thatι lowastcl-op is a coalgebra map one can check that the left-hand side of (328) is equal toλ(εop ι1 π1 ηop) π2 ηop for some λ = 0 This implies that π2 ηop = 0 Thusηop = sum

i ιi πi ηop = ι1 π1 ηop Hence we have

0 = π2 ι2 = π2 mop (ηop otimes ι2) = π2 mop ((ι1 π1 ηop)otimes ι2) (329)

However the right-hand side is zero by (327) This is a contradiction and hence Aopmust be simple

To formulate the next theorem we need the notion of the full centre of an algebra [FjDef 49] Recall that an algebra A in a braided tensor category has a left centre and aright centre [VZO] both of which are sub-algebras of A Of these two we will onlyneed the left centre The following definition is [Fr Def 231] which in our setting isequivalent to that of [VZO]

Definition 317 Let A be a symmetric special Frobenius algebra such that m A A =ζA idA

(i) The left centre Cl(A) of A is the image of the idempotent ζminus1A Pl

A(ii) The full centre Z(A) is Cl(R(A))

That ζminus1A Pl

A is an idempotent follows from [FRS Lem 52] when keeping track ofthe factors ζA ([FRS] assumes normalised-special ie ζA = 1) Note that Cl(A) is againan object of C while Z(A) is an object of C2plusmn Let el

A Cl(A) rarr A be the embeddingof Cl(A) into A The left centre is in fact the maximal subobject of A such that

m A cAA (elA otimes idA) = m A (330)

see [Fr Lem 232] This observation explains the name left centre and also makes theconnection to [O Def 15]

The full centre is by definition the image of the idempotent ζminus1A Pl

R(A) R(A) rarrR(A) Since C2plusmn is abelian the idempotent splits and we obtain the embedding andrestriction morphisms

e Z(A) rarr R(A) and r R(A) Z(A) (331)

which obey r e = idZ(A) and e r = ζminus1A Pl

R(A) It follows from Proposition 225 and

[Fr Prop 237] that Z(A) is a commutative symmetric Frobenius algebra in C2plusmn withstructure morphisms5

m Z(A) = r m R(A) (e otimes e) ηZ(A) = r ηR(A)

Z(A) = ζA middot (r otimes r) R(A) e εZ(A) = ζminus1A middot εR(A) e

(332)

Moreover if A is simple then Z(A) is simple and if A is simple and dim A = 0 thenZ(A) is simple and special The normalisation of the counit is such that

εZ(A) ηZ(A) = ζminus2A dim A DimC (333)

5 The normalisation of product and unit is the standard one The factors in the coproduct and counit haveto be included in order for (A|Z(A) e) to be a Cardy algebra see Theorem 318 below The normalisation ofthe counit enters the Cardy condition (314) through the definition of ( middot )lowast

Cardy Algebras and Sewing Constraints I 901

Theorem 318 Let A be a special symmetric Frobenius C-algebra Then (A|Z(A) e)is a Cardy C|C2plusmn-algebra

The proof of this theorem makes use the following two lemmas

Lemma 319 e Z(A) rarr R(A) is an algebra map and elowast = ζA middot r

Proof It follows from [Fr Lem 229] (or by direct calculation using in particularm R(A) R(A) = ζAidR(A)) that

m R(A) (e otimes e) = ζminus1A middot Pl

R(A) m R(A) (e otimes e) (334)

Substituting e r = ζminus1A Pl

R(A) shows that e is compatible with multiplication For theunit one finds

e ηZ(A) = e r ηR(A) = ζminus1A Pl

R(A) ηR(A) = ηR(A) (335)

Thus e is an algebra map For the second statement one computes

elowast (1)= ζA

R(A)

r

R(A)

R(A)

Z(A)

R(A)

Z(A)

r

e

e

(2)=

R(A)

r

R(A)

R(A)

R(A)

Z(A)

(3)=

R(A)

r

R(A)

Z(A)

(4)= ζA middot r (336)

where in (1) the definitions (230) and (332) have been substituted step (2) is e r = ζminus1

A PlR(A) step (3) uses that R(A) is symmetric Frobenius and step (4) is again

e r = ζminus1A Pl

R(A) Lemma 320 Let A be a symmetric Frobenius algebra in C The morphism

PlR(A) m R(A)

(Pl

R(A) otimes PlR(A)

) R(A)otimes R(A) minusrarr R(A) (337)

is S-invariant

The proof of this lemma is a slightly lengthy explicit calculation and has been deferredto Appendix A2

902 L Kong I Runkel

Proof of Theorem 318 That e is an algebra map was proved in Lemma 319 The centrecondition (313) holds by property (330) of the left centre The Cardy condition (314)also is an immediate consequence of Lemma 319

ιcl-op ιlowastcl-op = e (ζA r) = PlR(A) (338)

The full centre Z(A) is a commutative symmetric Frobenius algebra It remains to provemodular invariance That θZ(A) = idZ(A) is implied by commutativity and symmetryof Z(A) [Fr Prop 225] The S-invariance condition (33) follows from Lemma 320In (337) substitute Pl

R(A) = ζA e r and then put the resulting morphism into (33)Compose the resulting equation with e otimes idW from the right (ie from the bottom) andsubstitute the definition (332) of m Z(A) This results in the statement that m Z(A) isS-invariant

The following theorem is analogous to [LR Prop 29] and [Fj Thm 426] whichroughly speaking answer the question under which circumstances the restriction of atwo-dimensional conformal field theory to the boundary already determines the entireconformal field theory The first work is set in Minkowski space and uses operator alge-bras and subfactors while the second work is set in Euclidean space and uses modulartensor categories

Theorem 321 Let (A|Acl ιcl-op) be a Cardy C|C2plusmn-algebra such that dim A = 0 andAcl is simple Then A is special and (A|Acl ιcl-op) sim= (A|Z(A) e) as Cardy algebras

Proof By Proposition 316 A is simple and special Since Acl is simple the algebramap ιcl-op Acl rarr R(A) is either zero or a monomorphism But ιcl-op ηcl = ηR(A)and so ιcl-op = 0 Thus ιcl-op is monic By Lemma 218 (ii) ζminus1

A ιlowastcl-op is epi The Cardycondition (314) implies

ιcl-op ζminus1A ιlowastcl-op = ζminus1

A PlR(A) = e r (339)

Composing this with e r from the left yields e r ιcl-op ζminus1A ιlowastcl-op = e r =

ιcl-op ζminus1A ιlowastcl-op Since ζminus1

A ιlowastcl-op is epi we have

e r ιcl-op = ιcl-op (340)

Actually (340) also follows from (313) and specialness of R(A) We will prove that

( fop fcl) (A|Acl ιcl-op) minusrarr (A|Z(A) e) where fop = idA fcl = r ιcl-op

(341)

is an isomorphism of Cardy algebrasfcl is an algebra map Compatibility with the units follows since ιcl-op is an algebramap

fcl ηcl = r ιcl-op ηcl = r ηR(A) = ηZ(A) (342)

Compatibility with the multiplication also follows since ιcl-op is an algebra map

m Z(A) ( fcl otimes fcl) = r m R(A) (e otimes e) (r otimes r) (ιcl-op otimes ιcl-op)

= r m R(A) (ιcl-op otimes ιcl-op) = r ιcl-op mcl = fcl mcl (343)

where in the second step we used (340)

Cardy Algebras and Sewing Constraints I 903

fcl is an isomorphism As above since fcl is an algebra map and since Acl is simple fcl

has to be monic By Lemma 319 rlowast = ζminus1A e Thus f lowast

cl = ιlowastcl-op rlowast = ζminus1A ιlowastcl-op e and

fcl f lowastcl = r ιcl-op ζminus1

A ιlowastcl-op e = r e r e = idZ(A) (344)

and so fcl is also epi and hence isofcl is a coalgebra map Since fcl is an algebra map so is f minus1

cl By (344) f minus1cl = f lowast

cland by Lemma 218 (iii) this implies that fcl is a also coalgebra mapThe diagram (315) commutes Commutativity of (315) is equivalent to e fcl = ιcl-opwhich holds by (340)

Let A be a special symmetric Frobenius algebra So far we have seen that (A|Z(A) e)is a Cardy algebra and that all Cardy algebras with Aop = A and simple Acl are of thisform It is now natural to ask if every simple Acl does occur as part of a Cardy algebraThe following theorem provides an affirmative answer Recall that for an A-left moduleM the object Mor otimesA M is an algebra (see eg [KR Lem 42])

Theorem 322 If Acl is a simple modular invariant commutative symmetric FrobeniusC2plusmn-algebra then there exist a simple special symmetric Frobenius C-algebra A and amorphism ιcl-op Acl rarr R(A) such that

(i) Acl sim= Z(A) as Frobenius algebras(ii) (A|Acl ιcl-op) is a Cardy C|C2plusmn-algebra

(iii) T (Acl) sim= oplusκisinJ Morκ otimesA Mκ as algebras where MκκisinJ is a set of representa-

tives of the isomorphism classes of simple A-left modules

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis haploid It then follows from Theorem 34 that Acl is special By Proposition 224T (Acl) is a special symmetric Frobenius algebra in C Thus T (Acl) = oplusi Ai where theAi are simple symmetric Frobenius algebras We will show that at least one of the Aiis special Since T (Acl) is special we have mT (Acl) T (Acl) = ζ idT (Acl) for someζ isin C

times Restricting this to the summand Ai shows mi i = ζ idAi FurthermoreεT (Acl) ηT (Acl) = ξ id1 for some ξ isin C

times But εT (Acl) ηT (Acl) = sumi εi ηi and so

at least one of the εi ηi has to be nonzero Therefore at least one of the Ai is speciallet A equiv Ai be this summand We denote the embedding A rarr T (Acl) by e0 and therestriction T (Acl) A by r0 Notice that r0 is an algebra homomorphism Define

ιcl-op = χ (r0) Acl minusrarr R(A) (345)

By Proposition 216 ιcl-op is an algebra homomorphism Next we verify the centre con-dition (313) or rather its equivalent form (320) By substituting the definitions onecan convince oneself that the commutativity mcl cAclAcl = mcl of Acl in C2plusmn implies thecondition mT (Acl) = mT (Acl) in C see [K2 Prop 36] Here T (Acl)otimes T (Acl) rarrT (Acl)otimes T (Acl) is given by

=oplusmn

(idCl

notimes cCl

m Crnotimes idCr

m

)

(cCl

m Clnotimes cminus1

Crn C

rm

)

(idCl

motimes cminus1

Cln C

rm

otimes idCrn

)

(346)

and we decomposed Acl as Acl = oplusnCln times Cr

n As a consequence we obtain the identity

r0 mT (Acl) (idAcl otimes e0) = r0 mT (Acl) (idAcl otimes e0) (347)

904 L Kong I Runkel

Using that r0 is an algebra map and that by definition ιcl-op = r0 we obtain (320)In order to show that (A|Acl ιcl-op) is a Cardy algebra it remains to show that the

Cardy condition (314) is satisfied We will demonstrate this via a detour by first provingthat Acl sim= Z(A) as Frobenius algebras

Recall the notations e and r given in (331) Using the centre condition (320) onecan check that Pl

R(A) ιcl-op = m R(A) R(A) ιcl-op By specialness of A we have

m A A = ζA idA and so together with e r = ζminus1A Pl

R(A) we get

e r ιcl-op = ιcl-op (348)

Next consider the morphism

fcl = r ιcl-op Acl minusrarr Z(A) (349)

By the same derivation as in (342) and (343) one sees that fcl is an algebra map Inparticular fcl ηcl = ηZ(A) = 0 and so fcl = 0 Since Acl is simple fcl has to be amonomorphism

By the same argument as used in the proof of Theorem 34 (ii) up to isomorphismAcl is the unique simple local Acl-(left-)module The algebra monomorphism fcl turnsZ(A) into an Acl-module Since Z(A) is commutative it is local as an Acl-module andso Z(A) sim= AoplusN

cl for some N ge 1 By construction A is a simple special symmetricFrobenius algebra Proposition 225 and Corollary 227 show that R(A) inherits all theseproperties and thus Z(A) is simple (see the comment below Eq (332)) By Theorem318 Z(A) is modular invariant and then by Theorem 34 (i) dim Z(A) = Dim C Thisimplies that N = 1 in Z(A) sim= AoplusN

cl and so fcl is in fact an isomorphismSince Acl and Z(A) are both haploid we have εZ(A) fcl = ξ εcl for some ξ isin C

timesThe counit uniquely determines the Frobenius structure on Acl and Z(A) (see eg [FRSLemma 37]) so that fcl is a coalgebra isomorphism iff ξ = 1 To compute ξ we composethe above identity with ηcl from the right Defining ζcl via εcl ηcl = ζminus1

cl Dim C middot id1 andusing (333) gives ξ = dim A ζclζ

2A By rescaling the comultiplication and the counit

of A and consequently changing ζA we can always achieve ξ = 1 This proves part (i)of the theorem

Equation (348) implies that ιcl-op = e fcl Since fcl is an isomorphism of Frobeniusalgebras by Lemmas 218 and 319 we have

ιcl-op ιlowastcl-op = e fcl f lowastcl elowast = ζA e r = Pl

R(A) (350)

Thus (A|Acl ιcl-op) is a Cardy algebra This proves part (ii) of the theoremPart (iii) can be seen as follows By [KR Prop 43] T Z(A) sim= oplusκisinJ Mor

κ otimesA Mκ asalgebras Together with the observation that T ( fcl) T (Acl) rarr T Z(A) is an isomor-phism of algebras this proves part (iii) Remark 323 Part (i) of Theorem 322 was announced by Muumlger [Muuml2] We provide anindependent proof in the setting of Cardy algebras

The above theorem together with Lemma 211 and Theorem 34 shows that a sim-ple commutative symmetric Frobenius C2plusmn-algebra Acl with dim Acl = Dim C is alwayspart of a Cardy algebra (Aop|Acl ιcl-op) for some simple special symmetric Frobeniusalgebra Aop in C However the above proof also illustrates that Aop is not unique Thisraises the question how two Cardy algebras with a given Acl can differ This question isanswered by [KR Thm 11] which in the present framework can be restated as follows

Cardy Algebras and Sewing Constraints I 905

Theorem 324 If (A(i)op |A(i)cl ι(i)cl-op) i = 1 2 are two Cardy C|C2plusmn-algebras such that

A(i)cl is simple and dim A(i)op = 0 for i = 1 2 then A(1)clsim= A(2)cl as algebras if and only

if A(1)op and A(2)op are Morita equivalent

Proof Theorem 11 in [KR] is stated for A(i)op being non-degenerate algebras and A(i)cl =Z(A(i)op ) for i = 1 2 By Proposition 316 A(i)op are simple and special for i = 1 2

Then by [KR Lem 21] A(i)op are non-degenerate algebras By Theorem 321 we have

A(i)clsim= Z(A(i)op ) as Frobenius algebras Finally by [KR Thm 11] Z(A(1)op ) sim= Z(A(2)op )

as algebras iff A(1)op and A(2)op are Morita equivalent Let Cmax(C2plusmn) be the set of equivalence classes [B] of simple modular invariant

commutative symmetric Frobenius algebras B in C2plusmn Two such algebras B and B prime areequivalent if B and B prime are isomorphic as algebras (but not necessarily as Frobenius alge-bras) Let Msimp(C) be the set of Morita classes of simple special symmetric Frobeniusalgebras in C Define the map z Msimp(C) rarr Cmax(C2plusmn) by z A rarr [Z(A)] whereA denotes the Morita class of A From Theorem 322 (i) and [KR Thm 11] we learn

Corollary 325 The map z Msimp(C) rarr Cmax(C2plusmn) is a bijection

A Appendix

A1 Proof of Lemma 27 We will show that if (F ψ F2 ψ

F0 ) is a colax tensor functor

from C1 to C2 then (G φG2 φ

G0 ) is a lax tensor functor from C2 to C1 Applying this

result to the opposed categories then gives the converse statementWe need to show that φG

0 and φG2 make the diagrams (21) and (22) commute We

first prove the commutativity of (21) Consider the following diagram

G(A)otimes (G(B)otimes G(C))

GψF2 δ

idG(A)otimesφG2 G(A)otimes G(B otimes C)

GψF2 δ

G(FG(A)otimes F(G(B)otimes G(C)))

G(F(idG(A))otimesF(φG2 ))

G(idFG(A)otimesψ F2 )

G(FG(A)otimes FG(B otimes C))

G(ρAotimesρBotimesC )

G(FG(A)otimes (FG(B)otimes FG(C)))

G(ρAotimes(ρBotimesρC )) G(A otimes (B otimes C))

(A1)

The top subdiagram is commutative because of the naturality of Gψ F2 δ The commu-

tativity of the bottom subdiagram follows from the following identities

(ρB otimes ρC ) ψ F2 = (ρB otimes ρC ) ψ F

2 ρF Fδ

= ρBotimesC FG(ρB otimes ρC ) FG(ψ F2 ) Fδ

= ρBotimesC F(φG2 ) (A2)

as a map F(G(B) otimes G(C)) rarr B otimes C The commutativity of (A1) implies that thecomposition of maps in the left column in (21) can be replaced by

G(ρA otimes (ρB otimes ρC )) G((idFG(A) otimes ψ F

2 ) ψ F2

) δ (A3)

906 L Kong I Runkel

Similarly we can show that the composition of maps in the right column in (21) can bereplaced by

G((ρA otimes ρB)otimes ρC ) G((ψ F

2 otimes idFG(C)) ψ F2

) δ (A4)

Using the commutativity of (23) it is easy to see that (21) with the left and right col-umns of (21) replaced by (A3) and (A4) respectively is commutative Hence (21) iscommutative

Now we prove the commutativity of the first diagram in (22)

φG2 (φG

0 otimes idG(A))

(1)= G(ρ12 otimes ρA) Gψ F2 δ

[(Gψ F

0 δ11)otimes idG(A)

]

(2)= G(ρ12 otimes ρA) Gψ F2 G F(Gψ0 otimes idG(A)) G F(δ11 otimes idG(A)) δ

(3)= G(ρ12 otimes ρA) G(FG(ψ F0 )otimes idFG(A)) Gψ F

2 G F(δ11 otimes idG(A)) δ(4)= G(id12 otimes ρA) G

([ψ F

0 ρF(11) (Fδ)11] otimes idFG(A)

) Gψ F

2 δ(5)= G(id12 otimes ρA) G(ψ F

0 otimes idFG(A)) Gψ F2 δ

(6)= G(id12 otimes ρA) G(lminus1FG(A)) G F(lG(A)) δ

(7)= G(lminus1A ) GρA δG lG(A)

(8)= G(lminus1A ) lG(A) (A5)

where in step (1) we substituted the definition of φG0 φ

G2 given in (210) in step (2)

we used the naturality of δ in step (3) we used the naturality of Gψ F2 in step (4) we

switched the position between Gρ12 and G FG(ψ F0 ) and the position between Gψ2 and

G F(δ12 otimes idG(A)) using the naturality of ρ and FψG2 respectively in step (5) we applied

the second identity in (28) in step (6) we used (24) in step (7) we used the naturalityof lminus1 and δ in step (8) we used the first identity in (28)

The proof of the commutativity of the second diagram in (22) is similar Thus wehave shown that G is a lax tensor functor

A2 Proof of Lemma 320 To prepare the proof recall that for a given object B isin Cthe modular group P SL(2Z) acts on the space oplusi HomC(B otimes Ui Ui ) see eg [BKSect 31] and [K3 Eq (455)] We will only need the action of S and Sminus1 Let f isinoplusi HomC(B otimes Ui Ui ) Then

S oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A6)

Cardy Algebras and Sewing Constraints I 907

Sminus1 oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A7)

By Lemma 32 to establish that (337) is S-invariant it is enough to prove the identity(34) when f is given by (337) Using (A6) and (A7) we can see that Eq (34) simplysays that oplusi j [RHS of (34)] is invariant under the action of S times S Consider the elementg of oplus jkisinIHomC2plusmn(R(A)otimes (Uor

j times Uk)Uorj times Uk) given by

g =oplusjkisinI

sumα

R(A)

R(A)

R(A)

R(A)

Uorj times Uk

Uorj times Uk

α

α

(A8)

By the above arguments proving S-invariance of (337) is equivalent to proving invari-ance of g under the action of S times S

For i isin I we denote by gi the component of g in(oplus jisinIHomC(A otimes Uor

i otimes Uorj U

orj )

)otimes (opluskisinIHomC(Ui otimes UkUk))

We view the second Hom-space in the above tensor product as a Hom-space in C+ insteadof Cminus It is enough to show that gi is invariant under the action of S times Sminus1 Note thatthe action of Sminus1 in C+ is equivalent to that of S in Cminus

The morphism gi can be canonically identified with a bilinear pairing

( middot middot )i ⎛⎝oplus

jisinIHomC(Uor

j A otimes Uori otimes Uor

j )

⎞⎠

times(oplus

kisinIHomC(UkUi otimes Uk)

)minusrarr C (A9)

as follows For h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) we set

(h1 h2)i = (dim U j dim Uk)minus1 trUor

j timesUk[gi (h1 times h2)] (A10)

908 L Kong I Runkel

When substituting the explicit form of the product m R(A) of R(A) = (A times 1) otimes R(1)after a short calculation one finds

(h1 h2)i =sumα

1

dim U j

A

A

A

A

A A

A α

α

h1

h2

U jUk

Uorj

Ui

Uk (A11)

Here the top morphism PlR(A) has been simplified with the help of the identity

PR(A) m R(A) (PR(A) otimes PR(A))

=(oplus

iisinI((m A A)otimes idUor

i)times idUi

) m R(A) (PR(A) otimes PR(A)) (A12)

which can be checked by direct calculation along the same lines as in the proof of [FrLem 310]

The action of the modular transformation S on oplusiisinI(B otimes Ui Ui ) for B isin C nat-urally induces an action on oplusiisinI(Ui B otimes Ui ) [K3 Prop 514] which we denote bySlowast In the present case we get an action of Slowast on oplus jisinIHomC(Uor

j A otimes Uori otimes Uor

j ) and

opluskisinIHomC(UkUi otimes Uk) Then to show gi is invariant under the action of S times Sminus1

amounts to showing that

(h1 h2)i = ((Sminus1)lowast h1 Slowast h2)i (A13)

Cardy Algebras and Sewing Constraints I 909

for all h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) We have

((Sminus1)lowast h1 Slowast h2)i =sum

mnα

dim Un

DimC

A

A

A

A

A A

A

Uk

α

α

h1

h2

Ui

U j

UmUn

(A14)

Now drag the upper vertex indexed by α in the above graph along its Uorm -leg until it

meets the lower vertex also indexed by α then sum over α and m This gives

((Sminus1)lowast h1 Slowast h2)i =sum

n

dim Un

DimC

A

A

A

A

A A

A

Uk

Un

h1

h2

Ui

U j

(A15)

910 L Kong I Runkel

If we just look at the neighbourhood of the Un-loop in the above graph we see thefollowing subgraphs

sumn

dim Un

DimC

A Aor

Uorj U j

Uk Uork

Un=

sumα

1

dim U j

A Aor Uk Uork

Uorj U j

αα

(A16)

where we have applied [BK Cor 3111] Substituting this subgraph back to the originalgraph in (A15) we obtain

((Sminus1)lowast h1 Slowast h2)i =sumα

1

dim U j

A

A

A

A

A A

A

A

A

Uorj

α

α

h1

h2m A

Ui

U jUk

Uk

(A17)

The graph in (A17) is equal to that in (A11) In order to see this we first drag theldquobubblerdquo (m A A) along A lines and through the m A vertex (because m A A is abimodule map) until it reaches the lower-left leg of the upper vertex indexed by α Thendrag the m A vertex along the (red) dotted line in the above graph Finally we applythe associativity of A (A12) and [Fr Lem 311] Then we see that the graph in (A17)exactly matches with the one in (A11)

Acknowledgements We would like to thank the organisers of the Oberwolfach Arbeitsgemeinschaft ldquoAlge-braic structures in conformal field theoriesrdquo (April 2007) where this work was started for an inspiring meetingWe would further like to thank the Hausdorff Institute for Mathematics in Bonn and the organisers of the stim-ulating meeting ldquoGeometry and Physicsrdquo (May 2008) We are indebted to Alexei Davydov Jens FjelstadJuumlrgen Fuchs Yi-Zhi Huang Alexei Kitaev Urs Schreiber Christoph Schweigert Stephan Stolz and PeterTeichner for helpful discussions andor comments on a draft of this paper The research of IR was partiallysupported by the EPSRC First Grant EPE0050471 the PPARC rolling grant PPC5071451 and the MarieCurie network lsquoSuperstring Theoryrsquo (MRTN-CT-2004-512194)

Cardy Algebras and Sewing Constraints I 911

References

[AN] Alexeevski A Natanzon SM Noncommutative two-dimensional topological field theories andhurwitz numbers for real algebraic curves Sel Math New Ser 12 307ndash377 (2006)

[Bi] Bichon J Cosovereign hopf algebras J Pure Appl Alg 157 121ndash133 (2001)[BK] Bakalov B Kirillov AA Lectures on Tensor Categories and Modular Functors Providence

RI Amer Math Soc 2001[C1] Cardy JL Operator content of two-dimensional conformal invariant theories Nucl Phys

B 270 186ndash204 (1986)[C2] Cardy JL Boundary conditions fusion rules and the verlinde formula Nucl Phys B 324

581ndash596 (1989)[CL] Cardy JL Lewellen DC Bulk and boundary operators in conformal field theory Phys Lett

B 259 274ndash278 (1991)[DMZ] Dong C-Y Mason G Zhu Y-C Discrete series of the Virasoro algebra and the moonshine

module In Algebraic Groups and Their Generalizations Quantum and infinite-dimensional Meth-ods Proc Symp Pure Math 56 Part 2 Providence RI Amer MathSoc 1994 pp 295ndash316

[DP] Day B Pastro C Note on frobenius monoidal functors New York J Math 14 733ndash742 (2008)[ENO1] Etingof PI Nikshych D Ostrik V On fusion categories Ann Math 162 581ndash642 (2005)[ENO2] Etingof PI Nikshych D Ostrik V An analogue of radfordrsquos s4 formula for finite tensor

categories Int Math Research Notices 54 2915ndash2933 (2004)[Fe] Felder G Froumlhlich J Fuchs J Schweigert C Correlation functions and boundary conditions in

rational conformal field theory and three-dimensional topology Comp Math 131 189ndash237 (2002)[FHL] Frenkel IB Huang Y-Z Lepowsky J On axiomatic approaches to vertex operator algebras

and modules Mem Amer Math Soc 104 (1993)[Fj] Fjelstad J Fuchs J Runkel I Schweigert C Uniqueness of openclosed rational cft with

given algebra of open states Adv Theor Math Phys 12 1283ndash1375 (2008)[FK] Froumlhlich J King C The chern-simons theory and knot polynomials Commun Math

Phys 126 167ndash199 (1989)[Fr] Froumlhlich J Fuchs J Runkel I Schweigert C Correspondences of ribbon categories Adv

Math 199 192ndash329 (2006)[FRS] Fuchs J Runkel I Schweigert C Tft construction of rcft correlators I partition functions Nucl

Phys B 646 353ndash497 (2002)[FS] Fuchs J Schweigert C Category theory for conformal boundary conditions Fields Inst Com-

mun 39 25ndash71 (2003)[H1] Huang Y-Z Two-dimensional conformal geometry and vertex operator algebras Progress in

Mathematics Vol 148 Boston Birkhaumluser 1997[H2] Huang Y-Z Rigidity and modularity of vertex tensor categories Comm Contemp Math 10

871ndash911 (2008)[HK1] Huang Y-Z Kong L Open-string vertex algebra category and operads Commun Math

Phys 250 433ndash471 (2004)[HK2] Huang Y-Z Kong L Full field algebras Commun Math Phys 272 345ndash396 (2007)[HK3] Huang Y-Z Kong L Modular invariance for conformal full field algebras httparxivorgabs

math0609570v2[mathQA] 2006[HL] Huang Y-Z Lepowsky J Tensor products of modules for a vertex operator algebra and vertex

tensor categories In Lie Theory and Geometry in honor of Bertram Kostant ed R BrylinskiJ-L Brylinski V Guillemin V Kac Boston Birkhaumluser 1994 pp 349ndash383

[JS] Joyal A Street R Braided tensor categories Adv Math 102 20ndash78 (1993)[K1] Kong L Full field algebras operads and tensor categories Adv Math 213 271ndash340 (2007)[K2] Kong L Open-closed field algebras Commun Math Phys 280 207ndash261 (2008)[K3] Kong L Cardy condition for open-closed field algebras Commun Math Phys 283 25ndash92 (2008)[Ki] Kitaev A Private communication[KO] Kirillov AA Ostrik V On q-analog of mckay correspondence and ade classification of sl(2)

conformal field theories Adv Math 171 183ndash227 (2002)[KR] Kong L Runkel I Morita classes of algebras in modular tensor categories Adv Math 219

1548ndash1576 (2008)[Ld] Lauda AD Frobenius algebras and ambidextrous adjunctions Theo Appl Cat 16 84ndash

122 (2006)[Li1] Li H-S Regular representations of vertex operator algebras Commun Contemp Math 4

639ndash683 (2002)[Li2] Li H-S Regular representations and huang-lepowsky tensor functors for vertex operator

algebras J Alge 255 423ndash462 (2002)

Cardy Algebras and Sewing Constraints I 899

Definition 314 (CardyC|C2plusmn-algebra II) A CardyC-algebra is a triple (Aop|Acl ιcl-op)where Acl is a commutative symmetric Frobenius C2plusmn-algebra satisfying property (34)with f = mcl Aop is a symmetric Frobenius C-algebra and ιcl-op T (Acl) rarr Aop isan algebra homomorphism satisfying the conditions (320) and (322)

Remark 315 Up to a choice of normalisation Definition 314 is the same as the orig-inal one in [K3 Def 613] The difference between the two definitions is the factorDim C dim Ui on the left hand side of (322) which in [K3 Def 613] is given byradic

Dim C dim Ui The two definitions are related by rescaling the coproduct cl andcounit εcl of Acl by 1

radicDim C and

radicDim C respectively We chose the convention in

(322) to remove all dimension factors from the expression (314) for the Cardy condition

33 Uniqueness and existence theorems In this subsection we investigate the structureof Cardy algebras We start with the following proposition which when combined withthe results of part II provides an alternative proof of [Fj Prop 422]

Proposition 316 Let (Aop|Acl ιcl-op) be a Cardy C|C2plusmn-algebra If Acl is simple anddim Aop = 0 then Aop is simple and special

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis therefore haploid Restricting the Cardy condition (322) to the case Ui = 1 andcomposing both sides with εop from the left we see that εop kills all terms associated toU j times 1 isin Acl in the sum except for a single 1 times 1 term Thus we obtain the followingidentity

β εop = dAop (mop otimes idAorop) (idAop otimes bAop) (326)

where β isin C Composing with ηop from the right in turn implies that βεop ηop =dim Aop which is nonzero by assumption Thus also β = 0 and εop is a nonzero multipleof the morphism on the right hand side of (326) By [FRS Lem 311] Aop is special

Since Aop is a special Frobenius algebra Aop is semi-simple as an Aop-bimodule(apply [FS Prop 524] to Aop tensored with its opposite algebra) Suppose Aop is not

simple so that we can write Aop = A(1)op oplus A(2)op for nonzero Aop-bimodules A(1)op and A(2)op We denote the canonical embeddings and projections associated to this decompositionas ι12 and π12 We have the identities

mop (ι1 otimes ι2) = 0 εop ηop =2sum

i=1

εop ιi πi ηop (327)

The first identity follows since π1 mop (ι1 otimes ι2) = 0 (as mop gives the left action of

Aop on Aop and hence it preserves A(2)op ) and similarly π2 mop (ι1 otimes ι2) = 0 Thesecond identity is just the completeness of ι12 π12

Since εop ηop = 0 without losing generality we can assume εop ι1 π1 ηop = 0Using that π2 is a bimodule map we compute

π2 [LHS of (322)]Ui =1 ι1 π1 ηop = π2 [RHS of (322)]Ui =1 ι1 π1 ηop

= PlA(2)op

π2 ι1 π1 ηop = 0 (328)

900 L Kong I Runkel

On the other hand using that Acl is haploid that ιcl-op is an algebra map and thatι lowastcl-op is a coalgebra map one can check that the left-hand side of (328) is equal toλ(εop ι1 π1 ηop) π2 ηop for some λ = 0 This implies that π2 ηop = 0 Thusηop = sum

i ιi πi ηop = ι1 π1 ηop Hence we have

0 = π2 ι2 = π2 mop (ηop otimes ι2) = π2 mop ((ι1 π1 ηop)otimes ι2) (329)

However the right-hand side is zero by (327) This is a contradiction and hence Aopmust be simple

To formulate the next theorem we need the notion of the full centre of an algebra [FjDef 49] Recall that an algebra A in a braided tensor category has a left centre and aright centre [VZO] both of which are sub-algebras of A Of these two we will onlyneed the left centre The following definition is [Fr Def 231] which in our setting isequivalent to that of [VZO]

Definition 317 Let A be a symmetric special Frobenius algebra such that m A A =ζA idA

(i) The left centre Cl(A) of A is the image of the idempotent ζminus1A Pl

A(ii) The full centre Z(A) is Cl(R(A))

That ζminus1A Pl

A is an idempotent follows from [FRS Lem 52] when keeping track ofthe factors ζA ([FRS] assumes normalised-special ie ζA = 1) Note that Cl(A) is againan object of C while Z(A) is an object of C2plusmn Let el

A Cl(A) rarr A be the embeddingof Cl(A) into A The left centre is in fact the maximal subobject of A such that

m A cAA (elA otimes idA) = m A (330)

see [Fr Lem 232] This observation explains the name left centre and also makes theconnection to [O Def 15]

The full centre is by definition the image of the idempotent ζminus1A Pl

R(A) R(A) rarrR(A) Since C2plusmn is abelian the idempotent splits and we obtain the embedding andrestriction morphisms

e Z(A) rarr R(A) and r R(A) Z(A) (331)

which obey r e = idZ(A) and e r = ζminus1A Pl

R(A) It follows from Proposition 225 and

[Fr Prop 237] that Z(A) is a commutative symmetric Frobenius algebra in C2plusmn withstructure morphisms5

m Z(A) = r m R(A) (e otimes e) ηZ(A) = r ηR(A)

Z(A) = ζA middot (r otimes r) R(A) e εZ(A) = ζminus1A middot εR(A) e

(332)

Moreover if A is simple then Z(A) is simple and if A is simple and dim A = 0 thenZ(A) is simple and special The normalisation of the counit is such that

εZ(A) ηZ(A) = ζminus2A dim A DimC (333)

5 The normalisation of product and unit is the standard one The factors in the coproduct and counit haveto be included in order for (A|Z(A) e) to be a Cardy algebra see Theorem 318 below The normalisation ofthe counit enters the Cardy condition (314) through the definition of ( middot )lowast

Cardy Algebras and Sewing Constraints I 901

Theorem 318 Let A be a special symmetric Frobenius C-algebra Then (A|Z(A) e)is a Cardy C|C2plusmn-algebra

The proof of this theorem makes use the following two lemmas

Lemma 319 e Z(A) rarr R(A) is an algebra map and elowast = ζA middot r

Proof It follows from [Fr Lem 229] (or by direct calculation using in particularm R(A) R(A) = ζAidR(A)) that

m R(A) (e otimes e) = ζminus1A middot Pl

R(A) m R(A) (e otimes e) (334)

Substituting e r = ζminus1A Pl

R(A) shows that e is compatible with multiplication For theunit one finds

e ηZ(A) = e r ηR(A) = ζminus1A Pl

R(A) ηR(A) = ηR(A) (335)

Thus e is an algebra map For the second statement one computes

elowast (1)= ζA

R(A)

r

R(A)

R(A)

Z(A)

R(A)

Z(A)

r

e

e

(2)=

R(A)

r

R(A)

R(A)

R(A)

Z(A)

(3)=

R(A)

r

R(A)

Z(A)

(4)= ζA middot r (336)

where in (1) the definitions (230) and (332) have been substituted step (2) is e r = ζminus1

A PlR(A) step (3) uses that R(A) is symmetric Frobenius and step (4) is again

e r = ζminus1A Pl

R(A) Lemma 320 Let A be a symmetric Frobenius algebra in C The morphism

PlR(A) m R(A)

(Pl

R(A) otimes PlR(A)

) R(A)otimes R(A) minusrarr R(A) (337)

is S-invariant

The proof of this lemma is a slightly lengthy explicit calculation and has been deferredto Appendix A2

902 L Kong I Runkel

Proof of Theorem 318 That e is an algebra map was proved in Lemma 319 The centrecondition (313) holds by property (330) of the left centre The Cardy condition (314)also is an immediate consequence of Lemma 319

ιcl-op ιlowastcl-op = e (ζA r) = PlR(A) (338)

The full centre Z(A) is a commutative symmetric Frobenius algebra It remains to provemodular invariance That θZ(A) = idZ(A) is implied by commutativity and symmetryof Z(A) [Fr Prop 225] The S-invariance condition (33) follows from Lemma 320In (337) substitute Pl

R(A) = ζA e r and then put the resulting morphism into (33)Compose the resulting equation with e otimes idW from the right (ie from the bottom) andsubstitute the definition (332) of m Z(A) This results in the statement that m Z(A) isS-invariant

The following theorem is analogous to [LR Prop 29] and [Fj Thm 426] whichroughly speaking answer the question under which circumstances the restriction of atwo-dimensional conformal field theory to the boundary already determines the entireconformal field theory The first work is set in Minkowski space and uses operator alge-bras and subfactors while the second work is set in Euclidean space and uses modulartensor categories

Theorem 321 Let (A|Acl ιcl-op) be a Cardy C|C2plusmn-algebra such that dim A = 0 andAcl is simple Then A is special and (A|Acl ιcl-op) sim= (A|Z(A) e) as Cardy algebras

Proof By Proposition 316 A is simple and special Since Acl is simple the algebramap ιcl-op Acl rarr R(A) is either zero or a monomorphism But ιcl-op ηcl = ηR(A)and so ιcl-op = 0 Thus ιcl-op is monic By Lemma 218 (ii) ζminus1

A ιlowastcl-op is epi The Cardycondition (314) implies

ιcl-op ζminus1A ιlowastcl-op = ζminus1

A PlR(A) = e r (339)

Composing this with e r from the left yields e r ιcl-op ζminus1A ιlowastcl-op = e r =

ιcl-op ζminus1A ιlowastcl-op Since ζminus1

A ιlowastcl-op is epi we have

e r ιcl-op = ιcl-op (340)

Actually (340) also follows from (313) and specialness of R(A) We will prove that

( fop fcl) (A|Acl ιcl-op) minusrarr (A|Z(A) e) where fop = idA fcl = r ιcl-op

(341)

is an isomorphism of Cardy algebrasfcl is an algebra map Compatibility with the units follows since ιcl-op is an algebramap

fcl ηcl = r ιcl-op ηcl = r ηR(A) = ηZ(A) (342)

Compatibility with the multiplication also follows since ιcl-op is an algebra map

m Z(A) ( fcl otimes fcl) = r m R(A) (e otimes e) (r otimes r) (ιcl-op otimes ιcl-op)

= r m R(A) (ιcl-op otimes ιcl-op) = r ιcl-op mcl = fcl mcl (343)

where in the second step we used (340)

Cardy Algebras and Sewing Constraints I 903

fcl is an isomorphism As above since fcl is an algebra map and since Acl is simple fcl

has to be monic By Lemma 319 rlowast = ζminus1A e Thus f lowast

cl = ιlowastcl-op rlowast = ζminus1A ιlowastcl-op e and

fcl f lowastcl = r ιcl-op ζminus1

A ιlowastcl-op e = r e r e = idZ(A) (344)

and so fcl is also epi and hence isofcl is a coalgebra map Since fcl is an algebra map so is f minus1

cl By (344) f minus1cl = f lowast

cland by Lemma 218 (iii) this implies that fcl is a also coalgebra mapThe diagram (315) commutes Commutativity of (315) is equivalent to e fcl = ιcl-opwhich holds by (340)

Let A be a special symmetric Frobenius algebra So far we have seen that (A|Z(A) e)is a Cardy algebra and that all Cardy algebras with Aop = A and simple Acl are of thisform It is now natural to ask if every simple Acl does occur as part of a Cardy algebraThe following theorem provides an affirmative answer Recall that for an A-left moduleM the object Mor otimesA M is an algebra (see eg [KR Lem 42])

Theorem 322 If Acl is a simple modular invariant commutative symmetric FrobeniusC2plusmn-algebra then there exist a simple special symmetric Frobenius C-algebra A and amorphism ιcl-op Acl rarr R(A) such that

(i) Acl sim= Z(A) as Frobenius algebras(ii) (A|Acl ιcl-op) is a Cardy C|C2plusmn-algebra

(iii) T (Acl) sim= oplusκisinJ Morκ otimesA Mκ as algebras where MκκisinJ is a set of representa-

tives of the isomorphism classes of simple A-left modules

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis haploid It then follows from Theorem 34 that Acl is special By Proposition 224T (Acl) is a special symmetric Frobenius algebra in C Thus T (Acl) = oplusi Ai where theAi are simple symmetric Frobenius algebras We will show that at least one of the Aiis special Since T (Acl) is special we have mT (Acl) T (Acl) = ζ idT (Acl) for someζ isin C

times Restricting this to the summand Ai shows mi i = ζ idAi FurthermoreεT (Acl) ηT (Acl) = ξ id1 for some ξ isin C

times But εT (Acl) ηT (Acl) = sumi εi ηi and so

at least one of the εi ηi has to be nonzero Therefore at least one of the Ai is speciallet A equiv Ai be this summand We denote the embedding A rarr T (Acl) by e0 and therestriction T (Acl) A by r0 Notice that r0 is an algebra homomorphism Define

ιcl-op = χ (r0) Acl minusrarr R(A) (345)

By Proposition 216 ιcl-op is an algebra homomorphism Next we verify the centre con-dition (313) or rather its equivalent form (320) By substituting the definitions onecan convince oneself that the commutativity mcl cAclAcl = mcl of Acl in C2plusmn implies thecondition mT (Acl) = mT (Acl) in C see [K2 Prop 36] Here T (Acl)otimes T (Acl) rarrT (Acl)otimes T (Acl) is given by

=oplusmn

(idCl

notimes cCl

m Crnotimes idCr

m

)

(cCl

m Clnotimes cminus1

Crn C

rm

)

(idCl

motimes cminus1

Cln C

rm

otimes idCrn

)

(346)

and we decomposed Acl as Acl = oplusnCln times Cr

n As a consequence we obtain the identity

r0 mT (Acl) (idAcl otimes e0) = r0 mT (Acl) (idAcl otimes e0) (347)

904 L Kong I Runkel

Using that r0 is an algebra map and that by definition ιcl-op = r0 we obtain (320)In order to show that (A|Acl ιcl-op) is a Cardy algebra it remains to show that the

Cardy condition (314) is satisfied We will demonstrate this via a detour by first provingthat Acl sim= Z(A) as Frobenius algebras

Recall the notations e and r given in (331) Using the centre condition (320) onecan check that Pl

R(A) ιcl-op = m R(A) R(A) ιcl-op By specialness of A we have

m A A = ζA idA and so together with e r = ζminus1A Pl

R(A) we get

e r ιcl-op = ιcl-op (348)

Next consider the morphism

fcl = r ιcl-op Acl minusrarr Z(A) (349)

By the same derivation as in (342) and (343) one sees that fcl is an algebra map Inparticular fcl ηcl = ηZ(A) = 0 and so fcl = 0 Since Acl is simple fcl has to be amonomorphism

By the same argument as used in the proof of Theorem 34 (ii) up to isomorphismAcl is the unique simple local Acl-(left-)module The algebra monomorphism fcl turnsZ(A) into an Acl-module Since Z(A) is commutative it is local as an Acl-module andso Z(A) sim= AoplusN

cl for some N ge 1 By construction A is a simple special symmetricFrobenius algebra Proposition 225 and Corollary 227 show that R(A) inherits all theseproperties and thus Z(A) is simple (see the comment below Eq (332)) By Theorem318 Z(A) is modular invariant and then by Theorem 34 (i) dim Z(A) = Dim C Thisimplies that N = 1 in Z(A) sim= AoplusN

cl and so fcl is in fact an isomorphismSince Acl and Z(A) are both haploid we have εZ(A) fcl = ξ εcl for some ξ isin C

timesThe counit uniquely determines the Frobenius structure on Acl and Z(A) (see eg [FRSLemma 37]) so that fcl is a coalgebra isomorphism iff ξ = 1 To compute ξ we composethe above identity with ηcl from the right Defining ζcl via εcl ηcl = ζminus1

cl Dim C middot id1 andusing (333) gives ξ = dim A ζclζ

2A By rescaling the comultiplication and the counit

of A and consequently changing ζA we can always achieve ξ = 1 This proves part (i)of the theorem

Equation (348) implies that ιcl-op = e fcl Since fcl is an isomorphism of Frobeniusalgebras by Lemmas 218 and 319 we have

ιcl-op ιlowastcl-op = e fcl f lowastcl elowast = ζA e r = Pl

R(A) (350)

Thus (A|Acl ιcl-op) is a Cardy algebra This proves part (ii) of the theoremPart (iii) can be seen as follows By [KR Prop 43] T Z(A) sim= oplusκisinJ Mor

κ otimesA Mκ asalgebras Together with the observation that T ( fcl) T (Acl) rarr T Z(A) is an isomor-phism of algebras this proves part (iii) Remark 323 Part (i) of Theorem 322 was announced by Muumlger [Muuml2] We provide anindependent proof in the setting of Cardy algebras

The above theorem together with Lemma 211 and Theorem 34 shows that a sim-ple commutative symmetric Frobenius C2plusmn-algebra Acl with dim Acl = Dim C is alwayspart of a Cardy algebra (Aop|Acl ιcl-op) for some simple special symmetric Frobeniusalgebra Aop in C However the above proof also illustrates that Aop is not unique Thisraises the question how two Cardy algebras with a given Acl can differ This question isanswered by [KR Thm 11] which in the present framework can be restated as follows

Cardy Algebras and Sewing Constraints I 905

Theorem 324 If (A(i)op |A(i)cl ι(i)cl-op) i = 1 2 are two Cardy C|C2plusmn-algebras such that

A(i)cl is simple and dim A(i)op = 0 for i = 1 2 then A(1)clsim= A(2)cl as algebras if and only

if A(1)op and A(2)op are Morita equivalent

Proof Theorem 11 in [KR] is stated for A(i)op being non-degenerate algebras and A(i)cl =Z(A(i)op ) for i = 1 2 By Proposition 316 A(i)op are simple and special for i = 1 2

Then by [KR Lem 21] A(i)op are non-degenerate algebras By Theorem 321 we have

A(i)clsim= Z(A(i)op ) as Frobenius algebras Finally by [KR Thm 11] Z(A(1)op ) sim= Z(A(2)op )

as algebras iff A(1)op and A(2)op are Morita equivalent Let Cmax(C2plusmn) be the set of equivalence classes [B] of simple modular invariant

commutative symmetric Frobenius algebras B in C2plusmn Two such algebras B and B prime areequivalent if B and B prime are isomorphic as algebras (but not necessarily as Frobenius alge-bras) Let Msimp(C) be the set of Morita classes of simple special symmetric Frobeniusalgebras in C Define the map z Msimp(C) rarr Cmax(C2plusmn) by z A rarr [Z(A)] whereA denotes the Morita class of A From Theorem 322 (i) and [KR Thm 11] we learn

Corollary 325 The map z Msimp(C) rarr Cmax(C2plusmn) is a bijection

A Appendix

A1 Proof of Lemma 27 We will show that if (F ψ F2 ψ

F0 ) is a colax tensor functor

from C1 to C2 then (G φG2 φ

G0 ) is a lax tensor functor from C2 to C1 Applying this

result to the opposed categories then gives the converse statementWe need to show that φG

0 and φG2 make the diagrams (21) and (22) commute We

first prove the commutativity of (21) Consider the following diagram

G(A)otimes (G(B)otimes G(C))

GψF2 δ

idG(A)otimesφG2 G(A)otimes G(B otimes C)

GψF2 δ

G(FG(A)otimes F(G(B)otimes G(C)))

G(F(idG(A))otimesF(φG2 ))

G(idFG(A)otimesψ F2 )

G(FG(A)otimes FG(B otimes C))

G(ρAotimesρBotimesC )

G(FG(A)otimes (FG(B)otimes FG(C)))

G(ρAotimes(ρBotimesρC )) G(A otimes (B otimes C))

(A1)

The top subdiagram is commutative because of the naturality of Gψ F2 δ The commu-

tativity of the bottom subdiagram follows from the following identities

(ρB otimes ρC ) ψ F2 = (ρB otimes ρC ) ψ F

2 ρF Fδ

= ρBotimesC FG(ρB otimes ρC ) FG(ψ F2 ) Fδ

= ρBotimesC F(φG2 ) (A2)

as a map F(G(B) otimes G(C)) rarr B otimes C The commutativity of (A1) implies that thecomposition of maps in the left column in (21) can be replaced by

G(ρA otimes (ρB otimes ρC )) G((idFG(A) otimes ψ F

2 ) ψ F2

) δ (A3)

906 L Kong I Runkel

Similarly we can show that the composition of maps in the right column in (21) can bereplaced by

G((ρA otimes ρB)otimes ρC ) G((ψ F

2 otimes idFG(C)) ψ F2

) δ (A4)

Using the commutativity of (23) it is easy to see that (21) with the left and right col-umns of (21) replaced by (A3) and (A4) respectively is commutative Hence (21) iscommutative

Now we prove the commutativity of the first diagram in (22)

φG2 (φG

0 otimes idG(A))

(1)= G(ρ12 otimes ρA) Gψ F2 δ

[(Gψ F

0 δ11)otimes idG(A)

]

(2)= G(ρ12 otimes ρA) Gψ F2 G F(Gψ0 otimes idG(A)) G F(δ11 otimes idG(A)) δ

(3)= G(ρ12 otimes ρA) G(FG(ψ F0 )otimes idFG(A)) Gψ F

2 G F(δ11 otimes idG(A)) δ(4)= G(id12 otimes ρA) G

([ψ F

0 ρF(11) (Fδ)11] otimes idFG(A)

) Gψ F

2 δ(5)= G(id12 otimes ρA) G(ψ F

0 otimes idFG(A)) Gψ F2 δ

(6)= G(id12 otimes ρA) G(lminus1FG(A)) G F(lG(A)) δ

(7)= G(lminus1A ) GρA δG lG(A)

(8)= G(lminus1A ) lG(A) (A5)

where in step (1) we substituted the definition of φG0 φ

G2 given in (210) in step (2)

we used the naturality of δ in step (3) we used the naturality of Gψ F2 in step (4) we

switched the position between Gρ12 and G FG(ψ F0 ) and the position between Gψ2 and

G F(δ12 otimes idG(A)) using the naturality of ρ and FψG2 respectively in step (5) we applied

the second identity in (28) in step (6) we used (24) in step (7) we used the naturalityof lminus1 and δ in step (8) we used the first identity in (28)

The proof of the commutativity of the second diagram in (22) is similar Thus wehave shown that G is a lax tensor functor

A2 Proof of Lemma 320 To prepare the proof recall that for a given object B isin Cthe modular group P SL(2Z) acts on the space oplusi HomC(B otimes Ui Ui ) see eg [BKSect 31] and [K3 Eq (455)] We will only need the action of S and Sminus1 Let f isinoplusi HomC(B otimes Ui Ui ) Then

S oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A6)

Cardy Algebras and Sewing Constraints I 907

Sminus1 oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A7)

By Lemma 32 to establish that (337) is S-invariant it is enough to prove the identity(34) when f is given by (337) Using (A6) and (A7) we can see that Eq (34) simplysays that oplusi j [RHS of (34)] is invariant under the action of S times S Consider the elementg of oplus jkisinIHomC2plusmn(R(A)otimes (Uor

j times Uk)Uorj times Uk) given by

g =oplusjkisinI

sumα

R(A)

R(A)

R(A)

R(A)

Uorj times Uk

Uorj times Uk

α

α

(A8)

By the above arguments proving S-invariance of (337) is equivalent to proving invari-ance of g under the action of S times S

For i isin I we denote by gi the component of g in(oplus jisinIHomC(A otimes Uor

i otimes Uorj U

orj )

)otimes (opluskisinIHomC(Ui otimes UkUk))

We view the second Hom-space in the above tensor product as a Hom-space in C+ insteadof Cminus It is enough to show that gi is invariant under the action of S times Sminus1 Note thatthe action of Sminus1 in C+ is equivalent to that of S in Cminus

The morphism gi can be canonically identified with a bilinear pairing

( middot middot )i ⎛⎝oplus

jisinIHomC(Uor

j A otimes Uori otimes Uor

j )

⎞⎠

times(oplus

kisinIHomC(UkUi otimes Uk)

)minusrarr C (A9)

as follows For h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) we set

(h1 h2)i = (dim U j dim Uk)minus1 trUor

j timesUk[gi (h1 times h2)] (A10)

908 L Kong I Runkel

When substituting the explicit form of the product m R(A) of R(A) = (A times 1) otimes R(1)after a short calculation one finds

(h1 h2)i =sumα

1

dim U j

A

A

A

A

A A

A α

α

h1

h2

U jUk

Uorj

Ui

Uk (A11)

Here the top morphism PlR(A) has been simplified with the help of the identity

PR(A) m R(A) (PR(A) otimes PR(A))

=(oplus

iisinI((m A A)otimes idUor

i)times idUi

) m R(A) (PR(A) otimes PR(A)) (A12)

which can be checked by direct calculation along the same lines as in the proof of [FrLem 310]

The action of the modular transformation S on oplusiisinI(B otimes Ui Ui ) for B isin C nat-urally induces an action on oplusiisinI(Ui B otimes Ui ) [K3 Prop 514] which we denote bySlowast In the present case we get an action of Slowast on oplus jisinIHomC(Uor

j A otimes Uori otimes Uor

j ) and

opluskisinIHomC(UkUi otimes Uk) Then to show gi is invariant under the action of S times Sminus1

amounts to showing that

(h1 h2)i = ((Sminus1)lowast h1 Slowast h2)i (A13)

Cardy Algebras and Sewing Constraints I 909

for all h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) We have

((Sminus1)lowast h1 Slowast h2)i =sum

mnα

dim Un

DimC

A

A

A

A

A A

A

Uk

α

α

h1

h2

Ui

U j

UmUn

(A14)

Now drag the upper vertex indexed by α in the above graph along its Uorm -leg until it

meets the lower vertex also indexed by α then sum over α and m This gives

((Sminus1)lowast h1 Slowast h2)i =sum

n

dim Un

DimC

A

A

A

A

A A

A

Uk

Un

h1

h2

Ui

U j

(A15)

910 L Kong I Runkel

If we just look at the neighbourhood of the Un-loop in the above graph we see thefollowing subgraphs

sumn

dim Un

DimC

A Aor

Uorj U j

Uk Uork

Un=

sumα

1

dim U j

A Aor Uk Uork

Uorj U j

αα

(A16)

where we have applied [BK Cor 3111] Substituting this subgraph back to the originalgraph in (A15) we obtain

((Sminus1)lowast h1 Slowast h2)i =sumα

1

dim U j

A

A

A

A

A A

A

A

A

Uorj

α

α

h1

h2m A

Ui

U jUk

Uk

(A17)

The graph in (A17) is equal to that in (A11) In order to see this we first drag theldquobubblerdquo (m A A) along A lines and through the m A vertex (because m A A is abimodule map) until it reaches the lower-left leg of the upper vertex indexed by α Thendrag the m A vertex along the (red) dotted line in the above graph Finally we applythe associativity of A (A12) and [Fr Lem 311] Then we see that the graph in (A17)exactly matches with the one in (A11)

Acknowledgements We would like to thank the organisers of the Oberwolfach Arbeitsgemeinschaft ldquoAlge-braic structures in conformal field theoriesrdquo (April 2007) where this work was started for an inspiring meetingWe would further like to thank the Hausdorff Institute for Mathematics in Bonn and the organisers of the stim-ulating meeting ldquoGeometry and Physicsrdquo (May 2008) We are indebted to Alexei Davydov Jens FjelstadJuumlrgen Fuchs Yi-Zhi Huang Alexei Kitaev Urs Schreiber Christoph Schweigert Stephan Stolz and PeterTeichner for helpful discussions andor comments on a draft of this paper The research of IR was partiallysupported by the EPSRC First Grant EPE0050471 the PPARC rolling grant PPC5071451 and the MarieCurie network lsquoSuperstring Theoryrsquo (MRTN-CT-2004-512194)

Cardy Algebras and Sewing Constraints I 911

References

[AN] Alexeevski A Natanzon SM Noncommutative two-dimensional topological field theories andhurwitz numbers for real algebraic curves Sel Math New Ser 12 307ndash377 (2006)

[Bi] Bichon J Cosovereign hopf algebras J Pure Appl Alg 157 121ndash133 (2001)[BK] Bakalov B Kirillov AA Lectures on Tensor Categories and Modular Functors Providence

RI Amer Math Soc 2001[C1] Cardy JL Operator content of two-dimensional conformal invariant theories Nucl Phys

B 270 186ndash204 (1986)[C2] Cardy JL Boundary conditions fusion rules and the verlinde formula Nucl Phys B 324

581ndash596 (1989)[CL] Cardy JL Lewellen DC Bulk and boundary operators in conformal field theory Phys Lett

B 259 274ndash278 (1991)[DMZ] Dong C-Y Mason G Zhu Y-C Discrete series of the Virasoro algebra and the moonshine

module In Algebraic Groups and Their Generalizations Quantum and infinite-dimensional Meth-ods Proc Symp Pure Math 56 Part 2 Providence RI Amer MathSoc 1994 pp 295ndash316

[DP] Day B Pastro C Note on frobenius monoidal functors New York J Math 14 733ndash742 (2008)[ENO1] Etingof PI Nikshych D Ostrik V On fusion categories Ann Math 162 581ndash642 (2005)[ENO2] Etingof PI Nikshych D Ostrik V An analogue of radfordrsquos s4 formula for finite tensor

categories Int Math Research Notices 54 2915ndash2933 (2004)[Fe] Felder G Froumlhlich J Fuchs J Schweigert C Correlation functions and boundary conditions in

rational conformal field theory and three-dimensional topology Comp Math 131 189ndash237 (2002)[FHL] Frenkel IB Huang Y-Z Lepowsky J On axiomatic approaches to vertex operator algebras

and modules Mem Amer Math Soc 104 (1993)[Fj] Fjelstad J Fuchs J Runkel I Schweigert C Uniqueness of openclosed rational cft with

given algebra of open states Adv Theor Math Phys 12 1283ndash1375 (2008)[FK] Froumlhlich J King C The chern-simons theory and knot polynomials Commun Math

Phys 126 167ndash199 (1989)[Fr] Froumlhlich J Fuchs J Runkel I Schweigert C Correspondences of ribbon categories Adv

Math 199 192ndash329 (2006)[FRS] Fuchs J Runkel I Schweigert C Tft construction of rcft correlators I partition functions Nucl

Phys B 646 353ndash497 (2002)[FS] Fuchs J Schweigert C Category theory for conformal boundary conditions Fields Inst Com-

mun 39 25ndash71 (2003)[H1] Huang Y-Z Two-dimensional conformal geometry and vertex operator algebras Progress in

Mathematics Vol 148 Boston Birkhaumluser 1997[H2] Huang Y-Z Rigidity and modularity of vertex tensor categories Comm Contemp Math 10

871ndash911 (2008)[HK1] Huang Y-Z Kong L Open-string vertex algebra category and operads Commun Math

Phys 250 433ndash471 (2004)[HK2] Huang Y-Z Kong L Full field algebras Commun Math Phys 272 345ndash396 (2007)[HK3] Huang Y-Z Kong L Modular invariance for conformal full field algebras httparxivorgabs

math0609570v2[mathQA] 2006[HL] Huang Y-Z Lepowsky J Tensor products of modules for a vertex operator algebra and vertex

tensor categories In Lie Theory and Geometry in honor of Bertram Kostant ed R BrylinskiJ-L Brylinski V Guillemin V Kac Boston Birkhaumluser 1994 pp 349ndash383

[JS] Joyal A Street R Braided tensor categories Adv Math 102 20ndash78 (1993)[K1] Kong L Full field algebras operads and tensor categories Adv Math 213 271ndash340 (2007)[K2] Kong L Open-closed field algebras Commun Math Phys 280 207ndash261 (2008)[K3] Kong L Cardy condition for open-closed field algebras Commun Math Phys 283 25ndash92 (2008)[Ki] Kitaev A Private communication[KO] Kirillov AA Ostrik V On q-analog of mckay correspondence and ade classification of sl(2)

conformal field theories Adv Math 171 183ndash227 (2002)[KR] Kong L Runkel I Morita classes of algebras in modular tensor categories Adv Math 219

1548ndash1576 (2008)[Ld] Lauda AD Frobenius algebras and ambidextrous adjunctions Theo Appl Cat 16 84ndash

122 (2006)[Li1] Li H-S Regular representations of vertex operator algebras Commun Contemp Math 4

639ndash683 (2002)[Li2] Li H-S Regular representations and huang-lepowsky tensor functors for vertex operator

algebras J Alge 255 423ndash462 (2002)

900 L Kong I Runkel

On the other hand using that Acl is haploid that ιcl-op is an algebra map and thatι lowastcl-op is a coalgebra map one can check that the left-hand side of (328) is equal toλ(εop ι1 π1 ηop) π2 ηop for some λ = 0 This implies that π2 ηop = 0 Thusηop = sum

i ιi πi ηop = ι1 π1 ηop Hence we have

0 = π2 ι2 = π2 mop (ηop otimes ι2) = π2 mop ((ι1 π1 ηop)otimes ι2) (329)

However the right-hand side is zero by (327) This is a contradiction and hence Aopmust be simple

To formulate the next theorem we need the notion of the full centre of an algebra [FjDef 49] Recall that an algebra A in a braided tensor category has a left centre and aright centre [VZO] both of which are sub-algebras of A Of these two we will onlyneed the left centre The following definition is [Fr Def 231] which in our setting isequivalent to that of [VZO]

Definition 317 Let A be a symmetric special Frobenius algebra such that m A A =ζA idA

(i) The left centre Cl(A) of A is the image of the idempotent ζminus1A Pl

A(ii) The full centre Z(A) is Cl(R(A))

That ζminus1A Pl

A is an idempotent follows from [FRS Lem 52] when keeping track ofthe factors ζA ([FRS] assumes normalised-special ie ζA = 1) Note that Cl(A) is againan object of C while Z(A) is an object of C2plusmn Let el

A Cl(A) rarr A be the embeddingof Cl(A) into A The left centre is in fact the maximal subobject of A such that

m A cAA (elA otimes idA) = m A (330)

see [Fr Lem 232] This observation explains the name left centre and also makes theconnection to [O Def 15]

The full centre is by definition the image of the idempotent ζminus1A Pl

R(A) R(A) rarrR(A) Since C2plusmn is abelian the idempotent splits and we obtain the embedding andrestriction morphisms

e Z(A) rarr R(A) and r R(A) Z(A) (331)

which obey r e = idZ(A) and e r = ζminus1A Pl

R(A) It follows from Proposition 225 and

[Fr Prop 237] that Z(A) is a commutative symmetric Frobenius algebra in C2plusmn withstructure morphisms5

m Z(A) = r m R(A) (e otimes e) ηZ(A) = r ηR(A)

Z(A) = ζA middot (r otimes r) R(A) e εZ(A) = ζminus1A middot εR(A) e

(332)

Moreover if A is simple then Z(A) is simple and if A is simple and dim A = 0 thenZ(A) is simple and special The normalisation of the counit is such that

εZ(A) ηZ(A) = ζminus2A dim A DimC (333)

5 The normalisation of product and unit is the standard one The factors in the coproduct and counit haveto be included in order for (A|Z(A) e) to be a Cardy algebra see Theorem 318 below The normalisation ofthe counit enters the Cardy condition (314) through the definition of ( middot )lowast

Cardy Algebras and Sewing Constraints I 901

Theorem 318 Let A be a special symmetric Frobenius C-algebra Then (A|Z(A) e)is a Cardy C|C2plusmn-algebra

The proof of this theorem makes use the following two lemmas

Lemma 319 e Z(A) rarr R(A) is an algebra map and elowast = ζA middot r

Proof It follows from [Fr Lem 229] (or by direct calculation using in particularm R(A) R(A) = ζAidR(A)) that

m R(A) (e otimes e) = ζminus1A middot Pl

R(A) m R(A) (e otimes e) (334)

Substituting e r = ζminus1A Pl

R(A) shows that e is compatible with multiplication For theunit one finds

e ηZ(A) = e r ηR(A) = ζminus1A Pl

R(A) ηR(A) = ηR(A) (335)

Thus e is an algebra map For the second statement one computes

elowast (1)= ζA

R(A)

r

R(A)

R(A)

Z(A)

R(A)

Z(A)

r

e

e

(2)=

R(A)

r

R(A)

R(A)

R(A)

Z(A)

(3)=

R(A)

r

R(A)

Z(A)

(4)= ζA middot r (336)

where in (1) the definitions (230) and (332) have been substituted step (2) is e r = ζminus1

A PlR(A) step (3) uses that R(A) is symmetric Frobenius and step (4) is again

e r = ζminus1A Pl

R(A) Lemma 320 Let A be a symmetric Frobenius algebra in C The morphism

PlR(A) m R(A)

(Pl

R(A) otimes PlR(A)

) R(A)otimes R(A) minusrarr R(A) (337)

is S-invariant

The proof of this lemma is a slightly lengthy explicit calculation and has been deferredto Appendix A2

902 L Kong I Runkel

Proof of Theorem 318 That e is an algebra map was proved in Lemma 319 The centrecondition (313) holds by property (330) of the left centre The Cardy condition (314)also is an immediate consequence of Lemma 319

ιcl-op ιlowastcl-op = e (ζA r) = PlR(A) (338)

The full centre Z(A) is a commutative symmetric Frobenius algebra It remains to provemodular invariance That θZ(A) = idZ(A) is implied by commutativity and symmetryof Z(A) [Fr Prop 225] The S-invariance condition (33) follows from Lemma 320In (337) substitute Pl

R(A) = ζA e r and then put the resulting morphism into (33)Compose the resulting equation with e otimes idW from the right (ie from the bottom) andsubstitute the definition (332) of m Z(A) This results in the statement that m Z(A) isS-invariant

The following theorem is analogous to [LR Prop 29] and [Fj Thm 426] whichroughly speaking answer the question under which circumstances the restriction of atwo-dimensional conformal field theory to the boundary already determines the entireconformal field theory The first work is set in Minkowski space and uses operator alge-bras and subfactors while the second work is set in Euclidean space and uses modulartensor categories

Theorem 321 Let (A|Acl ιcl-op) be a Cardy C|C2plusmn-algebra such that dim A = 0 andAcl is simple Then A is special and (A|Acl ιcl-op) sim= (A|Z(A) e) as Cardy algebras

Proof By Proposition 316 A is simple and special Since Acl is simple the algebramap ιcl-op Acl rarr R(A) is either zero or a monomorphism But ιcl-op ηcl = ηR(A)and so ιcl-op = 0 Thus ιcl-op is monic By Lemma 218 (ii) ζminus1

A ιlowastcl-op is epi The Cardycondition (314) implies

ιcl-op ζminus1A ιlowastcl-op = ζminus1

A PlR(A) = e r (339)

Composing this with e r from the left yields e r ιcl-op ζminus1A ιlowastcl-op = e r =

ιcl-op ζminus1A ιlowastcl-op Since ζminus1

A ιlowastcl-op is epi we have

e r ιcl-op = ιcl-op (340)

Actually (340) also follows from (313) and specialness of R(A) We will prove that

( fop fcl) (A|Acl ιcl-op) minusrarr (A|Z(A) e) where fop = idA fcl = r ιcl-op

(341)

is an isomorphism of Cardy algebrasfcl is an algebra map Compatibility with the units follows since ιcl-op is an algebramap

fcl ηcl = r ιcl-op ηcl = r ηR(A) = ηZ(A) (342)

Compatibility with the multiplication also follows since ιcl-op is an algebra map

m Z(A) ( fcl otimes fcl) = r m R(A) (e otimes e) (r otimes r) (ιcl-op otimes ιcl-op)

= r m R(A) (ιcl-op otimes ιcl-op) = r ιcl-op mcl = fcl mcl (343)

where in the second step we used (340)

Cardy Algebras and Sewing Constraints I 903

fcl is an isomorphism As above since fcl is an algebra map and since Acl is simple fcl

has to be monic By Lemma 319 rlowast = ζminus1A e Thus f lowast

cl = ιlowastcl-op rlowast = ζminus1A ιlowastcl-op e and

fcl f lowastcl = r ιcl-op ζminus1

A ιlowastcl-op e = r e r e = idZ(A) (344)

and so fcl is also epi and hence isofcl is a coalgebra map Since fcl is an algebra map so is f minus1

cl By (344) f minus1cl = f lowast

cland by Lemma 218 (iii) this implies that fcl is a also coalgebra mapThe diagram (315) commutes Commutativity of (315) is equivalent to e fcl = ιcl-opwhich holds by (340)

Let A be a special symmetric Frobenius algebra So far we have seen that (A|Z(A) e)is a Cardy algebra and that all Cardy algebras with Aop = A and simple Acl are of thisform It is now natural to ask if every simple Acl does occur as part of a Cardy algebraThe following theorem provides an affirmative answer Recall that for an A-left moduleM the object Mor otimesA M is an algebra (see eg [KR Lem 42])

Theorem 322 If Acl is a simple modular invariant commutative symmetric FrobeniusC2plusmn-algebra then there exist a simple special symmetric Frobenius C-algebra A and amorphism ιcl-op Acl rarr R(A) such that

(i) Acl sim= Z(A) as Frobenius algebras(ii) (A|Acl ιcl-op) is a Cardy C|C2plusmn-algebra

(iii) T (Acl) sim= oplusκisinJ Morκ otimesA Mκ as algebras where MκκisinJ is a set of representa-

tives of the isomorphism classes of simple A-left modules

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis haploid It then follows from Theorem 34 that Acl is special By Proposition 224T (Acl) is a special symmetric Frobenius algebra in C Thus T (Acl) = oplusi Ai where theAi are simple symmetric Frobenius algebras We will show that at least one of the Aiis special Since T (Acl) is special we have mT (Acl) T (Acl) = ζ idT (Acl) for someζ isin C

times Restricting this to the summand Ai shows mi i = ζ idAi FurthermoreεT (Acl) ηT (Acl) = ξ id1 for some ξ isin C

times But εT (Acl) ηT (Acl) = sumi εi ηi and so

at least one of the εi ηi has to be nonzero Therefore at least one of the Ai is speciallet A equiv Ai be this summand We denote the embedding A rarr T (Acl) by e0 and therestriction T (Acl) A by r0 Notice that r0 is an algebra homomorphism Define

ιcl-op = χ (r0) Acl minusrarr R(A) (345)

By Proposition 216 ιcl-op is an algebra homomorphism Next we verify the centre con-dition (313) or rather its equivalent form (320) By substituting the definitions onecan convince oneself that the commutativity mcl cAclAcl = mcl of Acl in C2plusmn implies thecondition mT (Acl) = mT (Acl) in C see [K2 Prop 36] Here T (Acl)otimes T (Acl) rarrT (Acl)otimes T (Acl) is given by

=oplusmn

(idCl

notimes cCl

m Crnotimes idCr

m

)

(cCl

m Clnotimes cminus1

Crn C

rm

)

(idCl

motimes cminus1

Cln C

rm

otimes idCrn

)

(346)

and we decomposed Acl as Acl = oplusnCln times Cr

n As a consequence we obtain the identity

r0 mT (Acl) (idAcl otimes e0) = r0 mT (Acl) (idAcl otimes e0) (347)

904 L Kong I Runkel

Using that r0 is an algebra map and that by definition ιcl-op = r0 we obtain (320)In order to show that (A|Acl ιcl-op) is a Cardy algebra it remains to show that the

Cardy condition (314) is satisfied We will demonstrate this via a detour by first provingthat Acl sim= Z(A) as Frobenius algebras

Recall the notations e and r given in (331) Using the centre condition (320) onecan check that Pl

R(A) ιcl-op = m R(A) R(A) ιcl-op By specialness of A we have

m A A = ζA idA and so together with e r = ζminus1A Pl

R(A) we get

e r ιcl-op = ιcl-op (348)

Next consider the morphism

fcl = r ιcl-op Acl minusrarr Z(A) (349)

By the same derivation as in (342) and (343) one sees that fcl is an algebra map Inparticular fcl ηcl = ηZ(A) = 0 and so fcl = 0 Since Acl is simple fcl has to be amonomorphism

By the same argument as used in the proof of Theorem 34 (ii) up to isomorphismAcl is the unique simple local Acl-(left-)module The algebra monomorphism fcl turnsZ(A) into an Acl-module Since Z(A) is commutative it is local as an Acl-module andso Z(A) sim= AoplusN

cl for some N ge 1 By construction A is a simple special symmetricFrobenius algebra Proposition 225 and Corollary 227 show that R(A) inherits all theseproperties and thus Z(A) is simple (see the comment below Eq (332)) By Theorem318 Z(A) is modular invariant and then by Theorem 34 (i) dim Z(A) = Dim C Thisimplies that N = 1 in Z(A) sim= AoplusN

cl and so fcl is in fact an isomorphismSince Acl and Z(A) are both haploid we have εZ(A) fcl = ξ εcl for some ξ isin C

timesThe counit uniquely determines the Frobenius structure on Acl and Z(A) (see eg [FRSLemma 37]) so that fcl is a coalgebra isomorphism iff ξ = 1 To compute ξ we composethe above identity with ηcl from the right Defining ζcl via εcl ηcl = ζminus1

cl Dim C middot id1 andusing (333) gives ξ = dim A ζclζ

2A By rescaling the comultiplication and the counit

of A and consequently changing ζA we can always achieve ξ = 1 This proves part (i)of the theorem

Equation (348) implies that ιcl-op = e fcl Since fcl is an isomorphism of Frobeniusalgebras by Lemmas 218 and 319 we have

ιcl-op ιlowastcl-op = e fcl f lowastcl elowast = ζA e r = Pl

R(A) (350)

Thus (A|Acl ιcl-op) is a Cardy algebra This proves part (ii) of the theoremPart (iii) can be seen as follows By [KR Prop 43] T Z(A) sim= oplusκisinJ Mor

κ otimesA Mκ asalgebras Together with the observation that T ( fcl) T (Acl) rarr T Z(A) is an isomor-phism of algebras this proves part (iii) Remark 323 Part (i) of Theorem 322 was announced by Muumlger [Muuml2] We provide anindependent proof in the setting of Cardy algebras

The above theorem together with Lemma 211 and Theorem 34 shows that a sim-ple commutative symmetric Frobenius C2plusmn-algebra Acl with dim Acl = Dim C is alwayspart of a Cardy algebra (Aop|Acl ιcl-op) for some simple special symmetric Frobeniusalgebra Aop in C However the above proof also illustrates that Aop is not unique Thisraises the question how two Cardy algebras with a given Acl can differ This question isanswered by [KR Thm 11] which in the present framework can be restated as follows

Cardy Algebras and Sewing Constraints I 905

Theorem 324 If (A(i)op |A(i)cl ι(i)cl-op) i = 1 2 are two Cardy C|C2plusmn-algebras such that

A(i)cl is simple and dim A(i)op = 0 for i = 1 2 then A(1)clsim= A(2)cl as algebras if and only

if A(1)op and A(2)op are Morita equivalent

Proof Theorem 11 in [KR] is stated for A(i)op being non-degenerate algebras and A(i)cl =Z(A(i)op ) for i = 1 2 By Proposition 316 A(i)op are simple and special for i = 1 2

Then by [KR Lem 21] A(i)op are non-degenerate algebras By Theorem 321 we have

A(i)clsim= Z(A(i)op ) as Frobenius algebras Finally by [KR Thm 11] Z(A(1)op ) sim= Z(A(2)op )

as algebras iff A(1)op and A(2)op are Morita equivalent Let Cmax(C2plusmn) be the set of equivalence classes [B] of simple modular invariant

commutative symmetric Frobenius algebras B in C2plusmn Two such algebras B and B prime areequivalent if B and B prime are isomorphic as algebras (but not necessarily as Frobenius alge-bras) Let Msimp(C) be the set of Morita classes of simple special symmetric Frobeniusalgebras in C Define the map z Msimp(C) rarr Cmax(C2plusmn) by z A rarr [Z(A)] whereA denotes the Morita class of A From Theorem 322 (i) and [KR Thm 11] we learn

Corollary 325 The map z Msimp(C) rarr Cmax(C2plusmn) is a bijection

A Appendix

A1 Proof of Lemma 27 We will show that if (F ψ F2 ψ

F0 ) is a colax tensor functor

from C1 to C2 then (G φG2 φ

G0 ) is a lax tensor functor from C2 to C1 Applying this

result to the opposed categories then gives the converse statementWe need to show that φG

0 and φG2 make the diagrams (21) and (22) commute We

first prove the commutativity of (21) Consider the following diagram

G(A)otimes (G(B)otimes G(C))

GψF2 δ

idG(A)otimesφG2 G(A)otimes G(B otimes C)

GψF2 δ

G(FG(A)otimes F(G(B)otimes G(C)))

G(F(idG(A))otimesF(φG2 ))

G(idFG(A)otimesψ F2 )

G(FG(A)otimes FG(B otimes C))

G(ρAotimesρBotimesC )

G(FG(A)otimes (FG(B)otimes FG(C)))

G(ρAotimes(ρBotimesρC )) G(A otimes (B otimes C))

(A1)

The top subdiagram is commutative because of the naturality of Gψ F2 δ The commu-

tativity of the bottom subdiagram follows from the following identities

(ρB otimes ρC ) ψ F2 = (ρB otimes ρC ) ψ F

2 ρF Fδ

= ρBotimesC FG(ρB otimes ρC ) FG(ψ F2 ) Fδ

= ρBotimesC F(φG2 ) (A2)

as a map F(G(B) otimes G(C)) rarr B otimes C The commutativity of (A1) implies that thecomposition of maps in the left column in (21) can be replaced by

G(ρA otimes (ρB otimes ρC )) G((idFG(A) otimes ψ F

2 ) ψ F2

) δ (A3)

906 L Kong I Runkel

Similarly we can show that the composition of maps in the right column in (21) can bereplaced by

G((ρA otimes ρB)otimes ρC ) G((ψ F

2 otimes idFG(C)) ψ F2

) δ (A4)

Using the commutativity of (23) it is easy to see that (21) with the left and right col-umns of (21) replaced by (A3) and (A4) respectively is commutative Hence (21) iscommutative

Now we prove the commutativity of the first diagram in (22)

φG2 (φG

0 otimes idG(A))

(1)= G(ρ12 otimes ρA) Gψ F2 δ

[(Gψ F

0 δ11)otimes idG(A)

]

(2)= G(ρ12 otimes ρA) Gψ F2 G F(Gψ0 otimes idG(A)) G F(δ11 otimes idG(A)) δ

(3)= G(ρ12 otimes ρA) G(FG(ψ F0 )otimes idFG(A)) Gψ F

2 G F(δ11 otimes idG(A)) δ(4)= G(id12 otimes ρA) G

([ψ F

0 ρF(11) (Fδ)11] otimes idFG(A)

) Gψ F

2 δ(5)= G(id12 otimes ρA) G(ψ F

0 otimes idFG(A)) Gψ F2 δ

(6)= G(id12 otimes ρA) G(lminus1FG(A)) G F(lG(A)) δ

(7)= G(lminus1A ) GρA δG lG(A)

(8)= G(lminus1A ) lG(A) (A5)

where in step (1) we substituted the definition of φG0 φ

G2 given in (210) in step (2)

we used the naturality of δ in step (3) we used the naturality of Gψ F2 in step (4) we

switched the position between Gρ12 and G FG(ψ F0 ) and the position between Gψ2 and

G F(δ12 otimes idG(A)) using the naturality of ρ and FψG2 respectively in step (5) we applied

the second identity in (28) in step (6) we used (24) in step (7) we used the naturalityof lminus1 and δ in step (8) we used the first identity in (28)

The proof of the commutativity of the second diagram in (22) is similar Thus wehave shown that G is a lax tensor functor

A2 Proof of Lemma 320 To prepare the proof recall that for a given object B isin Cthe modular group P SL(2Z) acts on the space oplusi HomC(B otimes Ui Ui ) see eg [BKSect 31] and [K3 Eq (455)] We will only need the action of S and Sminus1 Let f isinoplusi HomC(B otimes Ui Ui ) Then

S oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A6)

Cardy Algebras and Sewing Constraints I 907

Sminus1 oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A7)

By Lemma 32 to establish that (337) is S-invariant it is enough to prove the identity(34) when f is given by (337) Using (A6) and (A7) we can see that Eq (34) simplysays that oplusi j [RHS of (34)] is invariant under the action of S times S Consider the elementg of oplus jkisinIHomC2plusmn(R(A)otimes (Uor

j times Uk)Uorj times Uk) given by

g =oplusjkisinI

sumα

R(A)

R(A)

R(A)

R(A)

Uorj times Uk

Uorj times Uk

α

α

(A8)

By the above arguments proving S-invariance of (337) is equivalent to proving invari-ance of g under the action of S times S

For i isin I we denote by gi the component of g in(oplus jisinIHomC(A otimes Uor

i otimes Uorj U

orj )

)otimes (opluskisinIHomC(Ui otimes UkUk))

We view the second Hom-space in the above tensor product as a Hom-space in C+ insteadof Cminus It is enough to show that gi is invariant under the action of S times Sminus1 Note thatthe action of Sminus1 in C+ is equivalent to that of S in Cminus

The morphism gi can be canonically identified with a bilinear pairing

( middot middot )i ⎛⎝oplus

jisinIHomC(Uor

j A otimes Uori otimes Uor

j )

⎞⎠

times(oplus

kisinIHomC(UkUi otimes Uk)

)minusrarr C (A9)

as follows For h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) we set

(h1 h2)i = (dim U j dim Uk)minus1 trUor

j timesUk[gi (h1 times h2)] (A10)

908 L Kong I Runkel

When substituting the explicit form of the product m R(A) of R(A) = (A times 1) otimes R(1)after a short calculation one finds

(h1 h2)i =sumα

1

dim U j

A

A

A

A

A A

A α

α

h1

h2

U jUk

Uorj

Ui

Uk (A11)

Here the top morphism PlR(A) has been simplified with the help of the identity

PR(A) m R(A) (PR(A) otimes PR(A))

=(oplus

iisinI((m A A)otimes idUor

i)times idUi

) m R(A) (PR(A) otimes PR(A)) (A12)

which can be checked by direct calculation along the same lines as in the proof of [FrLem 310]

The action of the modular transformation S on oplusiisinI(B otimes Ui Ui ) for B isin C nat-urally induces an action on oplusiisinI(Ui B otimes Ui ) [K3 Prop 514] which we denote bySlowast In the present case we get an action of Slowast on oplus jisinIHomC(Uor

j A otimes Uori otimes Uor

j ) and

opluskisinIHomC(UkUi otimes Uk) Then to show gi is invariant under the action of S times Sminus1

amounts to showing that

(h1 h2)i = ((Sminus1)lowast h1 Slowast h2)i (A13)

Cardy Algebras and Sewing Constraints I 909

for all h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) We have

((Sminus1)lowast h1 Slowast h2)i =sum

mnα

dim Un

DimC

A

A

A

A

A A

A

Uk

α

α

h1

h2

Ui

U j

UmUn

(A14)

Now drag the upper vertex indexed by α in the above graph along its Uorm -leg until it

meets the lower vertex also indexed by α then sum over α and m This gives

((Sminus1)lowast h1 Slowast h2)i =sum

n

dim Un

DimC

A

A

A

A

A A

A

Uk

Un

h1

h2

Ui

U j

(A15)

910 L Kong I Runkel

If we just look at the neighbourhood of the Un-loop in the above graph we see thefollowing subgraphs

sumn

dim Un

DimC

A Aor

Uorj U j

Uk Uork

Un=

sumα

1

dim U j

A Aor Uk Uork

Uorj U j

αα

(A16)

where we have applied [BK Cor 3111] Substituting this subgraph back to the originalgraph in (A15) we obtain

((Sminus1)lowast h1 Slowast h2)i =sumα

1

dim U j

A

A

A

A

A A

A

A

A

Uorj

α

α

h1

h2m A

Ui

U jUk

Uk

(A17)

The graph in (A17) is equal to that in (A11) In order to see this we first drag theldquobubblerdquo (m A A) along A lines and through the m A vertex (because m A A is abimodule map) until it reaches the lower-left leg of the upper vertex indexed by α Thendrag the m A vertex along the (red) dotted line in the above graph Finally we applythe associativity of A (A12) and [Fr Lem 311] Then we see that the graph in (A17)exactly matches with the one in (A11)

Acknowledgements We would like to thank the organisers of the Oberwolfach Arbeitsgemeinschaft ldquoAlge-braic structures in conformal field theoriesrdquo (April 2007) where this work was started for an inspiring meetingWe would further like to thank the Hausdorff Institute for Mathematics in Bonn and the organisers of the stim-ulating meeting ldquoGeometry and Physicsrdquo (May 2008) We are indebted to Alexei Davydov Jens FjelstadJuumlrgen Fuchs Yi-Zhi Huang Alexei Kitaev Urs Schreiber Christoph Schweigert Stephan Stolz and PeterTeichner for helpful discussions andor comments on a draft of this paper The research of IR was partiallysupported by the EPSRC First Grant EPE0050471 the PPARC rolling grant PPC5071451 and the MarieCurie network lsquoSuperstring Theoryrsquo (MRTN-CT-2004-512194)

Cardy Algebras and Sewing Constraints I 911

References

[AN] Alexeevski A Natanzon SM Noncommutative two-dimensional topological field theories andhurwitz numbers for real algebraic curves Sel Math New Ser 12 307ndash377 (2006)

[Bi] Bichon J Cosovereign hopf algebras J Pure Appl Alg 157 121ndash133 (2001)[BK] Bakalov B Kirillov AA Lectures on Tensor Categories and Modular Functors Providence

RI Amer Math Soc 2001[C1] Cardy JL Operator content of two-dimensional conformal invariant theories Nucl Phys

B 270 186ndash204 (1986)[C2] Cardy JL Boundary conditions fusion rules and the verlinde formula Nucl Phys B 324

581ndash596 (1989)[CL] Cardy JL Lewellen DC Bulk and boundary operators in conformal field theory Phys Lett

B 259 274ndash278 (1991)[DMZ] Dong C-Y Mason G Zhu Y-C Discrete series of the Virasoro algebra and the moonshine

module In Algebraic Groups and Their Generalizations Quantum and infinite-dimensional Meth-ods Proc Symp Pure Math 56 Part 2 Providence RI Amer MathSoc 1994 pp 295ndash316

[DP] Day B Pastro C Note on frobenius monoidal functors New York J Math 14 733ndash742 (2008)[ENO1] Etingof PI Nikshych D Ostrik V On fusion categories Ann Math 162 581ndash642 (2005)[ENO2] Etingof PI Nikshych D Ostrik V An analogue of radfordrsquos s4 formula for finite tensor

categories Int Math Research Notices 54 2915ndash2933 (2004)[Fe] Felder G Froumlhlich J Fuchs J Schweigert C Correlation functions and boundary conditions in

rational conformal field theory and three-dimensional topology Comp Math 131 189ndash237 (2002)[FHL] Frenkel IB Huang Y-Z Lepowsky J On axiomatic approaches to vertex operator algebras

and modules Mem Amer Math Soc 104 (1993)[Fj] Fjelstad J Fuchs J Runkel I Schweigert C Uniqueness of openclosed rational cft with

given algebra of open states Adv Theor Math Phys 12 1283ndash1375 (2008)[FK] Froumlhlich J King C The chern-simons theory and knot polynomials Commun Math

Phys 126 167ndash199 (1989)[Fr] Froumlhlich J Fuchs J Runkel I Schweigert C Correspondences of ribbon categories Adv

Math 199 192ndash329 (2006)[FRS] Fuchs J Runkel I Schweigert C Tft construction of rcft correlators I partition functions Nucl

Phys B 646 353ndash497 (2002)[FS] Fuchs J Schweigert C Category theory for conformal boundary conditions Fields Inst Com-

mun 39 25ndash71 (2003)[H1] Huang Y-Z Two-dimensional conformal geometry and vertex operator algebras Progress in

Mathematics Vol 148 Boston Birkhaumluser 1997[H2] Huang Y-Z Rigidity and modularity of vertex tensor categories Comm Contemp Math 10

871ndash911 (2008)[HK1] Huang Y-Z Kong L Open-string vertex algebra category and operads Commun Math

Phys 250 433ndash471 (2004)[HK2] Huang Y-Z Kong L Full field algebras Commun Math Phys 272 345ndash396 (2007)[HK3] Huang Y-Z Kong L Modular invariance for conformal full field algebras httparxivorgabs

math0609570v2[mathQA] 2006[HL] Huang Y-Z Lepowsky J Tensor products of modules for a vertex operator algebra and vertex

tensor categories In Lie Theory and Geometry in honor of Bertram Kostant ed R BrylinskiJ-L Brylinski V Guillemin V Kac Boston Birkhaumluser 1994 pp 349ndash383

[JS] Joyal A Street R Braided tensor categories Adv Math 102 20ndash78 (1993)[K1] Kong L Full field algebras operads and tensor categories Adv Math 213 271ndash340 (2007)[K2] Kong L Open-closed field algebras Commun Math Phys 280 207ndash261 (2008)[K3] Kong L Cardy condition for open-closed field algebras Commun Math Phys 283 25ndash92 (2008)[Ki] Kitaev A Private communication[KO] Kirillov AA Ostrik V On q-analog of mckay correspondence and ade classification of sl(2)

conformal field theories Adv Math 171 183ndash227 (2002)[KR] Kong L Runkel I Morita classes of algebras in modular tensor categories Adv Math 219

1548ndash1576 (2008)[Ld] Lauda AD Frobenius algebras and ambidextrous adjunctions Theo Appl Cat 16 84ndash

122 (2006)[Li1] Li H-S Regular representations of vertex operator algebras Commun Contemp Math 4

639ndash683 (2002)[Li2] Li H-S Regular representations and huang-lepowsky tensor functors for vertex operator

algebras J Alge 255 423ndash462 (2002)

Cardy Algebras and Sewing Constraints I 901

Theorem 318 Let A be a special symmetric Frobenius C-algebra Then (A|Z(A) e)is a Cardy C|C2plusmn-algebra

The proof of this theorem makes use the following two lemmas

Lemma 319 e Z(A) rarr R(A) is an algebra map and elowast = ζA middot r

Proof It follows from [Fr Lem 229] (or by direct calculation using in particularm R(A) R(A) = ζAidR(A)) that

m R(A) (e otimes e) = ζminus1A middot Pl

R(A) m R(A) (e otimes e) (334)

Substituting e r = ζminus1A Pl

R(A) shows that e is compatible with multiplication For theunit one finds

e ηZ(A) = e r ηR(A) = ζminus1A Pl

R(A) ηR(A) = ηR(A) (335)

Thus e is an algebra map For the second statement one computes

elowast (1)= ζA

R(A)

r

R(A)

R(A)

Z(A)

R(A)

Z(A)

r

e

e

(2)=

R(A)

r

R(A)

R(A)

R(A)

Z(A)

(3)=

R(A)

r

R(A)

Z(A)

(4)= ζA middot r (336)

where in (1) the definitions (230) and (332) have been substituted step (2) is e r = ζminus1

A PlR(A) step (3) uses that R(A) is symmetric Frobenius and step (4) is again

e r = ζminus1A Pl

R(A) Lemma 320 Let A be a symmetric Frobenius algebra in C The morphism

PlR(A) m R(A)

(Pl

R(A) otimes PlR(A)

) R(A)otimes R(A) minusrarr R(A) (337)

is S-invariant

The proof of this lemma is a slightly lengthy explicit calculation and has been deferredto Appendix A2

902 L Kong I Runkel

Proof of Theorem 318 That e is an algebra map was proved in Lemma 319 The centrecondition (313) holds by property (330) of the left centre The Cardy condition (314)also is an immediate consequence of Lemma 319

ιcl-op ιlowastcl-op = e (ζA r) = PlR(A) (338)

The full centre Z(A) is a commutative symmetric Frobenius algebra It remains to provemodular invariance That θZ(A) = idZ(A) is implied by commutativity and symmetryof Z(A) [Fr Prop 225] The S-invariance condition (33) follows from Lemma 320In (337) substitute Pl

R(A) = ζA e r and then put the resulting morphism into (33)Compose the resulting equation with e otimes idW from the right (ie from the bottom) andsubstitute the definition (332) of m Z(A) This results in the statement that m Z(A) isS-invariant

The following theorem is analogous to [LR Prop 29] and [Fj Thm 426] whichroughly speaking answer the question under which circumstances the restriction of atwo-dimensional conformal field theory to the boundary already determines the entireconformal field theory The first work is set in Minkowski space and uses operator alge-bras and subfactors while the second work is set in Euclidean space and uses modulartensor categories

Theorem 321 Let (A|Acl ιcl-op) be a Cardy C|C2plusmn-algebra such that dim A = 0 andAcl is simple Then A is special and (A|Acl ιcl-op) sim= (A|Z(A) e) as Cardy algebras

Proof By Proposition 316 A is simple and special Since Acl is simple the algebramap ιcl-op Acl rarr R(A) is either zero or a monomorphism But ιcl-op ηcl = ηR(A)and so ιcl-op = 0 Thus ιcl-op is monic By Lemma 218 (ii) ζminus1

A ιlowastcl-op is epi The Cardycondition (314) implies

ιcl-op ζminus1A ιlowastcl-op = ζminus1

A PlR(A) = e r (339)

Composing this with e r from the left yields e r ιcl-op ζminus1A ιlowastcl-op = e r =

ιcl-op ζminus1A ιlowastcl-op Since ζminus1

A ιlowastcl-op is epi we have

e r ιcl-op = ιcl-op (340)

Actually (340) also follows from (313) and specialness of R(A) We will prove that

( fop fcl) (A|Acl ιcl-op) minusrarr (A|Z(A) e) where fop = idA fcl = r ιcl-op

(341)

is an isomorphism of Cardy algebrasfcl is an algebra map Compatibility with the units follows since ιcl-op is an algebramap

fcl ηcl = r ιcl-op ηcl = r ηR(A) = ηZ(A) (342)

Compatibility with the multiplication also follows since ιcl-op is an algebra map

m Z(A) ( fcl otimes fcl) = r m R(A) (e otimes e) (r otimes r) (ιcl-op otimes ιcl-op)

= r m R(A) (ιcl-op otimes ιcl-op) = r ιcl-op mcl = fcl mcl (343)

where in the second step we used (340)

Cardy Algebras and Sewing Constraints I 903

fcl is an isomorphism As above since fcl is an algebra map and since Acl is simple fcl

has to be monic By Lemma 319 rlowast = ζminus1A e Thus f lowast

cl = ιlowastcl-op rlowast = ζminus1A ιlowastcl-op e and

fcl f lowastcl = r ιcl-op ζminus1

A ιlowastcl-op e = r e r e = idZ(A) (344)

and so fcl is also epi and hence isofcl is a coalgebra map Since fcl is an algebra map so is f minus1

cl By (344) f minus1cl = f lowast

cland by Lemma 218 (iii) this implies that fcl is a also coalgebra mapThe diagram (315) commutes Commutativity of (315) is equivalent to e fcl = ιcl-opwhich holds by (340)

Let A be a special symmetric Frobenius algebra So far we have seen that (A|Z(A) e)is a Cardy algebra and that all Cardy algebras with Aop = A and simple Acl are of thisform It is now natural to ask if every simple Acl does occur as part of a Cardy algebraThe following theorem provides an affirmative answer Recall that for an A-left moduleM the object Mor otimesA M is an algebra (see eg [KR Lem 42])

Theorem 322 If Acl is a simple modular invariant commutative symmetric FrobeniusC2plusmn-algebra then there exist a simple special symmetric Frobenius C-algebra A and amorphism ιcl-op Acl rarr R(A) such that

(i) Acl sim= Z(A) as Frobenius algebras(ii) (A|Acl ιcl-op) is a Cardy C|C2plusmn-algebra

(iii) T (Acl) sim= oplusκisinJ Morκ otimesA Mκ as algebras where MκκisinJ is a set of representa-

tives of the isomorphism classes of simple A-left modules

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis haploid It then follows from Theorem 34 that Acl is special By Proposition 224T (Acl) is a special symmetric Frobenius algebra in C Thus T (Acl) = oplusi Ai where theAi are simple symmetric Frobenius algebras We will show that at least one of the Aiis special Since T (Acl) is special we have mT (Acl) T (Acl) = ζ idT (Acl) for someζ isin C

times Restricting this to the summand Ai shows mi i = ζ idAi FurthermoreεT (Acl) ηT (Acl) = ξ id1 for some ξ isin C

times But εT (Acl) ηT (Acl) = sumi εi ηi and so

at least one of the εi ηi has to be nonzero Therefore at least one of the Ai is speciallet A equiv Ai be this summand We denote the embedding A rarr T (Acl) by e0 and therestriction T (Acl) A by r0 Notice that r0 is an algebra homomorphism Define

ιcl-op = χ (r0) Acl minusrarr R(A) (345)

By Proposition 216 ιcl-op is an algebra homomorphism Next we verify the centre con-dition (313) or rather its equivalent form (320) By substituting the definitions onecan convince oneself that the commutativity mcl cAclAcl = mcl of Acl in C2plusmn implies thecondition mT (Acl) = mT (Acl) in C see [K2 Prop 36] Here T (Acl)otimes T (Acl) rarrT (Acl)otimes T (Acl) is given by

=oplusmn

(idCl

notimes cCl

m Crnotimes idCr

m

)

(cCl

m Clnotimes cminus1

Crn C

rm

)

(idCl

motimes cminus1

Cln C

rm

otimes idCrn

)

(346)

and we decomposed Acl as Acl = oplusnCln times Cr

n As a consequence we obtain the identity

r0 mT (Acl) (idAcl otimes e0) = r0 mT (Acl) (idAcl otimes e0) (347)

904 L Kong I Runkel

Using that r0 is an algebra map and that by definition ιcl-op = r0 we obtain (320)In order to show that (A|Acl ιcl-op) is a Cardy algebra it remains to show that the

Cardy condition (314) is satisfied We will demonstrate this via a detour by first provingthat Acl sim= Z(A) as Frobenius algebras

Recall the notations e and r given in (331) Using the centre condition (320) onecan check that Pl

R(A) ιcl-op = m R(A) R(A) ιcl-op By specialness of A we have

m A A = ζA idA and so together with e r = ζminus1A Pl

R(A) we get

e r ιcl-op = ιcl-op (348)

Next consider the morphism

fcl = r ιcl-op Acl minusrarr Z(A) (349)

By the same derivation as in (342) and (343) one sees that fcl is an algebra map Inparticular fcl ηcl = ηZ(A) = 0 and so fcl = 0 Since Acl is simple fcl has to be amonomorphism

By the same argument as used in the proof of Theorem 34 (ii) up to isomorphismAcl is the unique simple local Acl-(left-)module The algebra monomorphism fcl turnsZ(A) into an Acl-module Since Z(A) is commutative it is local as an Acl-module andso Z(A) sim= AoplusN

cl for some N ge 1 By construction A is a simple special symmetricFrobenius algebra Proposition 225 and Corollary 227 show that R(A) inherits all theseproperties and thus Z(A) is simple (see the comment below Eq (332)) By Theorem318 Z(A) is modular invariant and then by Theorem 34 (i) dim Z(A) = Dim C Thisimplies that N = 1 in Z(A) sim= AoplusN

cl and so fcl is in fact an isomorphismSince Acl and Z(A) are both haploid we have εZ(A) fcl = ξ εcl for some ξ isin C

timesThe counit uniquely determines the Frobenius structure on Acl and Z(A) (see eg [FRSLemma 37]) so that fcl is a coalgebra isomorphism iff ξ = 1 To compute ξ we composethe above identity with ηcl from the right Defining ζcl via εcl ηcl = ζminus1

cl Dim C middot id1 andusing (333) gives ξ = dim A ζclζ

2A By rescaling the comultiplication and the counit

of A and consequently changing ζA we can always achieve ξ = 1 This proves part (i)of the theorem

Equation (348) implies that ιcl-op = e fcl Since fcl is an isomorphism of Frobeniusalgebras by Lemmas 218 and 319 we have

ιcl-op ιlowastcl-op = e fcl f lowastcl elowast = ζA e r = Pl

R(A) (350)

Thus (A|Acl ιcl-op) is a Cardy algebra This proves part (ii) of the theoremPart (iii) can be seen as follows By [KR Prop 43] T Z(A) sim= oplusκisinJ Mor

κ otimesA Mκ asalgebras Together with the observation that T ( fcl) T (Acl) rarr T Z(A) is an isomor-phism of algebras this proves part (iii) Remark 323 Part (i) of Theorem 322 was announced by Muumlger [Muuml2] We provide anindependent proof in the setting of Cardy algebras

The above theorem together with Lemma 211 and Theorem 34 shows that a sim-ple commutative symmetric Frobenius C2plusmn-algebra Acl with dim Acl = Dim C is alwayspart of a Cardy algebra (Aop|Acl ιcl-op) for some simple special symmetric Frobeniusalgebra Aop in C However the above proof also illustrates that Aop is not unique Thisraises the question how two Cardy algebras with a given Acl can differ This question isanswered by [KR Thm 11] which in the present framework can be restated as follows

Cardy Algebras and Sewing Constraints I 905

Theorem 324 If (A(i)op |A(i)cl ι(i)cl-op) i = 1 2 are two Cardy C|C2plusmn-algebras such that

A(i)cl is simple and dim A(i)op = 0 for i = 1 2 then A(1)clsim= A(2)cl as algebras if and only

if A(1)op and A(2)op are Morita equivalent

Proof Theorem 11 in [KR] is stated for A(i)op being non-degenerate algebras and A(i)cl =Z(A(i)op ) for i = 1 2 By Proposition 316 A(i)op are simple and special for i = 1 2

Then by [KR Lem 21] A(i)op are non-degenerate algebras By Theorem 321 we have

A(i)clsim= Z(A(i)op ) as Frobenius algebras Finally by [KR Thm 11] Z(A(1)op ) sim= Z(A(2)op )

as algebras iff A(1)op and A(2)op are Morita equivalent Let Cmax(C2plusmn) be the set of equivalence classes [B] of simple modular invariant

commutative symmetric Frobenius algebras B in C2plusmn Two such algebras B and B prime areequivalent if B and B prime are isomorphic as algebras (but not necessarily as Frobenius alge-bras) Let Msimp(C) be the set of Morita classes of simple special symmetric Frobeniusalgebras in C Define the map z Msimp(C) rarr Cmax(C2plusmn) by z A rarr [Z(A)] whereA denotes the Morita class of A From Theorem 322 (i) and [KR Thm 11] we learn

Corollary 325 The map z Msimp(C) rarr Cmax(C2plusmn) is a bijection

A Appendix

A1 Proof of Lemma 27 We will show that if (F ψ F2 ψ

F0 ) is a colax tensor functor

from C1 to C2 then (G φG2 φ

G0 ) is a lax tensor functor from C2 to C1 Applying this

result to the opposed categories then gives the converse statementWe need to show that φG

0 and φG2 make the diagrams (21) and (22) commute We

first prove the commutativity of (21) Consider the following diagram

G(A)otimes (G(B)otimes G(C))

GψF2 δ

idG(A)otimesφG2 G(A)otimes G(B otimes C)

GψF2 δ

G(FG(A)otimes F(G(B)otimes G(C)))

G(F(idG(A))otimesF(φG2 ))

G(idFG(A)otimesψ F2 )

G(FG(A)otimes FG(B otimes C))

G(ρAotimesρBotimesC )

G(FG(A)otimes (FG(B)otimes FG(C)))

G(ρAotimes(ρBotimesρC )) G(A otimes (B otimes C))

(A1)

The top subdiagram is commutative because of the naturality of Gψ F2 δ The commu-

tativity of the bottom subdiagram follows from the following identities

(ρB otimes ρC ) ψ F2 = (ρB otimes ρC ) ψ F

2 ρF Fδ

= ρBotimesC FG(ρB otimes ρC ) FG(ψ F2 ) Fδ

= ρBotimesC F(φG2 ) (A2)

as a map F(G(B) otimes G(C)) rarr B otimes C The commutativity of (A1) implies that thecomposition of maps in the left column in (21) can be replaced by

G(ρA otimes (ρB otimes ρC )) G((idFG(A) otimes ψ F

2 ) ψ F2

) δ (A3)

906 L Kong I Runkel

Similarly we can show that the composition of maps in the right column in (21) can bereplaced by

G((ρA otimes ρB)otimes ρC ) G((ψ F

2 otimes idFG(C)) ψ F2

) δ (A4)

Using the commutativity of (23) it is easy to see that (21) with the left and right col-umns of (21) replaced by (A3) and (A4) respectively is commutative Hence (21) iscommutative

Now we prove the commutativity of the first diagram in (22)

φG2 (φG

0 otimes idG(A))

(1)= G(ρ12 otimes ρA) Gψ F2 δ

[(Gψ F

0 δ11)otimes idG(A)

]

(2)= G(ρ12 otimes ρA) Gψ F2 G F(Gψ0 otimes idG(A)) G F(δ11 otimes idG(A)) δ

(3)= G(ρ12 otimes ρA) G(FG(ψ F0 )otimes idFG(A)) Gψ F

2 G F(δ11 otimes idG(A)) δ(4)= G(id12 otimes ρA) G

([ψ F

0 ρF(11) (Fδ)11] otimes idFG(A)

) Gψ F

2 δ(5)= G(id12 otimes ρA) G(ψ F

0 otimes idFG(A)) Gψ F2 δ

(6)= G(id12 otimes ρA) G(lminus1FG(A)) G F(lG(A)) δ

(7)= G(lminus1A ) GρA δG lG(A)

(8)= G(lminus1A ) lG(A) (A5)

where in step (1) we substituted the definition of φG0 φ

G2 given in (210) in step (2)

we used the naturality of δ in step (3) we used the naturality of Gψ F2 in step (4) we

switched the position between Gρ12 and G FG(ψ F0 ) and the position between Gψ2 and

G F(δ12 otimes idG(A)) using the naturality of ρ and FψG2 respectively in step (5) we applied

the second identity in (28) in step (6) we used (24) in step (7) we used the naturalityof lminus1 and δ in step (8) we used the first identity in (28)

The proof of the commutativity of the second diagram in (22) is similar Thus wehave shown that G is a lax tensor functor

A2 Proof of Lemma 320 To prepare the proof recall that for a given object B isin Cthe modular group P SL(2Z) acts on the space oplusi HomC(B otimes Ui Ui ) see eg [BKSect 31] and [K3 Eq (455)] We will only need the action of S and Sminus1 Let f isinoplusi HomC(B otimes Ui Ui ) Then

S oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A6)

Cardy Algebras and Sewing Constraints I 907

Sminus1 oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A7)

By Lemma 32 to establish that (337) is S-invariant it is enough to prove the identity(34) when f is given by (337) Using (A6) and (A7) we can see that Eq (34) simplysays that oplusi j [RHS of (34)] is invariant under the action of S times S Consider the elementg of oplus jkisinIHomC2plusmn(R(A)otimes (Uor

j times Uk)Uorj times Uk) given by

g =oplusjkisinI

sumα

R(A)

R(A)

R(A)

R(A)

Uorj times Uk

Uorj times Uk

α

α

(A8)

By the above arguments proving S-invariance of (337) is equivalent to proving invari-ance of g under the action of S times S

For i isin I we denote by gi the component of g in(oplus jisinIHomC(A otimes Uor

i otimes Uorj U

orj )

)otimes (opluskisinIHomC(Ui otimes UkUk))

We view the second Hom-space in the above tensor product as a Hom-space in C+ insteadof Cminus It is enough to show that gi is invariant under the action of S times Sminus1 Note thatthe action of Sminus1 in C+ is equivalent to that of S in Cminus

The morphism gi can be canonically identified with a bilinear pairing

( middot middot )i ⎛⎝oplus

jisinIHomC(Uor

j A otimes Uori otimes Uor

j )

⎞⎠

times(oplus

kisinIHomC(UkUi otimes Uk)

)minusrarr C (A9)

as follows For h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) we set

(h1 h2)i = (dim U j dim Uk)minus1 trUor

j timesUk[gi (h1 times h2)] (A10)

908 L Kong I Runkel

When substituting the explicit form of the product m R(A) of R(A) = (A times 1) otimes R(1)after a short calculation one finds

(h1 h2)i =sumα

1

dim U j

A

A

A

A

A A

A α

α

h1

h2

U jUk

Uorj

Ui

Uk (A11)

Here the top morphism PlR(A) has been simplified with the help of the identity

PR(A) m R(A) (PR(A) otimes PR(A))

=(oplus

iisinI((m A A)otimes idUor

i)times idUi

) m R(A) (PR(A) otimes PR(A)) (A12)

which can be checked by direct calculation along the same lines as in the proof of [FrLem 310]

The action of the modular transformation S on oplusiisinI(B otimes Ui Ui ) for B isin C nat-urally induces an action on oplusiisinI(Ui B otimes Ui ) [K3 Prop 514] which we denote bySlowast In the present case we get an action of Slowast on oplus jisinIHomC(Uor

j A otimes Uori otimes Uor

j ) and

opluskisinIHomC(UkUi otimes Uk) Then to show gi is invariant under the action of S times Sminus1

amounts to showing that

(h1 h2)i = ((Sminus1)lowast h1 Slowast h2)i (A13)

Cardy Algebras and Sewing Constraints I 909

for all h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) We have

((Sminus1)lowast h1 Slowast h2)i =sum

mnα

dim Un

DimC

A

A

A

A

A A

A

Uk

α

α

h1

h2

Ui

U j

UmUn

(A14)

Now drag the upper vertex indexed by α in the above graph along its Uorm -leg until it

meets the lower vertex also indexed by α then sum over α and m This gives

((Sminus1)lowast h1 Slowast h2)i =sum

n

dim Un

DimC

A

A

A

A

A A

A

Uk

Un

h1

h2

Ui

U j

(A15)

910 L Kong I Runkel

If we just look at the neighbourhood of the Un-loop in the above graph we see thefollowing subgraphs

sumn

dim Un

DimC

A Aor

Uorj U j

Uk Uork

Un=

sumα

1

dim U j

A Aor Uk Uork

Uorj U j

αα

(A16)

where we have applied [BK Cor 3111] Substituting this subgraph back to the originalgraph in (A15) we obtain

((Sminus1)lowast h1 Slowast h2)i =sumα

1

dim U j

A

A

A

A

A A

A

A

A

Uorj

α

α

h1

h2m A

Ui

U jUk

Uk

(A17)

The graph in (A17) is equal to that in (A11) In order to see this we first drag theldquobubblerdquo (m A A) along A lines and through the m A vertex (because m A A is abimodule map) until it reaches the lower-left leg of the upper vertex indexed by α Thendrag the m A vertex along the (red) dotted line in the above graph Finally we applythe associativity of A (A12) and [Fr Lem 311] Then we see that the graph in (A17)exactly matches with the one in (A11)

Acknowledgements We would like to thank the organisers of the Oberwolfach Arbeitsgemeinschaft ldquoAlge-braic structures in conformal field theoriesrdquo (April 2007) where this work was started for an inspiring meetingWe would further like to thank the Hausdorff Institute for Mathematics in Bonn and the organisers of the stim-ulating meeting ldquoGeometry and Physicsrdquo (May 2008) We are indebted to Alexei Davydov Jens FjelstadJuumlrgen Fuchs Yi-Zhi Huang Alexei Kitaev Urs Schreiber Christoph Schweigert Stephan Stolz and PeterTeichner for helpful discussions andor comments on a draft of this paper The research of IR was partiallysupported by the EPSRC First Grant EPE0050471 the PPARC rolling grant PPC5071451 and the MarieCurie network lsquoSuperstring Theoryrsquo (MRTN-CT-2004-512194)

Cardy Algebras and Sewing Constraints I 911

References

[AN] Alexeevski A Natanzon SM Noncommutative two-dimensional topological field theories andhurwitz numbers for real algebraic curves Sel Math New Ser 12 307ndash377 (2006)

[Bi] Bichon J Cosovereign hopf algebras J Pure Appl Alg 157 121ndash133 (2001)[BK] Bakalov B Kirillov AA Lectures on Tensor Categories and Modular Functors Providence

RI Amer Math Soc 2001[C1] Cardy JL Operator content of two-dimensional conformal invariant theories Nucl Phys

B 270 186ndash204 (1986)[C2] Cardy JL Boundary conditions fusion rules and the verlinde formula Nucl Phys B 324

581ndash596 (1989)[CL] Cardy JL Lewellen DC Bulk and boundary operators in conformal field theory Phys Lett

B 259 274ndash278 (1991)[DMZ] Dong C-Y Mason G Zhu Y-C Discrete series of the Virasoro algebra and the moonshine

module In Algebraic Groups and Their Generalizations Quantum and infinite-dimensional Meth-ods Proc Symp Pure Math 56 Part 2 Providence RI Amer MathSoc 1994 pp 295ndash316

[DP] Day B Pastro C Note on frobenius monoidal functors New York J Math 14 733ndash742 (2008)[ENO1] Etingof PI Nikshych D Ostrik V On fusion categories Ann Math 162 581ndash642 (2005)[ENO2] Etingof PI Nikshych D Ostrik V An analogue of radfordrsquos s4 formula for finite tensor

categories Int Math Research Notices 54 2915ndash2933 (2004)[Fe] Felder G Froumlhlich J Fuchs J Schweigert C Correlation functions and boundary conditions in

rational conformal field theory and three-dimensional topology Comp Math 131 189ndash237 (2002)[FHL] Frenkel IB Huang Y-Z Lepowsky J On axiomatic approaches to vertex operator algebras

and modules Mem Amer Math Soc 104 (1993)[Fj] Fjelstad J Fuchs J Runkel I Schweigert C Uniqueness of openclosed rational cft with

given algebra of open states Adv Theor Math Phys 12 1283ndash1375 (2008)[FK] Froumlhlich J King C The chern-simons theory and knot polynomials Commun Math

Phys 126 167ndash199 (1989)[Fr] Froumlhlich J Fuchs J Runkel I Schweigert C Correspondences of ribbon categories Adv

Math 199 192ndash329 (2006)[FRS] Fuchs J Runkel I Schweigert C Tft construction of rcft correlators I partition functions Nucl

Phys B 646 353ndash497 (2002)[FS] Fuchs J Schweigert C Category theory for conformal boundary conditions Fields Inst Com-

mun 39 25ndash71 (2003)[H1] Huang Y-Z Two-dimensional conformal geometry and vertex operator algebras Progress in

Mathematics Vol 148 Boston Birkhaumluser 1997[H2] Huang Y-Z Rigidity and modularity of vertex tensor categories Comm Contemp Math 10

871ndash911 (2008)[HK1] Huang Y-Z Kong L Open-string vertex algebra category and operads Commun Math

Phys 250 433ndash471 (2004)[HK2] Huang Y-Z Kong L Full field algebras Commun Math Phys 272 345ndash396 (2007)[HK3] Huang Y-Z Kong L Modular invariance for conformal full field algebras httparxivorgabs

math0609570v2[mathQA] 2006[HL] Huang Y-Z Lepowsky J Tensor products of modules for a vertex operator algebra and vertex

tensor categories In Lie Theory and Geometry in honor of Bertram Kostant ed R BrylinskiJ-L Brylinski V Guillemin V Kac Boston Birkhaumluser 1994 pp 349ndash383

[JS] Joyal A Street R Braided tensor categories Adv Math 102 20ndash78 (1993)[K1] Kong L Full field algebras operads and tensor categories Adv Math 213 271ndash340 (2007)[K2] Kong L Open-closed field algebras Commun Math Phys 280 207ndash261 (2008)[K3] Kong L Cardy condition for open-closed field algebras Commun Math Phys 283 25ndash92 (2008)[Ki] Kitaev A Private communication[KO] Kirillov AA Ostrik V On q-analog of mckay correspondence and ade classification of sl(2)

conformal field theories Adv Math 171 183ndash227 (2002)[KR] Kong L Runkel I Morita classes of algebras in modular tensor categories Adv Math 219

1548ndash1576 (2008)[Ld] Lauda AD Frobenius algebras and ambidextrous adjunctions Theo Appl Cat 16 84ndash

122 (2006)[Li1] Li H-S Regular representations of vertex operator algebras Commun Contemp Math 4

639ndash683 (2002)[Li2] Li H-S Regular representations and huang-lepowsky tensor functors for vertex operator

algebras J Alge 255 423ndash462 (2002)

902 L Kong I Runkel

Proof of Theorem 318 That e is an algebra map was proved in Lemma 319 The centrecondition (313) holds by property (330) of the left centre The Cardy condition (314)also is an immediate consequence of Lemma 319

ιcl-op ιlowastcl-op = e (ζA r) = PlR(A) (338)

The full centre Z(A) is a commutative symmetric Frobenius algebra It remains to provemodular invariance That θZ(A) = idZ(A) is implied by commutativity and symmetryof Z(A) [Fr Prop 225] The S-invariance condition (33) follows from Lemma 320In (337) substitute Pl

R(A) = ζA e r and then put the resulting morphism into (33)Compose the resulting equation with e otimes idW from the right (ie from the bottom) andsubstitute the definition (332) of m Z(A) This results in the statement that m Z(A) isS-invariant

The following theorem is analogous to [LR Prop 29] and [Fj Thm 426] whichroughly speaking answer the question under which circumstances the restriction of atwo-dimensional conformal field theory to the boundary already determines the entireconformal field theory The first work is set in Minkowski space and uses operator alge-bras and subfactors while the second work is set in Euclidean space and uses modulartensor categories

Theorem 321 Let (A|Acl ιcl-op) be a Cardy C|C2plusmn-algebra such that dim A = 0 andAcl is simple Then A is special and (A|Acl ιcl-op) sim= (A|Z(A) e) as Cardy algebras

Proof By Proposition 316 A is simple and special Since Acl is simple the algebramap ιcl-op Acl rarr R(A) is either zero or a monomorphism But ιcl-op ηcl = ηR(A)and so ιcl-op = 0 Thus ιcl-op is monic By Lemma 218 (ii) ζminus1

A ιlowastcl-op is epi The Cardycondition (314) implies

ιcl-op ζminus1A ιlowastcl-op = ζminus1

A PlR(A) = e r (339)

Composing this with e r from the left yields e r ιcl-op ζminus1A ιlowastcl-op = e r =

ιcl-op ζminus1A ιlowastcl-op Since ζminus1

A ιlowastcl-op is epi we have

e r ιcl-op = ιcl-op (340)

Actually (340) also follows from (313) and specialness of R(A) We will prove that

( fop fcl) (A|Acl ιcl-op) minusrarr (A|Z(A) e) where fop = idA fcl = r ιcl-op

(341)

is an isomorphism of Cardy algebrasfcl is an algebra map Compatibility with the units follows since ιcl-op is an algebramap

fcl ηcl = r ιcl-op ηcl = r ηR(A) = ηZ(A) (342)

Compatibility with the multiplication also follows since ιcl-op is an algebra map

m Z(A) ( fcl otimes fcl) = r m R(A) (e otimes e) (r otimes r) (ιcl-op otimes ιcl-op)

= r m R(A) (ιcl-op otimes ιcl-op) = r ιcl-op mcl = fcl mcl (343)

where in the second step we used (340)

Cardy Algebras and Sewing Constraints I 903

fcl is an isomorphism As above since fcl is an algebra map and since Acl is simple fcl

has to be monic By Lemma 319 rlowast = ζminus1A e Thus f lowast

cl = ιlowastcl-op rlowast = ζminus1A ιlowastcl-op e and

fcl f lowastcl = r ιcl-op ζminus1

A ιlowastcl-op e = r e r e = idZ(A) (344)

and so fcl is also epi and hence isofcl is a coalgebra map Since fcl is an algebra map so is f minus1

cl By (344) f minus1cl = f lowast

cland by Lemma 218 (iii) this implies that fcl is a also coalgebra mapThe diagram (315) commutes Commutativity of (315) is equivalent to e fcl = ιcl-opwhich holds by (340)

Let A be a special symmetric Frobenius algebra So far we have seen that (A|Z(A) e)is a Cardy algebra and that all Cardy algebras with Aop = A and simple Acl are of thisform It is now natural to ask if every simple Acl does occur as part of a Cardy algebraThe following theorem provides an affirmative answer Recall that for an A-left moduleM the object Mor otimesA M is an algebra (see eg [KR Lem 42])

Theorem 322 If Acl is a simple modular invariant commutative symmetric FrobeniusC2plusmn-algebra then there exist a simple special symmetric Frobenius C-algebra A and amorphism ιcl-op Acl rarr R(A) such that

(i) Acl sim= Z(A) as Frobenius algebras(ii) (A|Acl ιcl-op) is a Cardy C|C2plusmn-algebra

(iii) T (Acl) sim= oplusκisinJ Morκ otimesA Mκ as algebras where MκκisinJ is a set of representa-

tives of the isomorphism classes of simple A-left modules

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis haploid It then follows from Theorem 34 that Acl is special By Proposition 224T (Acl) is a special symmetric Frobenius algebra in C Thus T (Acl) = oplusi Ai where theAi are simple symmetric Frobenius algebras We will show that at least one of the Aiis special Since T (Acl) is special we have mT (Acl) T (Acl) = ζ idT (Acl) for someζ isin C

times Restricting this to the summand Ai shows mi i = ζ idAi FurthermoreεT (Acl) ηT (Acl) = ξ id1 for some ξ isin C

times But εT (Acl) ηT (Acl) = sumi εi ηi and so

at least one of the εi ηi has to be nonzero Therefore at least one of the Ai is speciallet A equiv Ai be this summand We denote the embedding A rarr T (Acl) by e0 and therestriction T (Acl) A by r0 Notice that r0 is an algebra homomorphism Define

ιcl-op = χ (r0) Acl minusrarr R(A) (345)

By Proposition 216 ιcl-op is an algebra homomorphism Next we verify the centre con-dition (313) or rather its equivalent form (320) By substituting the definitions onecan convince oneself that the commutativity mcl cAclAcl = mcl of Acl in C2plusmn implies thecondition mT (Acl) = mT (Acl) in C see [K2 Prop 36] Here T (Acl)otimes T (Acl) rarrT (Acl)otimes T (Acl) is given by

=oplusmn

(idCl

notimes cCl

m Crnotimes idCr

m

)

(cCl

m Clnotimes cminus1

Crn C

rm

)

(idCl

motimes cminus1

Cln C

rm

otimes idCrn

)

(346)

and we decomposed Acl as Acl = oplusnCln times Cr

n As a consequence we obtain the identity

r0 mT (Acl) (idAcl otimes e0) = r0 mT (Acl) (idAcl otimes e0) (347)

904 L Kong I Runkel

Using that r0 is an algebra map and that by definition ιcl-op = r0 we obtain (320)In order to show that (A|Acl ιcl-op) is a Cardy algebra it remains to show that the

Cardy condition (314) is satisfied We will demonstrate this via a detour by first provingthat Acl sim= Z(A) as Frobenius algebras

Recall the notations e and r given in (331) Using the centre condition (320) onecan check that Pl

R(A) ιcl-op = m R(A) R(A) ιcl-op By specialness of A we have

m A A = ζA idA and so together with e r = ζminus1A Pl

R(A) we get

e r ιcl-op = ιcl-op (348)

Next consider the morphism

fcl = r ιcl-op Acl minusrarr Z(A) (349)

By the same derivation as in (342) and (343) one sees that fcl is an algebra map Inparticular fcl ηcl = ηZ(A) = 0 and so fcl = 0 Since Acl is simple fcl has to be amonomorphism

By the same argument as used in the proof of Theorem 34 (ii) up to isomorphismAcl is the unique simple local Acl-(left-)module The algebra monomorphism fcl turnsZ(A) into an Acl-module Since Z(A) is commutative it is local as an Acl-module andso Z(A) sim= AoplusN

cl for some N ge 1 By construction A is a simple special symmetricFrobenius algebra Proposition 225 and Corollary 227 show that R(A) inherits all theseproperties and thus Z(A) is simple (see the comment below Eq (332)) By Theorem318 Z(A) is modular invariant and then by Theorem 34 (i) dim Z(A) = Dim C Thisimplies that N = 1 in Z(A) sim= AoplusN

cl and so fcl is in fact an isomorphismSince Acl and Z(A) are both haploid we have εZ(A) fcl = ξ εcl for some ξ isin C

timesThe counit uniquely determines the Frobenius structure on Acl and Z(A) (see eg [FRSLemma 37]) so that fcl is a coalgebra isomorphism iff ξ = 1 To compute ξ we composethe above identity with ηcl from the right Defining ζcl via εcl ηcl = ζminus1

cl Dim C middot id1 andusing (333) gives ξ = dim A ζclζ

2A By rescaling the comultiplication and the counit

of A and consequently changing ζA we can always achieve ξ = 1 This proves part (i)of the theorem

Equation (348) implies that ιcl-op = e fcl Since fcl is an isomorphism of Frobeniusalgebras by Lemmas 218 and 319 we have

ιcl-op ιlowastcl-op = e fcl f lowastcl elowast = ζA e r = Pl

R(A) (350)

Thus (A|Acl ιcl-op) is a Cardy algebra This proves part (ii) of the theoremPart (iii) can be seen as follows By [KR Prop 43] T Z(A) sim= oplusκisinJ Mor

κ otimesA Mκ asalgebras Together with the observation that T ( fcl) T (Acl) rarr T Z(A) is an isomor-phism of algebras this proves part (iii) Remark 323 Part (i) of Theorem 322 was announced by Muumlger [Muuml2] We provide anindependent proof in the setting of Cardy algebras

The above theorem together with Lemma 211 and Theorem 34 shows that a sim-ple commutative symmetric Frobenius C2plusmn-algebra Acl with dim Acl = Dim C is alwayspart of a Cardy algebra (Aop|Acl ιcl-op) for some simple special symmetric Frobeniusalgebra Aop in C However the above proof also illustrates that Aop is not unique Thisraises the question how two Cardy algebras with a given Acl can differ This question isanswered by [KR Thm 11] which in the present framework can be restated as follows

Cardy Algebras and Sewing Constraints I 905

Theorem 324 If (A(i)op |A(i)cl ι(i)cl-op) i = 1 2 are two Cardy C|C2plusmn-algebras such that

A(i)cl is simple and dim A(i)op = 0 for i = 1 2 then A(1)clsim= A(2)cl as algebras if and only

if A(1)op and A(2)op are Morita equivalent

Proof Theorem 11 in [KR] is stated for A(i)op being non-degenerate algebras and A(i)cl =Z(A(i)op ) for i = 1 2 By Proposition 316 A(i)op are simple and special for i = 1 2

Then by [KR Lem 21] A(i)op are non-degenerate algebras By Theorem 321 we have

A(i)clsim= Z(A(i)op ) as Frobenius algebras Finally by [KR Thm 11] Z(A(1)op ) sim= Z(A(2)op )

as algebras iff A(1)op and A(2)op are Morita equivalent Let Cmax(C2plusmn) be the set of equivalence classes [B] of simple modular invariant

commutative symmetric Frobenius algebras B in C2plusmn Two such algebras B and B prime areequivalent if B and B prime are isomorphic as algebras (but not necessarily as Frobenius alge-bras) Let Msimp(C) be the set of Morita classes of simple special symmetric Frobeniusalgebras in C Define the map z Msimp(C) rarr Cmax(C2plusmn) by z A rarr [Z(A)] whereA denotes the Morita class of A From Theorem 322 (i) and [KR Thm 11] we learn

Corollary 325 The map z Msimp(C) rarr Cmax(C2plusmn) is a bijection

A Appendix

A1 Proof of Lemma 27 We will show that if (F ψ F2 ψ

F0 ) is a colax tensor functor

from C1 to C2 then (G φG2 φ

G0 ) is a lax tensor functor from C2 to C1 Applying this

result to the opposed categories then gives the converse statementWe need to show that φG

0 and φG2 make the diagrams (21) and (22) commute We

first prove the commutativity of (21) Consider the following diagram

G(A)otimes (G(B)otimes G(C))

GψF2 δ

idG(A)otimesφG2 G(A)otimes G(B otimes C)

GψF2 δ

G(FG(A)otimes F(G(B)otimes G(C)))

G(F(idG(A))otimesF(φG2 ))

G(idFG(A)otimesψ F2 )

G(FG(A)otimes FG(B otimes C))

G(ρAotimesρBotimesC )

G(FG(A)otimes (FG(B)otimes FG(C)))

G(ρAotimes(ρBotimesρC )) G(A otimes (B otimes C))

(A1)

The top subdiagram is commutative because of the naturality of Gψ F2 δ The commu-

tativity of the bottom subdiagram follows from the following identities

(ρB otimes ρC ) ψ F2 = (ρB otimes ρC ) ψ F

2 ρF Fδ

= ρBotimesC FG(ρB otimes ρC ) FG(ψ F2 ) Fδ

= ρBotimesC F(φG2 ) (A2)

as a map F(G(B) otimes G(C)) rarr B otimes C The commutativity of (A1) implies that thecomposition of maps in the left column in (21) can be replaced by

G(ρA otimes (ρB otimes ρC )) G((idFG(A) otimes ψ F

2 ) ψ F2

) δ (A3)

906 L Kong I Runkel

Similarly we can show that the composition of maps in the right column in (21) can bereplaced by

G((ρA otimes ρB)otimes ρC ) G((ψ F

2 otimes idFG(C)) ψ F2

) δ (A4)

Using the commutativity of (23) it is easy to see that (21) with the left and right col-umns of (21) replaced by (A3) and (A4) respectively is commutative Hence (21) iscommutative

Now we prove the commutativity of the first diagram in (22)

φG2 (φG

0 otimes idG(A))

(1)= G(ρ12 otimes ρA) Gψ F2 δ

[(Gψ F

0 δ11)otimes idG(A)

]

(2)= G(ρ12 otimes ρA) Gψ F2 G F(Gψ0 otimes idG(A)) G F(δ11 otimes idG(A)) δ

(3)= G(ρ12 otimes ρA) G(FG(ψ F0 )otimes idFG(A)) Gψ F

2 G F(δ11 otimes idG(A)) δ(4)= G(id12 otimes ρA) G

([ψ F

0 ρF(11) (Fδ)11] otimes idFG(A)

) Gψ F

2 δ(5)= G(id12 otimes ρA) G(ψ F

0 otimes idFG(A)) Gψ F2 δ

(6)= G(id12 otimes ρA) G(lminus1FG(A)) G F(lG(A)) δ

(7)= G(lminus1A ) GρA δG lG(A)

(8)= G(lminus1A ) lG(A) (A5)

where in step (1) we substituted the definition of φG0 φ

G2 given in (210) in step (2)

we used the naturality of δ in step (3) we used the naturality of Gψ F2 in step (4) we

switched the position between Gρ12 and G FG(ψ F0 ) and the position between Gψ2 and

G F(δ12 otimes idG(A)) using the naturality of ρ and FψG2 respectively in step (5) we applied

the second identity in (28) in step (6) we used (24) in step (7) we used the naturalityof lminus1 and δ in step (8) we used the first identity in (28)

The proof of the commutativity of the second diagram in (22) is similar Thus wehave shown that G is a lax tensor functor

A2 Proof of Lemma 320 To prepare the proof recall that for a given object B isin Cthe modular group P SL(2Z) acts on the space oplusi HomC(B otimes Ui Ui ) see eg [BKSect 31] and [K3 Eq (455)] We will only need the action of S and Sminus1 Let f isinoplusi HomC(B otimes Ui Ui ) Then

S oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A6)

Cardy Algebras and Sewing Constraints I 907

Sminus1 oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A7)

By Lemma 32 to establish that (337) is S-invariant it is enough to prove the identity(34) when f is given by (337) Using (A6) and (A7) we can see that Eq (34) simplysays that oplusi j [RHS of (34)] is invariant under the action of S times S Consider the elementg of oplus jkisinIHomC2plusmn(R(A)otimes (Uor

j times Uk)Uorj times Uk) given by

g =oplusjkisinI

sumα

R(A)

R(A)

R(A)

R(A)

Uorj times Uk

Uorj times Uk

α

α

(A8)

By the above arguments proving S-invariance of (337) is equivalent to proving invari-ance of g under the action of S times S

For i isin I we denote by gi the component of g in(oplus jisinIHomC(A otimes Uor

i otimes Uorj U

orj )

)otimes (opluskisinIHomC(Ui otimes UkUk))

We view the second Hom-space in the above tensor product as a Hom-space in C+ insteadof Cminus It is enough to show that gi is invariant under the action of S times Sminus1 Note thatthe action of Sminus1 in C+ is equivalent to that of S in Cminus

The morphism gi can be canonically identified with a bilinear pairing

( middot middot )i ⎛⎝oplus

jisinIHomC(Uor

j A otimes Uori otimes Uor

j )

⎞⎠

times(oplus

kisinIHomC(UkUi otimes Uk)

)minusrarr C (A9)

as follows For h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) we set

(h1 h2)i = (dim U j dim Uk)minus1 trUor

j timesUk[gi (h1 times h2)] (A10)

908 L Kong I Runkel

When substituting the explicit form of the product m R(A) of R(A) = (A times 1) otimes R(1)after a short calculation one finds

(h1 h2)i =sumα

1

dim U j

A

A

A

A

A A

A α

α

h1

h2

U jUk

Uorj

Ui

Uk (A11)

Here the top morphism PlR(A) has been simplified with the help of the identity

PR(A) m R(A) (PR(A) otimes PR(A))

=(oplus

iisinI((m A A)otimes idUor

i)times idUi

) m R(A) (PR(A) otimes PR(A)) (A12)

which can be checked by direct calculation along the same lines as in the proof of [FrLem 310]

The action of the modular transformation S on oplusiisinI(B otimes Ui Ui ) for B isin C nat-urally induces an action on oplusiisinI(Ui B otimes Ui ) [K3 Prop 514] which we denote bySlowast In the present case we get an action of Slowast on oplus jisinIHomC(Uor

j A otimes Uori otimes Uor

j ) and

opluskisinIHomC(UkUi otimes Uk) Then to show gi is invariant under the action of S times Sminus1

amounts to showing that

(h1 h2)i = ((Sminus1)lowast h1 Slowast h2)i (A13)

Cardy Algebras and Sewing Constraints I 909

for all h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) We have

((Sminus1)lowast h1 Slowast h2)i =sum

mnα

dim Un

DimC

A

A

A

A

A A

A

Uk

α

α

h1

h2

Ui

U j

UmUn

(A14)

Now drag the upper vertex indexed by α in the above graph along its Uorm -leg until it

meets the lower vertex also indexed by α then sum over α and m This gives

((Sminus1)lowast h1 Slowast h2)i =sum

n

dim Un

DimC

A

A

A

A

A A

A

Uk

Un

h1

h2

Ui

U j

(A15)

910 L Kong I Runkel

If we just look at the neighbourhood of the Un-loop in the above graph we see thefollowing subgraphs

sumn

dim Un

DimC

A Aor

Uorj U j

Uk Uork

Un=

sumα

1

dim U j

A Aor Uk Uork

Uorj U j

αα

(A16)

where we have applied [BK Cor 3111] Substituting this subgraph back to the originalgraph in (A15) we obtain

((Sminus1)lowast h1 Slowast h2)i =sumα

1

dim U j

A

A

A

A

A A

A

A

A

Uorj

α

α

h1

h2m A

Ui

U jUk

Uk

(A17)

The graph in (A17) is equal to that in (A11) In order to see this we first drag theldquobubblerdquo (m A A) along A lines and through the m A vertex (because m A A is abimodule map) until it reaches the lower-left leg of the upper vertex indexed by α Thendrag the m A vertex along the (red) dotted line in the above graph Finally we applythe associativity of A (A12) and [Fr Lem 311] Then we see that the graph in (A17)exactly matches with the one in (A11)

Acknowledgements We would like to thank the organisers of the Oberwolfach Arbeitsgemeinschaft ldquoAlge-braic structures in conformal field theoriesrdquo (April 2007) where this work was started for an inspiring meetingWe would further like to thank the Hausdorff Institute for Mathematics in Bonn and the organisers of the stim-ulating meeting ldquoGeometry and Physicsrdquo (May 2008) We are indebted to Alexei Davydov Jens FjelstadJuumlrgen Fuchs Yi-Zhi Huang Alexei Kitaev Urs Schreiber Christoph Schweigert Stephan Stolz and PeterTeichner for helpful discussions andor comments on a draft of this paper The research of IR was partiallysupported by the EPSRC First Grant EPE0050471 the PPARC rolling grant PPC5071451 and the MarieCurie network lsquoSuperstring Theoryrsquo (MRTN-CT-2004-512194)

Cardy Algebras and Sewing Constraints I 911

References

[AN] Alexeevski A Natanzon SM Noncommutative two-dimensional topological field theories andhurwitz numbers for real algebraic curves Sel Math New Ser 12 307ndash377 (2006)

[Bi] Bichon J Cosovereign hopf algebras J Pure Appl Alg 157 121ndash133 (2001)[BK] Bakalov B Kirillov AA Lectures on Tensor Categories and Modular Functors Providence

RI Amer Math Soc 2001[C1] Cardy JL Operator content of two-dimensional conformal invariant theories Nucl Phys

B 270 186ndash204 (1986)[C2] Cardy JL Boundary conditions fusion rules and the verlinde formula Nucl Phys B 324

581ndash596 (1989)[CL] Cardy JL Lewellen DC Bulk and boundary operators in conformal field theory Phys Lett

B 259 274ndash278 (1991)[DMZ] Dong C-Y Mason G Zhu Y-C Discrete series of the Virasoro algebra and the moonshine

module In Algebraic Groups and Their Generalizations Quantum and infinite-dimensional Meth-ods Proc Symp Pure Math 56 Part 2 Providence RI Amer MathSoc 1994 pp 295ndash316

[DP] Day B Pastro C Note on frobenius monoidal functors New York J Math 14 733ndash742 (2008)[ENO1] Etingof PI Nikshych D Ostrik V On fusion categories Ann Math 162 581ndash642 (2005)[ENO2] Etingof PI Nikshych D Ostrik V An analogue of radfordrsquos s4 formula for finite tensor

categories Int Math Research Notices 54 2915ndash2933 (2004)[Fe] Felder G Froumlhlich J Fuchs J Schweigert C Correlation functions and boundary conditions in

rational conformal field theory and three-dimensional topology Comp Math 131 189ndash237 (2002)[FHL] Frenkel IB Huang Y-Z Lepowsky J On axiomatic approaches to vertex operator algebras

and modules Mem Amer Math Soc 104 (1993)[Fj] Fjelstad J Fuchs J Runkel I Schweigert C Uniqueness of openclosed rational cft with

given algebra of open states Adv Theor Math Phys 12 1283ndash1375 (2008)[FK] Froumlhlich J King C The chern-simons theory and knot polynomials Commun Math

Phys 126 167ndash199 (1989)[Fr] Froumlhlich J Fuchs J Runkel I Schweigert C Correspondences of ribbon categories Adv

Math 199 192ndash329 (2006)[FRS] Fuchs J Runkel I Schweigert C Tft construction of rcft correlators I partition functions Nucl

Phys B 646 353ndash497 (2002)[FS] Fuchs J Schweigert C Category theory for conformal boundary conditions Fields Inst Com-

mun 39 25ndash71 (2003)[H1] Huang Y-Z Two-dimensional conformal geometry and vertex operator algebras Progress in

Mathematics Vol 148 Boston Birkhaumluser 1997[H2] Huang Y-Z Rigidity and modularity of vertex tensor categories Comm Contemp Math 10

871ndash911 (2008)[HK1] Huang Y-Z Kong L Open-string vertex algebra category and operads Commun Math

Phys 250 433ndash471 (2004)[HK2] Huang Y-Z Kong L Full field algebras Commun Math Phys 272 345ndash396 (2007)[HK3] Huang Y-Z Kong L Modular invariance for conformal full field algebras httparxivorgabs

math0609570v2[mathQA] 2006[HL] Huang Y-Z Lepowsky J Tensor products of modules for a vertex operator algebra and vertex

tensor categories In Lie Theory and Geometry in honor of Bertram Kostant ed R BrylinskiJ-L Brylinski V Guillemin V Kac Boston Birkhaumluser 1994 pp 349ndash383

[JS] Joyal A Street R Braided tensor categories Adv Math 102 20ndash78 (1993)[K1] Kong L Full field algebras operads and tensor categories Adv Math 213 271ndash340 (2007)[K2] Kong L Open-closed field algebras Commun Math Phys 280 207ndash261 (2008)[K3] Kong L Cardy condition for open-closed field algebras Commun Math Phys 283 25ndash92 (2008)[Ki] Kitaev A Private communication[KO] Kirillov AA Ostrik V On q-analog of mckay correspondence and ade classification of sl(2)

conformal field theories Adv Math 171 183ndash227 (2002)[KR] Kong L Runkel I Morita classes of algebras in modular tensor categories Adv Math 219

1548ndash1576 (2008)[Ld] Lauda AD Frobenius algebras and ambidextrous adjunctions Theo Appl Cat 16 84ndash

122 (2006)[Li1] Li H-S Regular representations of vertex operator algebras Commun Contemp Math 4

639ndash683 (2002)[Li2] Li H-S Regular representations and huang-lepowsky tensor functors for vertex operator

algebras J Alge 255 423ndash462 (2002)

Cardy Algebras and Sewing Constraints I 903

fcl is an isomorphism As above since fcl is an algebra map and since Acl is simple fcl

has to be monic By Lemma 319 rlowast = ζminus1A e Thus f lowast

cl = ιlowastcl-op rlowast = ζminus1A ιlowastcl-op e and

fcl f lowastcl = r ιcl-op ζminus1

A ιlowastcl-op e = r e r e = idZ(A) (344)

and so fcl is also epi and hence isofcl is a coalgebra map Since fcl is an algebra map so is f minus1

cl By (344) f minus1cl = f lowast

cland by Lemma 218 (iii) this implies that fcl is a also coalgebra mapThe diagram (315) commutes Commutativity of (315) is equivalent to e fcl = ιcl-opwhich holds by (340)

Let A be a special symmetric Frobenius algebra So far we have seen that (A|Z(A) e)is a Cardy algebra and that all Cardy algebras with Aop = A and simple Acl are of thisform It is now natural to ask if every simple Acl does occur as part of a Cardy algebraThe following theorem provides an affirmative answer Recall that for an A-left moduleM the object Mor otimesA M is an algebra (see eg [KR Lem 42])

Theorem 322 If Acl is a simple modular invariant commutative symmetric FrobeniusC2plusmn-algebra then there exist a simple special symmetric Frobenius C-algebra A and amorphism ιcl-op Acl rarr R(A) such that

(i) Acl sim= Z(A) as Frobenius algebras(ii) (A|Acl ιcl-op) is a Cardy C|C2plusmn-algebra

(iii) T (Acl) sim= oplusκisinJ Morκ otimesA Mκ as algebras where MκκisinJ is a set of representa-

tives of the isomorphism classes of simple A-left modules

Proof By Remark 36 we have dim Acl = Z00 Dim C = 0 and by Lemma 211 Aclis haploid It then follows from Theorem 34 that Acl is special By Proposition 224T (Acl) is a special symmetric Frobenius algebra in C Thus T (Acl) = oplusi Ai where theAi are simple symmetric Frobenius algebras We will show that at least one of the Aiis special Since T (Acl) is special we have mT (Acl) T (Acl) = ζ idT (Acl) for someζ isin C

times Restricting this to the summand Ai shows mi i = ζ idAi FurthermoreεT (Acl) ηT (Acl) = ξ id1 for some ξ isin C

times But εT (Acl) ηT (Acl) = sumi εi ηi and so

at least one of the εi ηi has to be nonzero Therefore at least one of the Ai is speciallet A equiv Ai be this summand We denote the embedding A rarr T (Acl) by e0 and therestriction T (Acl) A by r0 Notice that r0 is an algebra homomorphism Define

ιcl-op = χ (r0) Acl minusrarr R(A) (345)

By Proposition 216 ιcl-op is an algebra homomorphism Next we verify the centre con-dition (313) or rather its equivalent form (320) By substituting the definitions onecan convince oneself that the commutativity mcl cAclAcl = mcl of Acl in C2plusmn implies thecondition mT (Acl) = mT (Acl) in C see [K2 Prop 36] Here T (Acl)otimes T (Acl) rarrT (Acl)otimes T (Acl) is given by

=oplusmn

(idCl

notimes cCl

m Crnotimes idCr

m

)

(cCl

m Clnotimes cminus1

Crn C

rm

)

(idCl

motimes cminus1

Cln C

rm

otimes idCrn

)

(346)

and we decomposed Acl as Acl = oplusnCln times Cr

n As a consequence we obtain the identity

r0 mT (Acl) (idAcl otimes e0) = r0 mT (Acl) (idAcl otimes e0) (347)

904 L Kong I Runkel

Using that r0 is an algebra map and that by definition ιcl-op = r0 we obtain (320)In order to show that (A|Acl ιcl-op) is a Cardy algebra it remains to show that the

Cardy condition (314) is satisfied We will demonstrate this via a detour by first provingthat Acl sim= Z(A) as Frobenius algebras

Recall the notations e and r given in (331) Using the centre condition (320) onecan check that Pl

R(A) ιcl-op = m R(A) R(A) ιcl-op By specialness of A we have

m A A = ζA idA and so together with e r = ζminus1A Pl

R(A) we get

e r ιcl-op = ιcl-op (348)

Next consider the morphism

fcl = r ιcl-op Acl minusrarr Z(A) (349)

By the same derivation as in (342) and (343) one sees that fcl is an algebra map Inparticular fcl ηcl = ηZ(A) = 0 and so fcl = 0 Since Acl is simple fcl has to be amonomorphism

By the same argument as used in the proof of Theorem 34 (ii) up to isomorphismAcl is the unique simple local Acl-(left-)module The algebra monomorphism fcl turnsZ(A) into an Acl-module Since Z(A) is commutative it is local as an Acl-module andso Z(A) sim= AoplusN

cl for some N ge 1 By construction A is a simple special symmetricFrobenius algebra Proposition 225 and Corollary 227 show that R(A) inherits all theseproperties and thus Z(A) is simple (see the comment below Eq (332)) By Theorem318 Z(A) is modular invariant and then by Theorem 34 (i) dim Z(A) = Dim C Thisimplies that N = 1 in Z(A) sim= AoplusN

cl and so fcl is in fact an isomorphismSince Acl and Z(A) are both haploid we have εZ(A) fcl = ξ εcl for some ξ isin C

timesThe counit uniquely determines the Frobenius structure on Acl and Z(A) (see eg [FRSLemma 37]) so that fcl is a coalgebra isomorphism iff ξ = 1 To compute ξ we composethe above identity with ηcl from the right Defining ζcl via εcl ηcl = ζminus1

cl Dim C middot id1 andusing (333) gives ξ = dim A ζclζ

2A By rescaling the comultiplication and the counit

of A and consequently changing ζA we can always achieve ξ = 1 This proves part (i)of the theorem

Equation (348) implies that ιcl-op = e fcl Since fcl is an isomorphism of Frobeniusalgebras by Lemmas 218 and 319 we have

ιcl-op ιlowastcl-op = e fcl f lowastcl elowast = ζA e r = Pl

R(A) (350)

Thus (A|Acl ιcl-op) is a Cardy algebra This proves part (ii) of the theoremPart (iii) can be seen as follows By [KR Prop 43] T Z(A) sim= oplusκisinJ Mor

κ otimesA Mκ asalgebras Together with the observation that T ( fcl) T (Acl) rarr T Z(A) is an isomor-phism of algebras this proves part (iii) Remark 323 Part (i) of Theorem 322 was announced by Muumlger [Muuml2] We provide anindependent proof in the setting of Cardy algebras

The above theorem together with Lemma 211 and Theorem 34 shows that a sim-ple commutative symmetric Frobenius C2plusmn-algebra Acl with dim Acl = Dim C is alwayspart of a Cardy algebra (Aop|Acl ιcl-op) for some simple special symmetric Frobeniusalgebra Aop in C However the above proof also illustrates that Aop is not unique Thisraises the question how two Cardy algebras with a given Acl can differ This question isanswered by [KR Thm 11] which in the present framework can be restated as follows

Cardy Algebras and Sewing Constraints I 905

Theorem 324 If (A(i)op |A(i)cl ι(i)cl-op) i = 1 2 are two Cardy C|C2plusmn-algebras such that

A(i)cl is simple and dim A(i)op = 0 for i = 1 2 then A(1)clsim= A(2)cl as algebras if and only

if A(1)op and A(2)op are Morita equivalent

Proof Theorem 11 in [KR] is stated for A(i)op being non-degenerate algebras and A(i)cl =Z(A(i)op ) for i = 1 2 By Proposition 316 A(i)op are simple and special for i = 1 2

Then by [KR Lem 21] A(i)op are non-degenerate algebras By Theorem 321 we have

A(i)clsim= Z(A(i)op ) as Frobenius algebras Finally by [KR Thm 11] Z(A(1)op ) sim= Z(A(2)op )

as algebras iff A(1)op and A(2)op are Morita equivalent Let Cmax(C2plusmn) be the set of equivalence classes [B] of simple modular invariant

commutative symmetric Frobenius algebras B in C2plusmn Two such algebras B and B prime areequivalent if B and B prime are isomorphic as algebras (but not necessarily as Frobenius alge-bras) Let Msimp(C) be the set of Morita classes of simple special symmetric Frobeniusalgebras in C Define the map z Msimp(C) rarr Cmax(C2plusmn) by z A rarr [Z(A)] whereA denotes the Morita class of A From Theorem 322 (i) and [KR Thm 11] we learn

Corollary 325 The map z Msimp(C) rarr Cmax(C2plusmn) is a bijection

A Appendix

A1 Proof of Lemma 27 We will show that if (F ψ F2 ψ

F0 ) is a colax tensor functor

from C1 to C2 then (G φG2 φ

G0 ) is a lax tensor functor from C2 to C1 Applying this

result to the opposed categories then gives the converse statementWe need to show that φG

0 and φG2 make the diagrams (21) and (22) commute We

first prove the commutativity of (21) Consider the following diagram

G(A)otimes (G(B)otimes G(C))

GψF2 δ

idG(A)otimesφG2 G(A)otimes G(B otimes C)

GψF2 δ

G(FG(A)otimes F(G(B)otimes G(C)))

G(F(idG(A))otimesF(φG2 ))

G(idFG(A)otimesψ F2 )

G(FG(A)otimes FG(B otimes C))

G(ρAotimesρBotimesC )

G(FG(A)otimes (FG(B)otimes FG(C)))

G(ρAotimes(ρBotimesρC )) G(A otimes (B otimes C))

(A1)

The top subdiagram is commutative because of the naturality of Gψ F2 δ The commu-

tativity of the bottom subdiagram follows from the following identities

(ρB otimes ρC ) ψ F2 = (ρB otimes ρC ) ψ F

2 ρF Fδ

= ρBotimesC FG(ρB otimes ρC ) FG(ψ F2 ) Fδ

= ρBotimesC F(φG2 ) (A2)

as a map F(G(B) otimes G(C)) rarr B otimes C The commutativity of (A1) implies that thecomposition of maps in the left column in (21) can be replaced by

G(ρA otimes (ρB otimes ρC )) G((idFG(A) otimes ψ F

2 ) ψ F2

) δ (A3)

906 L Kong I Runkel

Similarly we can show that the composition of maps in the right column in (21) can bereplaced by

G((ρA otimes ρB)otimes ρC ) G((ψ F

2 otimes idFG(C)) ψ F2

) δ (A4)

Using the commutativity of (23) it is easy to see that (21) with the left and right col-umns of (21) replaced by (A3) and (A4) respectively is commutative Hence (21) iscommutative

Now we prove the commutativity of the first diagram in (22)

φG2 (φG

0 otimes idG(A))

(1)= G(ρ12 otimes ρA) Gψ F2 δ

[(Gψ F

0 δ11)otimes idG(A)

]

(2)= G(ρ12 otimes ρA) Gψ F2 G F(Gψ0 otimes idG(A)) G F(δ11 otimes idG(A)) δ

(3)= G(ρ12 otimes ρA) G(FG(ψ F0 )otimes idFG(A)) Gψ F

2 G F(δ11 otimes idG(A)) δ(4)= G(id12 otimes ρA) G

([ψ F

0 ρF(11) (Fδ)11] otimes idFG(A)

) Gψ F

2 δ(5)= G(id12 otimes ρA) G(ψ F

0 otimes idFG(A)) Gψ F2 δ

(6)= G(id12 otimes ρA) G(lminus1FG(A)) G F(lG(A)) δ

(7)= G(lminus1A ) GρA δG lG(A)

(8)= G(lminus1A ) lG(A) (A5)

where in step (1) we substituted the definition of φG0 φ

G2 given in (210) in step (2)

we used the naturality of δ in step (3) we used the naturality of Gψ F2 in step (4) we

switched the position between Gρ12 and G FG(ψ F0 ) and the position between Gψ2 and

G F(δ12 otimes idG(A)) using the naturality of ρ and FψG2 respectively in step (5) we applied

the second identity in (28) in step (6) we used (24) in step (7) we used the naturalityof lminus1 and δ in step (8) we used the first identity in (28)

The proof of the commutativity of the second diagram in (22) is similar Thus wehave shown that G is a lax tensor functor

A2 Proof of Lemma 320 To prepare the proof recall that for a given object B isin Cthe modular group P SL(2Z) acts on the space oplusi HomC(B otimes Ui Ui ) see eg [BKSect 31] and [K3 Eq (455)] We will only need the action of S and Sminus1 Let f isinoplusi HomC(B otimes Ui Ui ) Then

S oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A6)

Cardy Algebras and Sewing Constraints I 907

Sminus1 oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A7)

By Lemma 32 to establish that (337) is S-invariant it is enough to prove the identity(34) when f is given by (337) Using (A6) and (A7) we can see that Eq (34) simplysays that oplusi j [RHS of (34)] is invariant under the action of S times S Consider the elementg of oplus jkisinIHomC2plusmn(R(A)otimes (Uor

j times Uk)Uorj times Uk) given by

g =oplusjkisinI

sumα

R(A)

R(A)

R(A)

R(A)

Uorj times Uk

Uorj times Uk

α

α

(A8)

By the above arguments proving S-invariance of (337) is equivalent to proving invari-ance of g under the action of S times S

For i isin I we denote by gi the component of g in(oplus jisinIHomC(A otimes Uor

i otimes Uorj U

orj )

)otimes (opluskisinIHomC(Ui otimes UkUk))

We view the second Hom-space in the above tensor product as a Hom-space in C+ insteadof Cminus It is enough to show that gi is invariant under the action of S times Sminus1 Note thatthe action of Sminus1 in C+ is equivalent to that of S in Cminus

The morphism gi can be canonically identified with a bilinear pairing

( middot middot )i ⎛⎝oplus

jisinIHomC(Uor

j A otimes Uori otimes Uor

j )

⎞⎠

times(oplus

kisinIHomC(UkUi otimes Uk)

)minusrarr C (A9)

as follows For h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) we set

(h1 h2)i = (dim U j dim Uk)minus1 trUor

j timesUk[gi (h1 times h2)] (A10)

908 L Kong I Runkel

When substituting the explicit form of the product m R(A) of R(A) = (A times 1) otimes R(1)after a short calculation one finds

(h1 h2)i =sumα

1

dim U j

A

A

A

A

A A

A α

α

h1

h2

U jUk

Uorj

Ui

Uk (A11)

Here the top morphism PlR(A) has been simplified with the help of the identity

PR(A) m R(A) (PR(A) otimes PR(A))

=(oplus

iisinI((m A A)otimes idUor

i)times idUi

) m R(A) (PR(A) otimes PR(A)) (A12)

which can be checked by direct calculation along the same lines as in the proof of [FrLem 310]

The action of the modular transformation S on oplusiisinI(B otimes Ui Ui ) for B isin C nat-urally induces an action on oplusiisinI(Ui B otimes Ui ) [K3 Prop 514] which we denote bySlowast In the present case we get an action of Slowast on oplus jisinIHomC(Uor

j A otimes Uori otimes Uor

j ) and

opluskisinIHomC(UkUi otimes Uk) Then to show gi is invariant under the action of S times Sminus1

amounts to showing that

(h1 h2)i = ((Sminus1)lowast h1 Slowast h2)i (A13)

Cardy Algebras and Sewing Constraints I 909

for all h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) We have

((Sminus1)lowast h1 Slowast h2)i =sum

mnα

dim Un

DimC

A

A

A

A

A A

A

Uk

α

α

h1

h2

Ui

U j

UmUn

(A14)

Now drag the upper vertex indexed by α in the above graph along its Uorm -leg until it

meets the lower vertex also indexed by α then sum over α and m This gives

((Sminus1)lowast h1 Slowast h2)i =sum

n

dim Un

DimC

A

A

A

A

A A

A

Uk

Un

h1

h2

Ui

U j

(A15)

910 L Kong I Runkel

If we just look at the neighbourhood of the Un-loop in the above graph we see thefollowing subgraphs

sumn

dim Un

DimC

A Aor

Uorj U j

Uk Uork

Un=

sumα

1

dim U j

A Aor Uk Uork

Uorj U j

αα

(A16)

where we have applied [BK Cor 3111] Substituting this subgraph back to the originalgraph in (A15) we obtain

((Sminus1)lowast h1 Slowast h2)i =sumα

1

dim U j

A

A

A

A

A A

A

A

A

Uorj

α

α

h1

h2m A

Ui

U jUk

Uk

(A17)

The graph in (A17) is equal to that in (A11) In order to see this we first drag theldquobubblerdquo (m A A) along A lines and through the m A vertex (because m A A is abimodule map) until it reaches the lower-left leg of the upper vertex indexed by α Thendrag the m A vertex along the (red) dotted line in the above graph Finally we applythe associativity of A (A12) and [Fr Lem 311] Then we see that the graph in (A17)exactly matches with the one in (A11)

Acknowledgements We would like to thank the organisers of the Oberwolfach Arbeitsgemeinschaft ldquoAlge-braic structures in conformal field theoriesrdquo (April 2007) where this work was started for an inspiring meetingWe would further like to thank the Hausdorff Institute for Mathematics in Bonn and the organisers of the stim-ulating meeting ldquoGeometry and Physicsrdquo (May 2008) We are indebted to Alexei Davydov Jens FjelstadJuumlrgen Fuchs Yi-Zhi Huang Alexei Kitaev Urs Schreiber Christoph Schweigert Stephan Stolz and PeterTeichner for helpful discussions andor comments on a draft of this paper The research of IR was partiallysupported by the EPSRC First Grant EPE0050471 the PPARC rolling grant PPC5071451 and the MarieCurie network lsquoSuperstring Theoryrsquo (MRTN-CT-2004-512194)

Cardy Algebras and Sewing Constraints I 911

References

[AN] Alexeevski A Natanzon SM Noncommutative two-dimensional topological field theories andhurwitz numbers for real algebraic curves Sel Math New Ser 12 307ndash377 (2006)

[Bi] Bichon J Cosovereign hopf algebras J Pure Appl Alg 157 121ndash133 (2001)[BK] Bakalov B Kirillov AA Lectures on Tensor Categories and Modular Functors Providence

RI Amer Math Soc 2001[C1] Cardy JL Operator content of two-dimensional conformal invariant theories Nucl Phys

B 270 186ndash204 (1986)[C2] Cardy JL Boundary conditions fusion rules and the verlinde formula Nucl Phys B 324

581ndash596 (1989)[CL] Cardy JL Lewellen DC Bulk and boundary operators in conformal field theory Phys Lett

B 259 274ndash278 (1991)[DMZ] Dong C-Y Mason G Zhu Y-C Discrete series of the Virasoro algebra and the moonshine

module In Algebraic Groups and Their Generalizations Quantum and infinite-dimensional Meth-ods Proc Symp Pure Math 56 Part 2 Providence RI Amer MathSoc 1994 pp 295ndash316

[DP] Day B Pastro C Note on frobenius monoidal functors New York J Math 14 733ndash742 (2008)[ENO1] Etingof PI Nikshych D Ostrik V On fusion categories Ann Math 162 581ndash642 (2005)[ENO2] Etingof PI Nikshych D Ostrik V An analogue of radfordrsquos s4 formula for finite tensor

categories Int Math Research Notices 54 2915ndash2933 (2004)[Fe] Felder G Froumlhlich J Fuchs J Schweigert C Correlation functions and boundary conditions in

rational conformal field theory and three-dimensional topology Comp Math 131 189ndash237 (2002)[FHL] Frenkel IB Huang Y-Z Lepowsky J On axiomatic approaches to vertex operator algebras

and modules Mem Amer Math Soc 104 (1993)[Fj] Fjelstad J Fuchs J Runkel I Schweigert C Uniqueness of openclosed rational cft with

given algebra of open states Adv Theor Math Phys 12 1283ndash1375 (2008)[FK] Froumlhlich J King C The chern-simons theory and knot polynomials Commun Math

Phys 126 167ndash199 (1989)[Fr] Froumlhlich J Fuchs J Runkel I Schweigert C Correspondences of ribbon categories Adv

Math 199 192ndash329 (2006)[FRS] Fuchs J Runkel I Schweigert C Tft construction of rcft correlators I partition functions Nucl

Phys B 646 353ndash497 (2002)[FS] Fuchs J Schweigert C Category theory for conformal boundary conditions Fields Inst Com-

mun 39 25ndash71 (2003)[H1] Huang Y-Z Two-dimensional conformal geometry and vertex operator algebras Progress in

Mathematics Vol 148 Boston Birkhaumluser 1997[H2] Huang Y-Z Rigidity and modularity of vertex tensor categories Comm Contemp Math 10

871ndash911 (2008)[HK1] Huang Y-Z Kong L Open-string vertex algebra category and operads Commun Math

Phys 250 433ndash471 (2004)[HK2] Huang Y-Z Kong L Full field algebras Commun Math Phys 272 345ndash396 (2007)[HK3] Huang Y-Z Kong L Modular invariance for conformal full field algebras httparxivorgabs

math0609570v2[mathQA] 2006[HL] Huang Y-Z Lepowsky J Tensor products of modules for a vertex operator algebra and vertex

tensor categories In Lie Theory and Geometry in honor of Bertram Kostant ed R BrylinskiJ-L Brylinski V Guillemin V Kac Boston Birkhaumluser 1994 pp 349ndash383

[JS] Joyal A Street R Braided tensor categories Adv Math 102 20ndash78 (1993)[K1] Kong L Full field algebras operads and tensor categories Adv Math 213 271ndash340 (2007)[K2] Kong L Open-closed field algebras Commun Math Phys 280 207ndash261 (2008)[K3] Kong L Cardy condition for open-closed field algebras Commun Math Phys 283 25ndash92 (2008)[Ki] Kitaev A Private communication[KO] Kirillov AA Ostrik V On q-analog of mckay correspondence and ade classification of sl(2)

conformal field theories Adv Math 171 183ndash227 (2002)[KR] Kong L Runkel I Morita classes of algebras in modular tensor categories Adv Math 219

1548ndash1576 (2008)[Ld] Lauda AD Frobenius algebras and ambidextrous adjunctions Theo Appl Cat 16 84ndash

122 (2006)[Li1] Li H-S Regular representations of vertex operator algebras Commun Contemp Math 4

639ndash683 (2002)[Li2] Li H-S Regular representations and huang-lepowsky tensor functors for vertex operator

algebras J Alge 255 423ndash462 (2002)

904 L Kong I Runkel

Using that r0 is an algebra map and that by definition ιcl-op = r0 we obtain (320)In order to show that (A|Acl ιcl-op) is a Cardy algebra it remains to show that the

Cardy condition (314) is satisfied We will demonstrate this via a detour by first provingthat Acl sim= Z(A) as Frobenius algebras

Recall the notations e and r given in (331) Using the centre condition (320) onecan check that Pl

R(A) ιcl-op = m R(A) R(A) ιcl-op By specialness of A we have

m A A = ζA idA and so together with e r = ζminus1A Pl

R(A) we get

e r ιcl-op = ιcl-op (348)

Next consider the morphism

fcl = r ιcl-op Acl minusrarr Z(A) (349)

By the same derivation as in (342) and (343) one sees that fcl is an algebra map Inparticular fcl ηcl = ηZ(A) = 0 and so fcl = 0 Since Acl is simple fcl has to be amonomorphism

By the same argument as used in the proof of Theorem 34 (ii) up to isomorphismAcl is the unique simple local Acl-(left-)module The algebra monomorphism fcl turnsZ(A) into an Acl-module Since Z(A) is commutative it is local as an Acl-module andso Z(A) sim= AoplusN

cl for some N ge 1 By construction A is a simple special symmetricFrobenius algebra Proposition 225 and Corollary 227 show that R(A) inherits all theseproperties and thus Z(A) is simple (see the comment below Eq (332)) By Theorem318 Z(A) is modular invariant and then by Theorem 34 (i) dim Z(A) = Dim C Thisimplies that N = 1 in Z(A) sim= AoplusN

cl and so fcl is in fact an isomorphismSince Acl and Z(A) are both haploid we have εZ(A) fcl = ξ εcl for some ξ isin C

timesThe counit uniquely determines the Frobenius structure on Acl and Z(A) (see eg [FRSLemma 37]) so that fcl is a coalgebra isomorphism iff ξ = 1 To compute ξ we composethe above identity with ηcl from the right Defining ζcl via εcl ηcl = ζminus1

cl Dim C middot id1 andusing (333) gives ξ = dim A ζclζ

2A By rescaling the comultiplication and the counit

of A and consequently changing ζA we can always achieve ξ = 1 This proves part (i)of the theorem

Equation (348) implies that ιcl-op = e fcl Since fcl is an isomorphism of Frobeniusalgebras by Lemmas 218 and 319 we have

ιcl-op ιlowastcl-op = e fcl f lowastcl elowast = ζA e r = Pl

R(A) (350)

Thus (A|Acl ιcl-op) is a Cardy algebra This proves part (ii) of the theoremPart (iii) can be seen as follows By [KR Prop 43] T Z(A) sim= oplusκisinJ Mor

κ otimesA Mκ asalgebras Together with the observation that T ( fcl) T (Acl) rarr T Z(A) is an isomor-phism of algebras this proves part (iii) Remark 323 Part (i) of Theorem 322 was announced by Muumlger [Muuml2] We provide anindependent proof in the setting of Cardy algebras

The above theorem together with Lemma 211 and Theorem 34 shows that a sim-ple commutative symmetric Frobenius C2plusmn-algebra Acl with dim Acl = Dim C is alwayspart of a Cardy algebra (Aop|Acl ιcl-op) for some simple special symmetric Frobeniusalgebra Aop in C However the above proof also illustrates that Aop is not unique Thisraises the question how two Cardy algebras with a given Acl can differ This question isanswered by [KR Thm 11] which in the present framework can be restated as follows

Cardy Algebras and Sewing Constraints I 905

Theorem 324 If (A(i)op |A(i)cl ι(i)cl-op) i = 1 2 are two Cardy C|C2plusmn-algebras such that

A(i)cl is simple and dim A(i)op = 0 for i = 1 2 then A(1)clsim= A(2)cl as algebras if and only

if A(1)op and A(2)op are Morita equivalent

Proof Theorem 11 in [KR] is stated for A(i)op being non-degenerate algebras and A(i)cl =Z(A(i)op ) for i = 1 2 By Proposition 316 A(i)op are simple and special for i = 1 2

Then by [KR Lem 21] A(i)op are non-degenerate algebras By Theorem 321 we have

A(i)clsim= Z(A(i)op ) as Frobenius algebras Finally by [KR Thm 11] Z(A(1)op ) sim= Z(A(2)op )

as algebras iff A(1)op and A(2)op are Morita equivalent Let Cmax(C2plusmn) be the set of equivalence classes [B] of simple modular invariant

commutative symmetric Frobenius algebras B in C2plusmn Two such algebras B and B prime areequivalent if B and B prime are isomorphic as algebras (but not necessarily as Frobenius alge-bras) Let Msimp(C) be the set of Morita classes of simple special symmetric Frobeniusalgebras in C Define the map z Msimp(C) rarr Cmax(C2plusmn) by z A rarr [Z(A)] whereA denotes the Morita class of A From Theorem 322 (i) and [KR Thm 11] we learn

Corollary 325 The map z Msimp(C) rarr Cmax(C2plusmn) is a bijection

A Appendix

A1 Proof of Lemma 27 We will show that if (F ψ F2 ψ

F0 ) is a colax tensor functor

from C1 to C2 then (G φG2 φ

G0 ) is a lax tensor functor from C2 to C1 Applying this

result to the opposed categories then gives the converse statementWe need to show that φG

0 and φG2 make the diagrams (21) and (22) commute We

first prove the commutativity of (21) Consider the following diagram

G(A)otimes (G(B)otimes G(C))

GψF2 δ

idG(A)otimesφG2 G(A)otimes G(B otimes C)

GψF2 δ

G(FG(A)otimes F(G(B)otimes G(C)))

G(F(idG(A))otimesF(φG2 ))

G(idFG(A)otimesψ F2 )

G(FG(A)otimes FG(B otimes C))

G(ρAotimesρBotimesC )

G(FG(A)otimes (FG(B)otimes FG(C)))

G(ρAotimes(ρBotimesρC )) G(A otimes (B otimes C))

(A1)

The top subdiagram is commutative because of the naturality of Gψ F2 δ The commu-

tativity of the bottom subdiagram follows from the following identities

(ρB otimes ρC ) ψ F2 = (ρB otimes ρC ) ψ F

2 ρF Fδ

= ρBotimesC FG(ρB otimes ρC ) FG(ψ F2 ) Fδ

= ρBotimesC F(φG2 ) (A2)

as a map F(G(B) otimes G(C)) rarr B otimes C The commutativity of (A1) implies that thecomposition of maps in the left column in (21) can be replaced by

G(ρA otimes (ρB otimes ρC )) G((idFG(A) otimes ψ F

2 ) ψ F2

) δ (A3)

906 L Kong I Runkel

Similarly we can show that the composition of maps in the right column in (21) can bereplaced by

G((ρA otimes ρB)otimes ρC ) G((ψ F

2 otimes idFG(C)) ψ F2

) δ (A4)

Using the commutativity of (23) it is easy to see that (21) with the left and right col-umns of (21) replaced by (A3) and (A4) respectively is commutative Hence (21) iscommutative

Now we prove the commutativity of the first diagram in (22)

φG2 (φG

0 otimes idG(A))

(1)= G(ρ12 otimes ρA) Gψ F2 δ

[(Gψ F

0 δ11)otimes idG(A)

]

(2)= G(ρ12 otimes ρA) Gψ F2 G F(Gψ0 otimes idG(A)) G F(δ11 otimes idG(A)) δ

(3)= G(ρ12 otimes ρA) G(FG(ψ F0 )otimes idFG(A)) Gψ F

2 G F(δ11 otimes idG(A)) δ(4)= G(id12 otimes ρA) G

([ψ F

0 ρF(11) (Fδ)11] otimes idFG(A)

) Gψ F

2 δ(5)= G(id12 otimes ρA) G(ψ F

0 otimes idFG(A)) Gψ F2 δ

(6)= G(id12 otimes ρA) G(lminus1FG(A)) G F(lG(A)) δ

(7)= G(lminus1A ) GρA δG lG(A)

(8)= G(lminus1A ) lG(A) (A5)

where in step (1) we substituted the definition of φG0 φ

G2 given in (210) in step (2)

we used the naturality of δ in step (3) we used the naturality of Gψ F2 in step (4) we

switched the position between Gρ12 and G FG(ψ F0 ) and the position between Gψ2 and

G F(δ12 otimes idG(A)) using the naturality of ρ and FψG2 respectively in step (5) we applied

the second identity in (28) in step (6) we used (24) in step (7) we used the naturalityof lminus1 and δ in step (8) we used the first identity in (28)

The proof of the commutativity of the second diagram in (22) is similar Thus wehave shown that G is a lax tensor functor

A2 Proof of Lemma 320 To prepare the proof recall that for a given object B isin Cthe modular group P SL(2Z) acts on the space oplusi HomC(B otimes Ui Ui ) see eg [BKSect 31] and [K3 Eq (455)] We will only need the action of S and Sminus1 Let f isinoplusi HomC(B otimes Ui Ui ) Then

S oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A6)

Cardy Algebras and Sewing Constraints I 907

Sminus1 oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A7)

By Lemma 32 to establish that (337) is S-invariant it is enough to prove the identity(34) when f is given by (337) Using (A6) and (A7) we can see that Eq (34) simplysays that oplusi j [RHS of (34)] is invariant under the action of S times S Consider the elementg of oplus jkisinIHomC2plusmn(R(A)otimes (Uor

j times Uk)Uorj times Uk) given by

g =oplusjkisinI

sumα

R(A)

R(A)

R(A)

R(A)

Uorj times Uk

Uorj times Uk

α

α

(A8)

By the above arguments proving S-invariance of (337) is equivalent to proving invari-ance of g under the action of S times S

For i isin I we denote by gi the component of g in(oplus jisinIHomC(A otimes Uor

i otimes Uorj U

orj )

)otimes (opluskisinIHomC(Ui otimes UkUk))

We view the second Hom-space in the above tensor product as a Hom-space in C+ insteadof Cminus It is enough to show that gi is invariant under the action of S times Sminus1 Note thatthe action of Sminus1 in C+ is equivalent to that of S in Cminus

The morphism gi can be canonically identified with a bilinear pairing

( middot middot )i ⎛⎝oplus

jisinIHomC(Uor

j A otimes Uori otimes Uor

j )

⎞⎠

times(oplus

kisinIHomC(UkUi otimes Uk)

)minusrarr C (A9)

as follows For h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) we set

(h1 h2)i = (dim U j dim Uk)minus1 trUor

j timesUk[gi (h1 times h2)] (A10)

908 L Kong I Runkel

When substituting the explicit form of the product m R(A) of R(A) = (A times 1) otimes R(1)after a short calculation one finds

(h1 h2)i =sumα

1

dim U j

A

A

A

A

A A

A α

α

h1

h2

U jUk

Uorj

Ui

Uk (A11)

Here the top morphism PlR(A) has been simplified with the help of the identity

PR(A) m R(A) (PR(A) otimes PR(A))

=(oplus

iisinI((m A A)otimes idUor

i)times idUi

) m R(A) (PR(A) otimes PR(A)) (A12)

which can be checked by direct calculation along the same lines as in the proof of [FrLem 310]

The action of the modular transformation S on oplusiisinI(B otimes Ui Ui ) for B isin C nat-urally induces an action on oplusiisinI(Ui B otimes Ui ) [K3 Prop 514] which we denote bySlowast In the present case we get an action of Slowast on oplus jisinIHomC(Uor

j A otimes Uori otimes Uor

j ) and

opluskisinIHomC(UkUi otimes Uk) Then to show gi is invariant under the action of S times Sminus1

amounts to showing that

(h1 h2)i = ((Sminus1)lowast h1 Slowast h2)i (A13)

Cardy Algebras and Sewing Constraints I 909

for all h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) We have

((Sminus1)lowast h1 Slowast h2)i =sum

mnα

dim Un

DimC

A

A

A

A

A A

A

Uk

α

α

h1

h2

Ui

U j

UmUn

(A14)

Now drag the upper vertex indexed by α in the above graph along its Uorm -leg until it

meets the lower vertex also indexed by α then sum over α and m This gives

((Sminus1)lowast h1 Slowast h2)i =sum

n

dim Un

DimC

A

A

A

A

A A

A

Uk

Un

h1

h2

Ui

U j

(A15)

910 L Kong I Runkel

If we just look at the neighbourhood of the Un-loop in the above graph we see thefollowing subgraphs

sumn

dim Un

DimC

A Aor

Uorj U j

Uk Uork

Un=

sumα

1

dim U j

A Aor Uk Uork

Uorj U j

αα

(A16)

where we have applied [BK Cor 3111] Substituting this subgraph back to the originalgraph in (A15) we obtain

((Sminus1)lowast h1 Slowast h2)i =sumα

1

dim U j

A

A

A

A

A A

A

A

A

Uorj

α

α

h1

h2m A

Ui

U jUk

Uk

(A17)

The graph in (A17) is equal to that in (A11) In order to see this we first drag theldquobubblerdquo (m A A) along A lines and through the m A vertex (because m A A is abimodule map) until it reaches the lower-left leg of the upper vertex indexed by α Thendrag the m A vertex along the (red) dotted line in the above graph Finally we applythe associativity of A (A12) and [Fr Lem 311] Then we see that the graph in (A17)exactly matches with the one in (A11)

Acknowledgements We would like to thank the organisers of the Oberwolfach Arbeitsgemeinschaft ldquoAlge-braic structures in conformal field theoriesrdquo (April 2007) where this work was started for an inspiring meetingWe would further like to thank the Hausdorff Institute for Mathematics in Bonn and the organisers of the stim-ulating meeting ldquoGeometry and Physicsrdquo (May 2008) We are indebted to Alexei Davydov Jens FjelstadJuumlrgen Fuchs Yi-Zhi Huang Alexei Kitaev Urs Schreiber Christoph Schweigert Stephan Stolz and PeterTeichner for helpful discussions andor comments on a draft of this paper The research of IR was partiallysupported by the EPSRC First Grant EPE0050471 the PPARC rolling grant PPC5071451 and the MarieCurie network lsquoSuperstring Theoryrsquo (MRTN-CT-2004-512194)

Cardy Algebras and Sewing Constraints I 911

References

[AN] Alexeevski A Natanzon SM Noncommutative two-dimensional topological field theories andhurwitz numbers for real algebraic curves Sel Math New Ser 12 307ndash377 (2006)

[Bi] Bichon J Cosovereign hopf algebras J Pure Appl Alg 157 121ndash133 (2001)[BK] Bakalov B Kirillov AA Lectures on Tensor Categories and Modular Functors Providence

RI Amer Math Soc 2001[C1] Cardy JL Operator content of two-dimensional conformal invariant theories Nucl Phys

B 270 186ndash204 (1986)[C2] Cardy JL Boundary conditions fusion rules and the verlinde formula Nucl Phys B 324

581ndash596 (1989)[CL] Cardy JL Lewellen DC Bulk and boundary operators in conformal field theory Phys Lett

B 259 274ndash278 (1991)[DMZ] Dong C-Y Mason G Zhu Y-C Discrete series of the Virasoro algebra and the moonshine

module In Algebraic Groups and Their Generalizations Quantum and infinite-dimensional Meth-ods Proc Symp Pure Math 56 Part 2 Providence RI Amer MathSoc 1994 pp 295ndash316

[DP] Day B Pastro C Note on frobenius monoidal functors New York J Math 14 733ndash742 (2008)[ENO1] Etingof PI Nikshych D Ostrik V On fusion categories Ann Math 162 581ndash642 (2005)[ENO2] Etingof PI Nikshych D Ostrik V An analogue of radfordrsquos s4 formula for finite tensor

categories Int Math Research Notices 54 2915ndash2933 (2004)[Fe] Felder G Froumlhlich J Fuchs J Schweigert C Correlation functions and boundary conditions in

rational conformal field theory and three-dimensional topology Comp Math 131 189ndash237 (2002)[FHL] Frenkel IB Huang Y-Z Lepowsky J On axiomatic approaches to vertex operator algebras

and modules Mem Amer Math Soc 104 (1993)[Fj] Fjelstad J Fuchs J Runkel I Schweigert C Uniqueness of openclosed rational cft with

given algebra of open states Adv Theor Math Phys 12 1283ndash1375 (2008)[FK] Froumlhlich J King C The chern-simons theory and knot polynomials Commun Math

Phys 126 167ndash199 (1989)[Fr] Froumlhlich J Fuchs J Runkel I Schweigert C Correspondences of ribbon categories Adv

Math 199 192ndash329 (2006)[FRS] Fuchs J Runkel I Schweigert C Tft construction of rcft correlators I partition functions Nucl

Phys B 646 353ndash497 (2002)[FS] Fuchs J Schweigert C Category theory for conformal boundary conditions Fields Inst Com-

mun 39 25ndash71 (2003)[H1] Huang Y-Z Two-dimensional conformal geometry and vertex operator algebras Progress in

Mathematics Vol 148 Boston Birkhaumluser 1997[H2] Huang Y-Z Rigidity and modularity of vertex tensor categories Comm Contemp Math 10

871ndash911 (2008)[HK1] Huang Y-Z Kong L Open-string vertex algebra category and operads Commun Math

Phys 250 433ndash471 (2004)[HK2] Huang Y-Z Kong L Full field algebras Commun Math Phys 272 345ndash396 (2007)[HK3] Huang Y-Z Kong L Modular invariance for conformal full field algebras httparxivorgabs

math0609570v2[mathQA] 2006[HL] Huang Y-Z Lepowsky J Tensor products of modules for a vertex operator algebra and vertex

tensor categories In Lie Theory and Geometry in honor of Bertram Kostant ed R BrylinskiJ-L Brylinski V Guillemin V Kac Boston Birkhaumluser 1994 pp 349ndash383

[JS] Joyal A Street R Braided tensor categories Adv Math 102 20ndash78 (1993)[K1] Kong L Full field algebras operads and tensor categories Adv Math 213 271ndash340 (2007)[K2] Kong L Open-closed field algebras Commun Math Phys 280 207ndash261 (2008)[K3] Kong L Cardy condition for open-closed field algebras Commun Math Phys 283 25ndash92 (2008)[Ki] Kitaev A Private communication[KO] Kirillov AA Ostrik V On q-analog of mckay correspondence and ade classification of sl(2)

conformal field theories Adv Math 171 183ndash227 (2002)[KR] Kong L Runkel I Morita classes of algebras in modular tensor categories Adv Math 219

1548ndash1576 (2008)[Ld] Lauda AD Frobenius algebras and ambidextrous adjunctions Theo Appl Cat 16 84ndash

122 (2006)[Li1] Li H-S Regular representations of vertex operator algebras Commun Contemp Math 4

639ndash683 (2002)[Li2] Li H-S Regular representations and huang-lepowsky tensor functors for vertex operator

algebras J Alge 255 423ndash462 (2002)

Cardy Algebras and Sewing Constraints I 905

Theorem 324 If (A(i)op |A(i)cl ι(i)cl-op) i = 1 2 are two Cardy C|C2plusmn-algebras such that

A(i)cl is simple and dim A(i)op = 0 for i = 1 2 then A(1)clsim= A(2)cl as algebras if and only

if A(1)op and A(2)op are Morita equivalent

Proof Theorem 11 in [KR] is stated for A(i)op being non-degenerate algebras and A(i)cl =Z(A(i)op ) for i = 1 2 By Proposition 316 A(i)op are simple and special for i = 1 2

Then by [KR Lem 21] A(i)op are non-degenerate algebras By Theorem 321 we have

A(i)clsim= Z(A(i)op ) as Frobenius algebras Finally by [KR Thm 11] Z(A(1)op ) sim= Z(A(2)op )

as algebras iff A(1)op and A(2)op are Morita equivalent Let Cmax(C2plusmn) be the set of equivalence classes [B] of simple modular invariant

commutative symmetric Frobenius algebras B in C2plusmn Two such algebras B and B prime areequivalent if B and B prime are isomorphic as algebras (but not necessarily as Frobenius alge-bras) Let Msimp(C) be the set of Morita classes of simple special symmetric Frobeniusalgebras in C Define the map z Msimp(C) rarr Cmax(C2plusmn) by z A rarr [Z(A)] whereA denotes the Morita class of A From Theorem 322 (i) and [KR Thm 11] we learn

Corollary 325 The map z Msimp(C) rarr Cmax(C2plusmn) is a bijection

A Appendix

A1 Proof of Lemma 27 We will show that if (F ψ F2 ψ

F0 ) is a colax tensor functor

from C1 to C2 then (G φG2 φ

G0 ) is a lax tensor functor from C2 to C1 Applying this

result to the opposed categories then gives the converse statementWe need to show that φG

0 and φG2 make the diagrams (21) and (22) commute We

first prove the commutativity of (21) Consider the following diagram

G(A)otimes (G(B)otimes G(C))

GψF2 δ

idG(A)otimesφG2 G(A)otimes G(B otimes C)

GψF2 δ

G(FG(A)otimes F(G(B)otimes G(C)))

G(F(idG(A))otimesF(φG2 ))

G(idFG(A)otimesψ F2 )

G(FG(A)otimes FG(B otimes C))

G(ρAotimesρBotimesC )

G(FG(A)otimes (FG(B)otimes FG(C)))

G(ρAotimes(ρBotimesρC )) G(A otimes (B otimes C))

(A1)

The top subdiagram is commutative because of the naturality of Gψ F2 δ The commu-

tativity of the bottom subdiagram follows from the following identities

(ρB otimes ρC ) ψ F2 = (ρB otimes ρC ) ψ F

2 ρF Fδ

= ρBotimesC FG(ρB otimes ρC ) FG(ψ F2 ) Fδ

= ρBotimesC F(φG2 ) (A2)

as a map F(G(B) otimes G(C)) rarr B otimes C The commutativity of (A1) implies that thecomposition of maps in the left column in (21) can be replaced by

G(ρA otimes (ρB otimes ρC )) G((idFG(A) otimes ψ F

2 ) ψ F2

) δ (A3)

906 L Kong I Runkel

Similarly we can show that the composition of maps in the right column in (21) can bereplaced by

G((ρA otimes ρB)otimes ρC ) G((ψ F

2 otimes idFG(C)) ψ F2

) δ (A4)

Using the commutativity of (23) it is easy to see that (21) with the left and right col-umns of (21) replaced by (A3) and (A4) respectively is commutative Hence (21) iscommutative

Now we prove the commutativity of the first diagram in (22)

φG2 (φG

0 otimes idG(A))

(1)= G(ρ12 otimes ρA) Gψ F2 δ

[(Gψ F

0 δ11)otimes idG(A)

]

(2)= G(ρ12 otimes ρA) Gψ F2 G F(Gψ0 otimes idG(A)) G F(δ11 otimes idG(A)) δ

(3)= G(ρ12 otimes ρA) G(FG(ψ F0 )otimes idFG(A)) Gψ F

2 G F(δ11 otimes idG(A)) δ(4)= G(id12 otimes ρA) G

([ψ F

0 ρF(11) (Fδ)11] otimes idFG(A)

) Gψ F

2 δ(5)= G(id12 otimes ρA) G(ψ F

0 otimes idFG(A)) Gψ F2 δ

(6)= G(id12 otimes ρA) G(lminus1FG(A)) G F(lG(A)) δ

(7)= G(lminus1A ) GρA δG lG(A)

(8)= G(lminus1A ) lG(A) (A5)

where in step (1) we substituted the definition of φG0 φ

G2 given in (210) in step (2)

we used the naturality of δ in step (3) we used the naturality of Gψ F2 in step (4) we

switched the position between Gρ12 and G FG(ψ F0 ) and the position between Gψ2 and

G F(δ12 otimes idG(A)) using the naturality of ρ and FψG2 respectively in step (5) we applied

the second identity in (28) in step (6) we used (24) in step (7) we used the naturalityof lminus1 and δ in step (8) we used the first identity in (28)

The proof of the commutativity of the second diagram in (22) is similar Thus wehave shown that G is a lax tensor functor

A2 Proof of Lemma 320 To prepare the proof recall that for a given object B isin Cthe modular group P SL(2Z) acts on the space oplusi HomC(B otimes Ui Ui ) see eg [BKSect 31] and [K3 Eq (455)] We will only need the action of S and Sminus1 Let f isinoplusi HomC(B otimes Ui Ui ) Then

S oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A6)

Cardy Algebras and Sewing Constraints I 907

Sminus1 oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A7)

By Lemma 32 to establish that (337) is S-invariant it is enough to prove the identity(34) when f is given by (337) Using (A6) and (A7) we can see that Eq (34) simplysays that oplusi j [RHS of (34)] is invariant under the action of S times S Consider the elementg of oplus jkisinIHomC2plusmn(R(A)otimes (Uor

j times Uk)Uorj times Uk) given by

g =oplusjkisinI

sumα

R(A)

R(A)

R(A)

R(A)

Uorj times Uk

Uorj times Uk

α

α

(A8)

By the above arguments proving S-invariance of (337) is equivalent to proving invari-ance of g under the action of S times S

For i isin I we denote by gi the component of g in(oplus jisinIHomC(A otimes Uor

i otimes Uorj U

orj )

)otimes (opluskisinIHomC(Ui otimes UkUk))

We view the second Hom-space in the above tensor product as a Hom-space in C+ insteadof Cminus It is enough to show that gi is invariant under the action of S times Sminus1 Note thatthe action of Sminus1 in C+ is equivalent to that of S in Cminus

The morphism gi can be canonically identified with a bilinear pairing

( middot middot )i ⎛⎝oplus

jisinIHomC(Uor

j A otimes Uori otimes Uor

j )

⎞⎠

times(oplus

kisinIHomC(UkUi otimes Uk)

)minusrarr C (A9)

as follows For h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) we set

(h1 h2)i = (dim U j dim Uk)minus1 trUor

j timesUk[gi (h1 times h2)] (A10)

908 L Kong I Runkel

When substituting the explicit form of the product m R(A) of R(A) = (A times 1) otimes R(1)after a short calculation one finds

(h1 h2)i =sumα

1

dim U j

A

A

A

A

A A

A α

α

h1

h2

U jUk

Uorj

Ui

Uk (A11)

Here the top morphism PlR(A) has been simplified with the help of the identity

PR(A) m R(A) (PR(A) otimes PR(A))

=(oplus

iisinI((m A A)otimes idUor

i)times idUi

) m R(A) (PR(A) otimes PR(A)) (A12)

which can be checked by direct calculation along the same lines as in the proof of [FrLem 310]

The action of the modular transformation S on oplusiisinI(B otimes Ui Ui ) for B isin C nat-urally induces an action on oplusiisinI(Ui B otimes Ui ) [K3 Prop 514] which we denote bySlowast In the present case we get an action of Slowast on oplus jisinIHomC(Uor

j A otimes Uori otimes Uor

j ) and

opluskisinIHomC(UkUi otimes Uk) Then to show gi is invariant under the action of S times Sminus1

amounts to showing that

(h1 h2)i = ((Sminus1)lowast h1 Slowast h2)i (A13)

Cardy Algebras and Sewing Constraints I 909

for all h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) We have

((Sminus1)lowast h1 Slowast h2)i =sum

mnα

dim Un

DimC

A

A

A

A

A A

A

Uk

α

α

h1

h2

Ui

U j

UmUn

(A14)

Now drag the upper vertex indexed by α in the above graph along its Uorm -leg until it

meets the lower vertex also indexed by α then sum over α and m This gives

((Sminus1)lowast h1 Slowast h2)i =sum

n

dim Un

DimC

A

A

A

A

A A

A

Uk

Un

h1

h2

Ui

U j

(A15)

910 L Kong I Runkel

If we just look at the neighbourhood of the Un-loop in the above graph we see thefollowing subgraphs

sumn

dim Un

DimC

A Aor

Uorj U j

Uk Uork

Un=

sumα

1

dim U j

A Aor Uk Uork

Uorj U j

αα

(A16)

where we have applied [BK Cor 3111] Substituting this subgraph back to the originalgraph in (A15) we obtain

((Sminus1)lowast h1 Slowast h2)i =sumα

1

dim U j

A

A

A

A

A A

A

A

A

Uorj

α

α

h1

h2m A

Ui

U jUk

Uk

(A17)

The graph in (A17) is equal to that in (A11) In order to see this we first drag theldquobubblerdquo (m A A) along A lines and through the m A vertex (because m A A is abimodule map) until it reaches the lower-left leg of the upper vertex indexed by α Thendrag the m A vertex along the (red) dotted line in the above graph Finally we applythe associativity of A (A12) and [Fr Lem 311] Then we see that the graph in (A17)exactly matches with the one in (A11)

Acknowledgements We would like to thank the organisers of the Oberwolfach Arbeitsgemeinschaft ldquoAlge-braic structures in conformal field theoriesrdquo (April 2007) where this work was started for an inspiring meetingWe would further like to thank the Hausdorff Institute for Mathematics in Bonn and the organisers of the stim-ulating meeting ldquoGeometry and Physicsrdquo (May 2008) We are indebted to Alexei Davydov Jens FjelstadJuumlrgen Fuchs Yi-Zhi Huang Alexei Kitaev Urs Schreiber Christoph Schweigert Stephan Stolz and PeterTeichner for helpful discussions andor comments on a draft of this paper The research of IR was partiallysupported by the EPSRC First Grant EPE0050471 the PPARC rolling grant PPC5071451 and the MarieCurie network lsquoSuperstring Theoryrsquo (MRTN-CT-2004-512194)

Cardy Algebras and Sewing Constraints I 911

References

[AN] Alexeevski A Natanzon SM Noncommutative two-dimensional topological field theories andhurwitz numbers for real algebraic curves Sel Math New Ser 12 307ndash377 (2006)

[Bi] Bichon J Cosovereign hopf algebras J Pure Appl Alg 157 121ndash133 (2001)[BK] Bakalov B Kirillov AA Lectures on Tensor Categories and Modular Functors Providence

RI Amer Math Soc 2001[C1] Cardy JL Operator content of two-dimensional conformal invariant theories Nucl Phys

B 270 186ndash204 (1986)[C2] Cardy JL Boundary conditions fusion rules and the verlinde formula Nucl Phys B 324

581ndash596 (1989)[CL] Cardy JL Lewellen DC Bulk and boundary operators in conformal field theory Phys Lett

B 259 274ndash278 (1991)[DMZ] Dong C-Y Mason G Zhu Y-C Discrete series of the Virasoro algebra and the moonshine

module In Algebraic Groups and Their Generalizations Quantum and infinite-dimensional Meth-ods Proc Symp Pure Math 56 Part 2 Providence RI Amer MathSoc 1994 pp 295ndash316

[DP] Day B Pastro C Note on frobenius monoidal functors New York J Math 14 733ndash742 (2008)[ENO1] Etingof PI Nikshych D Ostrik V On fusion categories Ann Math 162 581ndash642 (2005)[ENO2] Etingof PI Nikshych D Ostrik V An analogue of radfordrsquos s4 formula for finite tensor

categories Int Math Research Notices 54 2915ndash2933 (2004)[Fe] Felder G Froumlhlich J Fuchs J Schweigert C Correlation functions and boundary conditions in

rational conformal field theory and three-dimensional topology Comp Math 131 189ndash237 (2002)[FHL] Frenkel IB Huang Y-Z Lepowsky J On axiomatic approaches to vertex operator algebras

and modules Mem Amer Math Soc 104 (1993)[Fj] Fjelstad J Fuchs J Runkel I Schweigert C Uniqueness of openclosed rational cft with

given algebra of open states Adv Theor Math Phys 12 1283ndash1375 (2008)[FK] Froumlhlich J King C The chern-simons theory and knot polynomials Commun Math

Phys 126 167ndash199 (1989)[Fr] Froumlhlich J Fuchs J Runkel I Schweigert C Correspondences of ribbon categories Adv

Math 199 192ndash329 (2006)[FRS] Fuchs J Runkel I Schweigert C Tft construction of rcft correlators I partition functions Nucl

Phys B 646 353ndash497 (2002)[FS] Fuchs J Schweigert C Category theory for conformal boundary conditions Fields Inst Com-

mun 39 25ndash71 (2003)[H1] Huang Y-Z Two-dimensional conformal geometry and vertex operator algebras Progress in

Mathematics Vol 148 Boston Birkhaumluser 1997[H2] Huang Y-Z Rigidity and modularity of vertex tensor categories Comm Contemp Math 10

871ndash911 (2008)[HK1] Huang Y-Z Kong L Open-string vertex algebra category and operads Commun Math

Phys 250 433ndash471 (2004)[HK2] Huang Y-Z Kong L Full field algebras Commun Math Phys 272 345ndash396 (2007)[HK3] Huang Y-Z Kong L Modular invariance for conformal full field algebras httparxivorgabs

math0609570v2[mathQA] 2006[HL] Huang Y-Z Lepowsky J Tensor products of modules for a vertex operator algebra and vertex

tensor categories In Lie Theory and Geometry in honor of Bertram Kostant ed R BrylinskiJ-L Brylinski V Guillemin V Kac Boston Birkhaumluser 1994 pp 349ndash383

[JS] Joyal A Street R Braided tensor categories Adv Math 102 20ndash78 (1993)[K1] Kong L Full field algebras operads and tensor categories Adv Math 213 271ndash340 (2007)[K2] Kong L Open-closed field algebras Commun Math Phys 280 207ndash261 (2008)[K3] Kong L Cardy condition for open-closed field algebras Commun Math Phys 283 25ndash92 (2008)[Ki] Kitaev A Private communication[KO] Kirillov AA Ostrik V On q-analog of mckay correspondence and ade classification of sl(2)

conformal field theories Adv Math 171 183ndash227 (2002)[KR] Kong L Runkel I Morita classes of algebras in modular tensor categories Adv Math 219

1548ndash1576 (2008)[Ld] Lauda AD Frobenius algebras and ambidextrous adjunctions Theo Appl Cat 16 84ndash

122 (2006)[Li1] Li H-S Regular representations of vertex operator algebras Commun Contemp Math 4

639ndash683 (2002)[Li2] Li H-S Regular representations and huang-lepowsky tensor functors for vertex operator

algebras J Alge 255 423ndash462 (2002)

906 L Kong I Runkel

Similarly we can show that the composition of maps in the right column in (21) can bereplaced by

G((ρA otimes ρB)otimes ρC ) G((ψ F

2 otimes idFG(C)) ψ F2

) δ (A4)

Using the commutativity of (23) it is easy to see that (21) with the left and right col-umns of (21) replaced by (A3) and (A4) respectively is commutative Hence (21) iscommutative

Now we prove the commutativity of the first diagram in (22)

φG2 (φG

0 otimes idG(A))

(1)= G(ρ12 otimes ρA) Gψ F2 δ

[(Gψ F

0 δ11)otimes idG(A)

]

(2)= G(ρ12 otimes ρA) Gψ F2 G F(Gψ0 otimes idG(A)) G F(δ11 otimes idG(A)) δ

(3)= G(ρ12 otimes ρA) G(FG(ψ F0 )otimes idFG(A)) Gψ F

2 G F(δ11 otimes idG(A)) δ(4)= G(id12 otimes ρA) G

([ψ F

0 ρF(11) (Fδ)11] otimes idFG(A)

) Gψ F

2 δ(5)= G(id12 otimes ρA) G(ψ F

0 otimes idFG(A)) Gψ F2 δ

(6)= G(id12 otimes ρA) G(lminus1FG(A)) G F(lG(A)) δ

(7)= G(lminus1A ) GρA δG lG(A)

(8)= G(lminus1A ) lG(A) (A5)

where in step (1) we substituted the definition of φG0 φ

G2 given in (210) in step (2)

we used the naturality of δ in step (3) we used the naturality of Gψ F2 in step (4) we

switched the position between Gρ12 and G FG(ψ F0 ) and the position between Gψ2 and

G F(δ12 otimes idG(A)) using the naturality of ρ and FψG2 respectively in step (5) we applied

the second identity in (28) in step (6) we used (24) in step (7) we used the naturalityof lminus1 and δ in step (8) we used the first identity in (28)

The proof of the commutativity of the second diagram in (22) is similar Thus wehave shown that G is a lax tensor functor

A2 Proof of Lemma 320 To prepare the proof recall that for a given object B isin Cthe modular group P SL(2Z) acts on the space oplusi HomC(B otimes Ui Ui ) see eg [BKSect 31] and [K3 Eq (455)] We will only need the action of S and Sminus1 Let f isinoplusi HomC(B otimes Ui Ui ) Then

S oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A6)

Cardy Algebras and Sewing Constraints I 907

Sminus1 oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A7)

By Lemma 32 to establish that (337) is S-invariant it is enough to prove the identity(34) when f is given by (337) Using (A6) and (A7) we can see that Eq (34) simplysays that oplusi j [RHS of (34)] is invariant under the action of S times S Consider the elementg of oplus jkisinIHomC2plusmn(R(A)otimes (Uor

j times Uk)Uorj times Uk) given by

g =oplusjkisinI

sumα

R(A)

R(A)

R(A)

R(A)

Uorj times Uk

Uorj times Uk

α

α

(A8)

By the above arguments proving S-invariance of (337) is equivalent to proving invari-ance of g under the action of S times S

For i isin I we denote by gi the component of g in(oplus jisinIHomC(A otimes Uor

i otimes Uorj U

orj )

)otimes (opluskisinIHomC(Ui otimes UkUk))

We view the second Hom-space in the above tensor product as a Hom-space in C+ insteadof Cminus It is enough to show that gi is invariant under the action of S times Sminus1 Note thatthe action of Sminus1 in C+ is equivalent to that of S in Cminus

The morphism gi can be canonically identified with a bilinear pairing

( middot middot )i ⎛⎝oplus

jisinIHomC(Uor

j A otimes Uori otimes Uor

j )

⎞⎠

times(oplus

kisinIHomC(UkUi otimes Uk)

)minusrarr C (A9)

as follows For h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) we set

(h1 h2)i = (dim U j dim Uk)minus1 trUor

j timesUk[gi (h1 times h2)] (A10)

908 L Kong I Runkel

When substituting the explicit form of the product m R(A) of R(A) = (A times 1) otimes R(1)after a short calculation one finds

(h1 h2)i =sumα

1

dim U j

A

A

A

A

A A

A α

α

h1

h2

U jUk

Uorj

Ui

Uk (A11)

Here the top morphism PlR(A) has been simplified with the help of the identity

PR(A) m R(A) (PR(A) otimes PR(A))

=(oplus

iisinI((m A A)otimes idUor

i)times idUi

) m R(A) (PR(A) otimes PR(A)) (A12)

which can be checked by direct calculation along the same lines as in the proof of [FrLem 310]

The action of the modular transformation S on oplusiisinI(B otimes Ui Ui ) for B isin C nat-urally induces an action on oplusiisinI(Ui B otimes Ui ) [K3 Prop 514] which we denote bySlowast In the present case we get an action of Slowast on oplus jisinIHomC(Uor

j A otimes Uori otimes Uor

j ) and

opluskisinIHomC(UkUi otimes Uk) Then to show gi is invariant under the action of S times Sminus1

amounts to showing that

(h1 h2)i = ((Sminus1)lowast h1 Slowast h2)i (A13)

Cardy Algebras and Sewing Constraints I 909

for all h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) We have

((Sminus1)lowast h1 Slowast h2)i =sum

mnα

dim Un

DimC

A

A

A

A

A A

A

Uk

α

α

h1

h2

Ui

U j

UmUn

(A14)

Now drag the upper vertex indexed by α in the above graph along its Uorm -leg until it

meets the lower vertex also indexed by α then sum over α and m This gives

((Sminus1)lowast h1 Slowast h2)i =sum

n

dim Un

DimC

A

A

A

A

A A

A

Uk

Un

h1

h2

Ui

U j

(A15)

910 L Kong I Runkel

If we just look at the neighbourhood of the Un-loop in the above graph we see thefollowing subgraphs

sumn

dim Un

DimC

A Aor

Uorj U j

Uk Uork

Un=

sumα

1

dim U j

A Aor Uk Uork

Uorj U j

αα

(A16)

where we have applied [BK Cor 3111] Substituting this subgraph back to the originalgraph in (A15) we obtain

((Sminus1)lowast h1 Slowast h2)i =sumα

1

dim U j

A

A

A

A

A A

A

A

A

Uorj

α

α

h1

h2m A

Ui

U jUk

Uk

(A17)

The graph in (A17) is equal to that in (A11) In order to see this we first drag theldquobubblerdquo (m A A) along A lines and through the m A vertex (because m A A is abimodule map) until it reaches the lower-left leg of the upper vertex indexed by α Thendrag the m A vertex along the (red) dotted line in the above graph Finally we applythe associativity of A (A12) and [Fr Lem 311] Then we see that the graph in (A17)exactly matches with the one in (A11)

Acknowledgements We would like to thank the organisers of the Oberwolfach Arbeitsgemeinschaft ldquoAlge-braic structures in conformal field theoriesrdquo (April 2007) where this work was started for an inspiring meetingWe would further like to thank the Hausdorff Institute for Mathematics in Bonn and the organisers of the stim-ulating meeting ldquoGeometry and Physicsrdquo (May 2008) We are indebted to Alexei Davydov Jens FjelstadJuumlrgen Fuchs Yi-Zhi Huang Alexei Kitaev Urs Schreiber Christoph Schweigert Stephan Stolz and PeterTeichner for helpful discussions andor comments on a draft of this paper The research of IR was partiallysupported by the EPSRC First Grant EPE0050471 the PPARC rolling grant PPC5071451 and the MarieCurie network lsquoSuperstring Theoryrsquo (MRTN-CT-2004-512194)

Cardy Algebras and Sewing Constraints I 911

References

[AN] Alexeevski A Natanzon SM Noncommutative two-dimensional topological field theories andhurwitz numbers for real algebraic curves Sel Math New Ser 12 307ndash377 (2006)

[Bi] Bichon J Cosovereign hopf algebras J Pure Appl Alg 157 121ndash133 (2001)[BK] Bakalov B Kirillov AA Lectures on Tensor Categories and Modular Functors Providence

RI Amer Math Soc 2001[C1] Cardy JL Operator content of two-dimensional conformal invariant theories Nucl Phys

B 270 186ndash204 (1986)[C2] Cardy JL Boundary conditions fusion rules and the verlinde formula Nucl Phys B 324

581ndash596 (1989)[CL] Cardy JL Lewellen DC Bulk and boundary operators in conformal field theory Phys Lett

B 259 274ndash278 (1991)[DMZ] Dong C-Y Mason G Zhu Y-C Discrete series of the Virasoro algebra and the moonshine

module In Algebraic Groups and Their Generalizations Quantum and infinite-dimensional Meth-ods Proc Symp Pure Math 56 Part 2 Providence RI Amer MathSoc 1994 pp 295ndash316

[DP] Day B Pastro C Note on frobenius monoidal functors New York J Math 14 733ndash742 (2008)[ENO1] Etingof PI Nikshych D Ostrik V On fusion categories Ann Math 162 581ndash642 (2005)[ENO2] Etingof PI Nikshych D Ostrik V An analogue of radfordrsquos s4 formula for finite tensor

categories Int Math Research Notices 54 2915ndash2933 (2004)[Fe] Felder G Froumlhlich J Fuchs J Schweigert C Correlation functions and boundary conditions in

rational conformal field theory and three-dimensional topology Comp Math 131 189ndash237 (2002)[FHL] Frenkel IB Huang Y-Z Lepowsky J On axiomatic approaches to vertex operator algebras

and modules Mem Amer Math Soc 104 (1993)[Fj] Fjelstad J Fuchs J Runkel I Schweigert C Uniqueness of openclosed rational cft with

given algebra of open states Adv Theor Math Phys 12 1283ndash1375 (2008)[FK] Froumlhlich J King C The chern-simons theory and knot polynomials Commun Math

Phys 126 167ndash199 (1989)[Fr] Froumlhlich J Fuchs J Runkel I Schweigert C Correspondences of ribbon categories Adv

Math 199 192ndash329 (2006)[FRS] Fuchs J Runkel I Schweigert C Tft construction of rcft correlators I partition functions Nucl

Phys B 646 353ndash497 (2002)[FS] Fuchs J Schweigert C Category theory for conformal boundary conditions Fields Inst Com-

mun 39 25ndash71 (2003)[H1] Huang Y-Z Two-dimensional conformal geometry and vertex operator algebras Progress in

Mathematics Vol 148 Boston Birkhaumluser 1997[H2] Huang Y-Z Rigidity and modularity of vertex tensor categories Comm Contemp Math 10

871ndash911 (2008)[HK1] Huang Y-Z Kong L Open-string vertex algebra category and operads Commun Math

Phys 250 433ndash471 (2004)[HK2] Huang Y-Z Kong L Full field algebras Commun Math Phys 272 345ndash396 (2007)[HK3] Huang Y-Z Kong L Modular invariance for conformal full field algebras httparxivorgabs

math0609570v2[mathQA] 2006[HL] Huang Y-Z Lepowsky J Tensor products of modules for a vertex operator algebra and vertex

tensor categories In Lie Theory and Geometry in honor of Bertram Kostant ed R BrylinskiJ-L Brylinski V Guillemin V Kac Boston Birkhaumluser 1994 pp 349ndash383

[JS] Joyal A Street R Braided tensor categories Adv Math 102 20ndash78 (1993)[K1] Kong L Full field algebras operads and tensor categories Adv Math 213 271ndash340 (2007)[K2] Kong L Open-closed field algebras Commun Math Phys 280 207ndash261 (2008)[K3] Kong L Cardy condition for open-closed field algebras Commun Math Phys 283 25ndash92 (2008)[Ki] Kitaev A Private communication[KO] Kirillov AA Ostrik V On q-analog of mckay correspondence and ade classification of sl(2)

conformal field theories Adv Math 171 183ndash227 (2002)[KR] Kong L Runkel I Morita classes of algebras in modular tensor categories Adv Math 219

1548ndash1576 (2008)[Ld] Lauda AD Frobenius algebras and ambidextrous adjunctions Theo Appl Cat 16 84ndash

122 (2006)[Li1] Li H-S Regular representations of vertex operator algebras Commun Contemp Math 4

639ndash683 (2002)[Li2] Li H-S Regular representations and huang-lepowsky tensor functors for vertex operator

algebras J Alge 255 423ndash462 (2002)

Cardy Algebras and Sewing Constraints I 907

Sminus1 oplusiisinI

B Ui

Ui

f minusrarroplusjisinI

dim U jradicDimC

sumiisinI

B

U j

U j

Ui

f (A7)

By Lemma 32 to establish that (337) is S-invariant it is enough to prove the identity(34) when f is given by (337) Using (A6) and (A7) we can see that Eq (34) simplysays that oplusi j [RHS of (34)] is invariant under the action of S times S Consider the elementg of oplus jkisinIHomC2plusmn(R(A)otimes (Uor

j times Uk)Uorj times Uk) given by

g =oplusjkisinI

sumα

R(A)

R(A)

R(A)

R(A)

Uorj times Uk

Uorj times Uk

α

α

(A8)

By the above arguments proving S-invariance of (337) is equivalent to proving invari-ance of g under the action of S times S

For i isin I we denote by gi the component of g in(oplus jisinIHomC(A otimes Uor

i otimes Uorj U

orj )

)otimes (opluskisinIHomC(Ui otimes UkUk))

We view the second Hom-space in the above tensor product as a Hom-space in C+ insteadof Cminus It is enough to show that gi is invariant under the action of S times Sminus1 Note thatthe action of Sminus1 in C+ is equivalent to that of S in Cminus

The morphism gi can be canonically identified with a bilinear pairing

( middot middot )i ⎛⎝oplus

jisinIHomC(Uor

j A otimes Uori otimes Uor

j )

⎞⎠

times(oplus

kisinIHomC(UkUi otimes Uk)

)minusrarr C (A9)

as follows For h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) we set

(h1 h2)i = (dim U j dim Uk)minus1 trUor

j timesUk[gi (h1 times h2)] (A10)

908 L Kong I Runkel

When substituting the explicit form of the product m R(A) of R(A) = (A times 1) otimes R(1)after a short calculation one finds

(h1 h2)i =sumα

1

dim U j

A

A

A

A

A A

A α

α

h1

h2

U jUk

Uorj

Ui

Uk (A11)

Here the top morphism PlR(A) has been simplified with the help of the identity

PR(A) m R(A) (PR(A) otimes PR(A))

=(oplus

iisinI((m A A)otimes idUor

i)times idUi

) m R(A) (PR(A) otimes PR(A)) (A12)

which can be checked by direct calculation along the same lines as in the proof of [FrLem 310]

The action of the modular transformation S on oplusiisinI(B otimes Ui Ui ) for B isin C nat-urally induces an action on oplusiisinI(Ui B otimes Ui ) [K3 Prop 514] which we denote bySlowast In the present case we get an action of Slowast on oplus jisinIHomC(Uor

j A otimes Uori otimes Uor

j ) and

opluskisinIHomC(UkUi otimes Uk) Then to show gi is invariant under the action of S times Sminus1

amounts to showing that

(h1 h2)i = ((Sminus1)lowast h1 Slowast h2)i (A13)

Cardy Algebras and Sewing Constraints I 909

for all h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) We have

((Sminus1)lowast h1 Slowast h2)i =sum

mnα

dim Un

DimC

A

A

A

A

A A

A

Uk

α

α

h1

h2

Ui

U j

UmUn

(A14)

Now drag the upper vertex indexed by α in the above graph along its Uorm -leg until it

meets the lower vertex also indexed by α then sum over α and m This gives

((Sminus1)lowast h1 Slowast h2)i =sum

n

dim Un

DimC

A

A

A

A

A A

A

Uk

Un

h1

h2

Ui

U j

(A15)

910 L Kong I Runkel

If we just look at the neighbourhood of the Un-loop in the above graph we see thefollowing subgraphs

sumn

dim Un

DimC

A Aor

Uorj U j

Uk Uork

Un=

sumα

1

dim U j

A Aor Uk Uork

Uorj U j

αα

(A16)

where we have applied [BK Cor 3111] Substituting this subgraph back to the originalgraph in (A15) we obtain

((Sminus1)lowast h1 Slowast h2)i =sumα

1

dim U j

A

A

A

A

A A

A

A

A

Uorj

α

α

h1

h2m A

Ui

U jUk

Uk

(A17)

The graph in (A17) is equal to that in (A11) In order to see this we first drag theldquobubblerdquo (m A A) along A lines and through the m A vertex (because m A A is abimodule map) until it reaches the lower-left leg of the upper vertex indexed by α Thendrag the m A vertex along the (red) dotted line in the above graph Finally we applythe associativity of A (A12) and [Fr Lem 311] Then we see that the graph in (A17)exactly matches with the one in (A11)

Acknowledgements We would like to thank the organisers of the Oberwolfach Arbeitsgemeinschaft ldquoAlge-braic structures in conformal field theoriesrdquo (April 2007) where this work was started for an inspiring meetingWe would further like to thank the Hausdorff Institute for Mathematics in Bonn and the organisers of the stim-ulating meeting ldquoGeometry and Physicsrdquo (May 2008) We are indebted to Alexei Davydov Jens FjelstadJuumlrgen Fuchs Yi-Zhi Huang Alexei Kitaev Urs Schreiber Christoph Schweigert Stephan Stolz and PeterTeichner for helpful discussions andor comments on a draft of this paper The research of IR was partiallysupported by the EPSRC First Grant EPE0050471 the PPARC rolling grant PPC5071451 and the MarieCurie network lsquoSuperstring Theoryrsquo (MRTN-CT-2004-512194)

Cardy Algebras and Sewing Constraints I 911

References

[AN] Alexeevski A Natanzon SM Noncommutative two-dimensional topological field theories andhurwitz numbers for real algebraic curves Sel Math New Ser 12 307ndash377 (2006)

[Bi] Bichon J Cosovereign hopf algebras J Pure Appl Alg 157 121ndash133 (2001)[BK] Bakalov B Kirillov AA Lectures on Tensor Categories and Modular Functors Providence

RI Amer Math Soc 2001[C1] Cardy JL Operator content of two-dimensional conformal invariant theories Nucl Phys

B 270 186ndash204 (1986)[C2] Cardy JL Boundary conditions fusion rules and the verlinde formula Nucl Phys B 324

581ndash596 (1989)[CL] Cardy JL Lewellen DC Bulk and boundary operators in conformal field theory Phys Lett

B 259 274ndash278 (1991)[DMZ] Dong C-Y Mason G Zhu Y-C Discrete series of the Virasoro algebra and the moonshine

module In Algebraic Groups and Their Generalizations Quantum and infinite-dimensional Meth-ods Proc Symp Pure Math 56 Part 2 Providence RI Amer MathSoc 1994 pp 295ndash316

[DP] Day B Pastro C Note on frobenius monoidal functors New York J Math 14 733ndash742 (2008)[ENO1] Etingof PI Nikshych D Ostrik V On fusion categories Ann Math 162 581ndash642 (2005)[ENO2] Etingof PI Nikshych D Ostrik V An analogue of radfordrsquos s4 formula for finite tensor

categories Int Math Research Notices 54 2915ndash2933 (2004)[Fe] Felder G Froumlhlich J Fuchs J Schweigert C Correlation functions and boundary conditions in

rational conformal field theory and three-dimensional topology Comp Math 131 189ndash237 (2002)[FHL] Frenkel IB Huang Y-Z Lepowsky J On axiomatic approaches to vertex operator algebras

and modules Mem Amer Math Soc 104 (1993)[Fj] Fjelstad J Fuchs J Runkel I Schweigert C Uniqueness of openclosed rational cft with

given algebra of open states Adv Theor Math Phys 12 1283ndash1375 (2008)[FK] Froumlhlich J King C The chern-simons theory and knot polynomials Commun Math

Phys 126 167ndash199 (1989)[Fr] Froumlhlich J Fuchs J Runkel I Schweigert C Correspondences of ribbon categories Adv

Math 199 192ndash329 (2006)[FRS] Fuchs J Runkel I Schweigert C Tft construction of rcft correlators I partition functions Nucl

Phys B 646 353ndash497 (2002)[FS] Fuchs J Schweigert C Category theory for conformal boundary conditions Fields Inst Com-

mun 39 25ndash71 (2003)[H1] Huang Y-Z Two-dimensional conformal geometry and vertex operator algebras Progress in

Mathematics Vol 148 Boston Birkhaumluser 1997[H2] Huang Y-Z Rigidity and modularity of vertex tensor categories Comm Contemp Math 10

871ndash911 (2008)[HK1] Huang Y-Z Kong L Open-string vertex algebra category and operads Commun Math

Phys 250 433ndash471 (2004)[HK2] Huang Y-Z Kong L Full field algebras Commun Math Phys 272 345ndash396 (2007)[HK3] Huang Y-Z Kong L Modular invariance for conformal full field algebras httparxivorgabs

math0609570v2[mathQA] 2006[HL] Huang Y-Z Lepowsky J Tensor products of modules for a vertex operator algebra and vertex

tensor categories In Lie Theory and Geometry in honor of Bertram Kostant ed R BrylinskiJ-L Brylinski V Guillemin V Kac Boston Birkhaumluser 1994 pp 349ndash383

[JS] Joyal A Street R Braided tensor categories Adv Math 102 20ndash78 (1993)[K1] Kong L Full field algebras operads and tensor categories Adv Math 213 271ndash340 (2007)[K2] Kong L Open-closed field algebras Commun Math Phys 280 207ndash261 (2008)[K3] Kong L Cardy condition for open-closed field algebras Commun Math Phys 283 25ndash92 (2008)[Ki] Kitaev A Private communication[KO] Kirillov AA Ostrik V On q-analog of mckay correspondence and ade classification of sl(2)

conformal field theories Adv Math 171 183ndash227 (2002)[KR] Kong L Runkel I Morita classes of algebras in modular tensor categories Adv Math 219

1548ndash1576 (2008)[Ld] Lauda AD Frobenius algebras and ambidextrous adjunctions Theo Appl Cat 16 84ndash

122 (2006)[Li1] Li H-S Regular representations of vertex operator algebras Commun Contemp Math 4

639ndash683 (2002)[Li2] Li H-S Regular representations and huang-lepowsky tensor functors for vertex operator

algebras J Alge 255 423ndash462 (2002)

908 L Kong I Runkel

When substituting the explicit form of the product m R(A) of R(A) = (A times 1) otimes R(1)after a short calculation one finds

(h1 h2)i =sumα

1

dim U j

A

A

A

A

A A

A α

α

h1

h2

U jUk

Uorj

Ui

Uk (A11)

Here the top morphism PlR(A) has been simplified with the help of the identity

PR(A) m R(A) (PR(A) otimes PR(A))

=(oplus

iisinI((m A A)otimes idUor

i)times idUi

) m R(A) (PR(A) otimes PR(A)) (A12)

which can be checked by direct calculation along the same lines as in the proof of [FrLem 310]

The action of the modular transformation S on oplusiisinI(B otimes Ui Ui ) for B isin C nat-urally induces an action on oplusiisinI(Ui B otimes Ui ) [K3 Prop 514] which we denote bySlowast In the present case we get an action of Slowast on oplus jisinIHomC(Uor

j A otimes Uori otimes Uor

j ) and

opluskisinIHomC(UkUi otimes Uk) Then to show gi is invariant under the action of S times Sminus1

amounts to showing that

(h1 h2)i = ((Sminus1)lowast h1 Slowast h2)i (A13)

Cardy Algebras and Sewing Constraints I 909

for all h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) We have

((Sminus1)lowast h1 Slowast h2)i =sum

mnα

dim Un

DimC

A

A

A

A

A A

A

Uk

α

α

h1

h2

Ui

U j

UmUn

(A14)

Now drag the upper vertex indexed by α in the above graph along its Uorm -leg until it

meets the lower vertex also indexed by α then sum over α and m This gives

((Sminus1)lowast h1 Slowast h2)i =sum

n

dim Un

DimC

A

A

A

A

A A

A

Uk

Un

h1

h2

Ui

U j

(A15)

910 L Kong I Runkel

If we just look at the neighbourhood of the Un-loop in the above graph we see thefollowing subgraphs

sumn

dim Un

DimC

A Aor

Uorj U j

Uk Uork

Un=

sumα

1

dim U j

A Aor Uk Uork

Uorj U j

αα

(A16)

where we have applied [BK Cor 3111] Substituting this subgraph back to the originalgraph in (A15) we obtain

((Sminus1)lowast h1 Slowast h2)i =sumα

1

dim U j

A

A

A

A

A A

A

A

A

Uorj

α

α

h1

h2m A

Ui

U jUk

Uk

(A17)

The graph in (A17) is equal to that in (A11) In order to see this we first drag theldquobubblerdquo (m A A) along A lines and through the m A vertex (because m A A is abimodule map) until it reaches the lower-left leg of the upper vertex indexed by α Thendrag the m A vertex along the (red) dotted line in the above graph Finally we applythe associativity of A (A12) and [Fr Lem 311] Then we see that the graph in (A17)exactly matches with the one in (A11)

Acknowledgements We would like to thank the organisers of the Oberwolfach Arbeitsgemeinschaft ldquoAlge-braic structures in conformal field theoriesrdquo (April 2007) where this work was started for an inspiring meetingWe would further like to thank the Hausdorff Institute for Mathematics in Bonn and the organisers of the stim-ulating meeting ldquoGeometry and Physicsrdquo (May 2008) We are indebted to Alexei Davydov Jens FjelstadJuumlrgen Fuchs Yi-Zhi Huang Alexei Kitaev Urs Schreiber Christoph Schweigert Stephan Stolz and PeterTeichner for helpful discussions andor comments on a draft of this paper The research of IR was partiallysupported by the EPSRC First Grant EPE0050471 the PPARC rolling grant PPC5071451 and the MarieCurie network lsquoSuperstring Theoryrsquo (MRTN-CT-2004-512194)

Cardy Algebras and Sewing Constraints I 911

References

[AN] Alexeevski A Natanzon SM Noncommutative two-dimensional topological field theories andhurwitz numbers for real algebraic curves Sel Math New Ser 12 307ndash377 (2006)

[Bi] Bichon J Cosovereign hopf algebras J Pure Appl Alg 157 121ndash133 (2001)[BK] Bakalov B Kirillov AA Lectures on Tensor Categories and Modular Functors Providence

RI Amer Math Soc 2001[C1] Cardy JL Operator content of two-dimensional conformal invariant theories Nucl Phys

B 270 186ndash204 (1986)[C2] Cardy JL Boundary conditions fusion rules and the verlinde formula Nucl Phys B 324

581ndash596 (1989)[CL] Cardy JL Lewellen DC Bulk and boundary operators in conformal field theory Phys Lett

B 259 274ndash278 (1991)[DMZ] Dong C-Y Mason G Zhu Y-C Discrete series of the Virasoro algebra and the moonshine

module In Algebraic Groups and Their Generalizations Quantum and infinite-dimensional Meth-ods Proc Symp Pure Math 56 Part 2 Providence RI Amer MathSoc 1994 pp 295ndash316

[DP] Day B Pastro C Note on frobenius monoidal functors New York J Math 14 733ndash742 (2008)[ENO1] Etingof PI Nikshych D Ostrik V On fusion categories Ann Math 162 581ndash642 (2005)[ENO2] Etingof PI Nikshych D Ostrik V An analogue of radfordrsquos s4 formula for finite tensor

categories Int Math Research Notices 54 2915ndash2933 (2004)[Fe] Felder G Froumlhlich J Fuchs J Schweigert C Correlation functions and boundary conditions in

rational conformal field theory and three-dimensional topology Comp Math 131 189ndash237 (2002)[FHL] Frenkel IB Huang Y-Z Lepowsky J On axiomatic approaches to vertex operator algebras

and modules Mem Amer Math Soc 104 (1993)[Fj] Fjelstad J Fuchs J Runkel I Schweigert C Uniqueness of openclosed rational cft with

given algebra of open states Adv Theor Math Phys 12 1283ndash1375 (2008)[FK] Froumlhlich J King C The chern-simons theory and knot polynomials Commun Math

Phys 126 167ndash199 (1989)[Fr] Froumlhlich J Fuchs J Runkel I Schweigert C Correspondences of ribbon categories Adv

Math 199 192ndash329 (2006)[FRS] Fuchs J Runkel I Schweigert C Tft construction of rcft correlators I partition functions Nucl

Phys B 646 353ndash497 (2002)[FS] Fuchs J Schweigert C Category theory for conformal boundary conditions Fields Inst Com-

mun 39 25ndash71 (2003)[H1] Huang Y-Z Two-dimensional conformal geometry and vertex operator algebras Progress in

Mathematics Vol 148 Boston Birkhaumluser 1997[H2] Huang Y-Z Rigidity and modularity of vertex tensor categories Comm Contemp Math 10

871ndash911 (2008)[HK1] Huang Y-Z Kong L Open-string vertex algebra category and operads Commun Math

Phys 250 433ndash471 (2004)[HK2] Huang Y-Z Kong L Full field algebras Commun Math Phys 272 345ndash396 (2007)[HK3] Huang Y-Z Kong L Modular invariance for conformal full field algebras httparxivorgabs

math0609570v2[mathQA] 2006[HL] Huang Y-Z Lepowsky J Tensor products of modules for a vertex operator algebra and vertex

tensor categories In Lie Theory and Geometry in honor of Bertram Kostant ed R BrylinskiJ-L Brylinski V Guillemin V Kac Boston Birkhaumluser 1994 pp 349ndash383

[JS] Joyal A Street R Braided tensor categories Adv Math 102 20ndash78 (1993)[K1] Kong L Full field algebras operads and tensor categories Adv Math 213 271ndash340 (2007)[K2] Kong L Open-closed field algebras Commun Math Phys 280 207ndash261 (2008)[K3] Kong L Cardy condition for open-closed field algebras Commun Math Phys 283 25ndash92 (2008)[Ki] Kitaev A Private communication[KO] Kirillov AA Ostrik V On q-analog of mckay correspondence and ade classification of sl(2)

conformal field theories Adv Math 171 183ndash227 (2002)[KR] Kong L Runkel I Morita classes of algebras in modular tensor categories Adv Math 219

1548ndash1576 (2008)[Ld] Lauda AD Frobenius algebras and ambidextrous adjunctions Theo Appl Cat 16 84ndash

122 (2006)[Li1] Li H-S Regular representations of vertex operator algebras Commun Contemp Math 4

639ndash683 (2002)[Li2] Li H-S Regular representations and huang-lepowsky tensor functors for vertex operator

algebras J Alge 255 423ndash462 (2002)

Cardy Algebras and Sewing Constraints I 909

for all h1 isin HomC(Uorj A otimes Uor

i otimes Uorj ) and h2 isin HomC(UkUi otimes Uk) We have

((Sminus1)lowast h1 Slowast h2)i =sum

mnα

dim Un

DimC

A

A

A

A

A A

A

Uk

α

α

h1

h2

Ui

U j

UmUn

(A14)

Now drag the upper vertex indexed by α in the above graph along its Uorm -leg until it

meets the lower vertex also indexed by α then sum over α and m This gives

((Sminus1)lowast h1 Slowast h2)i =sum

n

dim Un

DimC

A

A

A

A

A A

A

Uk

Un

h1

h2

Ui

U j

(A15)

910 L Kong I Runkel

If we just look at the neighbourhood of the Un-loop in the above graph we see thefollowing subgraphs

sumn

dim Un

DimC

A Aor

Uorj U j

Uk Uork

Un=

sumα

1

dim U j

A Aor Uk Uork

Uorj U j

αα

(A16)

where we have applied [BK Cor 3111] Substituting this subgraph back to the originalgraph in (A15) we obtain

((Sminus1)lowast h1 Slowast h2)i =sumα

1

dim U j

A

A

A

A

A A

A

A

A

Uorj

α

α

h1

h2m A

Ui

U jUk

Uk

(A17)

The graph in (A17) is equal to that in (A11) In order to see this we first drag theldquobubblerdquo (m A A) along A lines and through the m A vertex (because m A A is abimodule map) until it reaches the lower-left leg of the upper vertex indexed by α Thendrag the m A vertex along the (red) dotted line in the above graph Finally we applythe associativity of A (A12) and [Fr Lem 311] Then we see that the graph in (A17)exactly matches with the one in (A11)

Acknowledgements We would like to thank the organisers of the Oberwolfach Arbeitsgemeinschaft ldquoAlge-braic structures in conformal field theoriesrdquo (April 2007) where this work was started for an inspiring meetingWe would further like to thank the Hausdorff Institute for Mathematics in Bonn and the organisers of the stim-ulating meeting ldquoGeometry and Physicsrdquo (May 2008) We are indebted to Alexei Davydov Jens FjelstadJuumlrgen Fuchs Yi-Zhi Huang Alexei Kitaev Urs Schreiber Christoph Schweigert Stephan Stolz and PeterTeichner for helpful discussions andor comments on a draft of this paper The research of IR was partiallysupported by the EPSRC First Grant EPE0050471 the PPARC rolling grant PPC5071451 and the MarieCurie network lsquoSuperstring Theoryrsquo (MRTN-CT-2004-512194)

Cardy Algebras and Sewing Constraints I 911

References

[AN] Alexeevski A Natanzon SM Noncommutative two-dimensional topological field theories andhurwitz numbers for real algebraic curves Sel Math New Ser 12 307ndash377 (2006)

[Bi] Bichon J Cosovereign hopf algebras J Pure Appl Alg 157 121ndash133 (2001)[BK] Bakalov B Kirillov AA Lectures on Tensor Categories and Modular Functors Providence

RI Amer Math Soc 2001[C1] Cardy JL Operator content of two-dimensional conformal invariant theories Nucl Phys

B 270 186ndash204 (1986)[C2] Cardy JL Boundary conditions fusion rules and the verlinde formula Nucl Phys B 324

581ndash596 (1989)[CL] Cardy JL Lewellen DC Bulk and boundary operators in conformal field theory Phys Lett

B 259 274ndash278 (1991)[DMZ] Dong C-Y Mason G Zhu Y-C Discrete series of the Virasoro algebra and the moonshine

module In Algebraic Groups and Their Generalizations Quantum and infinite-dimensional Meth-ods Proc Symp Pure Math 56 Part 2 Providence RI Amer MathSoc 1994 pp 295ndash316

[DP] Day B Pastro C Note on frobenius monoidal functors New York J Math 14 733ndash742 (2008)[ENO1] Etingof PI Nikshych D Ostrik V On fusion categories Ann Math 162 581ndash642 (2005)[ENO2] Etingof PI Nikshych D Ostrik V An analogue of radfordrsquos s4 formula for finite tensor

categories Int Math Research Notices 54 2915ndash2933 (2004)[Fe] Felder G Froumlhlich J Fuchs J Schweigert C Correlation functions and boundary conditions in

rational conformal field theory and three-dimensional topology Comp Math 131 189ndash237 (2002)[FHL] Frenkel IB Huang Y-Z Lepowsky J On axiomatic approaches to vertex operator algebras

and modules Mem Amer Math Soc 104 (1993)[Fj] Fjelstad J Fuchs J Runkel I Schweigert C Uniqueness of openclosed rational cft with

given algebra of open states Adv Theor Math Phys 12 1283ndash1375 (2008)[FK] Froumlhlich J King C The chern-simons theory and knot polynomials Commun Math

Phys 126 167ndash199 (1989)[Fr] Froumlhlich J Fuchs J Runkel I Schweigert C Correspondences of ribbon categories Adv

Math 199 192ndash329 (2006)[FRS] Fuchs J Runkel I Schweigert C Tft construction of rcft correlators I partition functions Nucl

Phys B 646 353ndash497 (2002)[FS] Fuchs J Schweigert C Category theory for conformal boundary conditions Fields Inst Com-

mun 39 25ndash71 (2003)[H1] Huang Y-Z Two-dimensional conformal geometry and vertex operator algebras Progress in

Mathematics Vol 148 Boston Birkhaumluser 1997[H2] Huang Y-Z Rigidity and modularity of vertex tensor categories Comm Contemp Math 10

871ndash911 (2008)[HK1] Huang Y-Z Kong L Open-string vertex algebra category and operads Commun Math

Phys 250 433ndash471 (2004)[HK2] Huang Y-Z Kong L Full field algebras Commun Math Phys 272 345ndash396 (2007)[HK3] Huang Y-Z Kong L Modular invariance for conformal full field algebras httparxivorgabs

math0609570v2[mathQA] 2006[HL] Huang Y-Z Lepowsky J Tensor products of modules for a vertex operator algebra and vertex

tensor categories In Lie Theory and Geometry in honor of Bertram Kostant ed R BrylinskiJ-L Brylinski V Guillemin V Kac Boston Birkhaumluser 1994 pp 349ndash383

[JS] Joyal A Street R Braided tensor categories Adv Math 102 20ndash78 (1993)[K1] Kong L Full field algebras operads and tensor categories Adv Math 213 271ndash340 (2007)[K2] Kong L Open-closed field algebras Commun Math Phys 280 207ndash261 (2008)[K3] Kong L Cardy condition for open-closed field algebras Commun Math Phys 283 25ndash92 (2008)[Ki] Kitaev A Private communication[KO] Kirillov AA Ostrik V On q-analog of mckay correspondence and ade classification of sl(2)

conformal field theories Adv Math 171 183ndash227 (2002)[KR] Kong L Runkel I Morita classes of algebras in modular tensor categories Adv Math 219

1548ndash1576 (2008)[Ld] Lauda AD Frobenius algebras and ambidextrous adjunctions Theo Appl Cat 16 84ndash

122 (2006)[Li1] Li H-S Regular representations of vertex operator algebras Commun Contemp Math 4

639ndash683 (2002)[Li2] Li H-S Regular representations and huang-lepowsky tensor functors for vertex operator

algebras J Alge 255 423ndash462 (2002)

910 L Kong I Runkel

If we just look at the neighbourhood of the Un-loop in the above graph we see thefollowing subgraphs

sumn

dim Un

DimC

A Aor

Uorj U j

Uk Uork

Un=

sumα

1

dim U j

A Aor Uk Uork

Uorj U j

αα

(A16)

where we have applied [BK Cor 3111] Substituting this subgraph back to the originalgraph in (A15) we obtain

((Sminus1)lowast h1 Slowast h2)i =sumα

1

dim U j

A

A

A

A

A A

A

A

A

Uorj

α

α

h1

h2m A

Ui

U jUk

Uk

(A17)

The graph in (A17) is equal to that in (A11) In order to see this we first drag theldquobubblerdquo (m A A) along A lines and through the m A vertex (because m A A is abimodule map) until it reaches the lower-left leg of the upper vertex indexed by α Thendrag the m A vertex along the (red) dotted line in the above graph Finally we applythe associativity of A (A12) and [Fr Lem 311] Then we see that the graph in (A17)exactly matches with the one in (A11)

Acknowledgements We would like to thank the organisers of the Oberwolfach Arbeitsgemeinschaft ldquoAlge-braic structures in conformal field theoriesrdquo (April 2007) where this work was started for an inspiring meetingWe would further like to thank the Hausdorff Institute for Mathematics in Bonn and the organisers of the stim-ulating meeting ldquoGeometry and Physicsrdquo (May 2008) We are indebted to Alexei Davydov Jens FjelstadJuumlrgen Fuchs Yi-Zhi Huang Alexei Kitaev Urs Schreiber Christoph Schweigert Stephan Stolz and PeterTeichner for helpful discussions andor comments on a draft of this paper The research of IR was partiallysupported by the EPSRC First Grant EPE0050471 the PPARC rolling grant PPC5071451 and the MarieCurie network lsquoSuperstring Theoryrsquo (MRTN-CT-2004-512194)

Cardy Algebras and Sewing Constraints I 911

References

[AN] Alexeevski A Natanzon SM Noncommutative two-dimensional topological field theories andhurwitz numbers for real algebraic curves Sel Math New Ser 12 307ndash377 (2006)

[Bi] Bichon J Cosovereign hopf algebras J Pure Appl Alg 157 121ndash133 (2001)[BK] Bakalov B Kirillov AA Lectures on Tensor Categories and Modular Functors Providence

RI Amer Math Soc 2001[C1] Cardy JL Operator content of two-dimensional conformal invariant theories Nucl Phys

B 270 186ndash204 (1986)[C2] Cardy JL Boundary conditions fusion rules and the verlinde formula Nucl Phys B 324

581ndash596 (1989)[CL] Cardy JL Lewellen DC Bulk and boundary operators in conformal field theory Phys Lett

B 259 274ndash278 (1991)[DMZ] Dong C-Y Mason G Zhu Y-C Discrete series of the Virasoro algebra and the moonshine

module In Algebraic Groups and Their Generalizations Quantum and infinite-dimensional Meth-ods Proc Symp Pure Math 56 Part 2 Providence RI Amer MathSoc 1994 pp 295ndash316

[DP] Day B Pastro C Note on frobenius monoidal functors New York J Math 14 733ndash742 (2008)[ENO1] Etingof PI Nikshych D Ostrik V On fusion categories Ann Math 162 581ndash642 (2005)[ENO2] Etingof PI Nikshych D Ostrik V An analogue of radfordrsquos s4 formula for finite tensor

categories Int Math Research Notices 54 2915ndash2933 (2004)[Fe] Felder G Froumlhlich J Fuchs J Schweigert C Correlation functions and boundary conditions in

rational conformal field theory and three-dimensional topology Comp Math 131 189ndash237 (2002)[FHL] Frenkel IB Huang Y-Z Lepowsky J On axiomatic approaches to vertex operator algebras

and modules Mem Amer Math Soc 104 (1993)[Fj] Fjelstad J Fuchs J Runkel I Schweigert C Uniqueness of openclosed rational cft with

given algebra of open states Adv Theor Math Phys 12 1283ndash1375 (2008)[FK] Froumlhlich J King C The chern-simons theory and knot polynomials Commun Math

Phys 126 167ndash199 (1989)[Fr] Froumlhlich J Fuchs J Runkel I Schweigert C Correspondences of ribbon categories Adv

Math 199 192ndash329 (2006)[FRS] Fuchs J Runkel I Schweigert C Tft construction of rcft correlators I partition functions Nucl

Phys B 646 353ndash497 (2002)[FS] Fuchs J Schweigert C Category theory for conformal boundary conditions Fields Inst Com-

mun 39 25ndash71 (2003)[H1] Huang Y-Z Two-dimensional conformal geometry and vertex operator algebras Progress in

Mathematics Vol 148 Boston Birkhaumluser 1997[H2] Huang Y-Z Rigidity and modularity of vertex tensor categories Comm Contemp Math 10

871ndash911 (2008)[HK1] Huang Y-Z Kong L Open-string vertex algebra category and operads Commun Math

Phys 250 433ndash471 (2004)[HK2] Huang Y-Z Kong L Full field algebras Commun Math Phys 272 345ndash396 (2007)[HK3] Huang Y-Z Kong L Modular invariance for conformal full field algebras httparxivorgabs

math0609570v2[mathQA] 2006[HL] Huang Y-Z Lepowsky J Tensor products of modules for a vertex operator algebra and vertex

tensor categories In Lie Theory and Geometry in honor of Bertram Kostant ed R BrylinskiJ-L Brylinski V Guillemin V Kac Boston Birkhaumluser 1994 pp 349ndash383

[JS] Joyal A Street R Braided tensor categories Adv Math 102 20ndash78 (1993)[K1] Kong L Full field algebras operads and tensor categories Adv Math 213 271ndash340 (2007)[K2] Kong L Open-closed field algebras Commun Math Phys 280 207ndash261 (2008)[K3] Kong L Cardy condition for open-closed field algebras Commun Math Phys 283 25ndash92 (2008)[Ki] Kitaev A Private communication[KO] Kirillov AA Ostrik V On q-analog of mckay correspondence and ade classification of sl(2)

conformal field theories Adv Math 171 183ndash227 (2002)[KR] Kong L Runkel I Morita classes of algebras in modular tensor categories Adv Math 219

1548ndash1576 (2008)[Ld] Lauda AD Frobenius algebras and ambidextrous adjunctions Theo Appl Cat 16 84ndash

122 (2006)[Li1] Li H-S Regular representations of vertex operator algebras Commun Contemp Math 4

639ndash683 (2002)[Li2] Li H-S Regular representations and huang-lepowsky tensor functors for vertex operator

algebras J Alge 255 423ndash462 (2002)

Cardy Algebras and Sewing Constraints I 911

References

[AN] Alexeevski A Natanzon SM Noncommutative two-dimensional topological field theories andhurwitz numbers for real algebraic curves Sel Math New Ser 12 307ndash377 (2006)

[Bi] Bichon J Cosovereign hopf algebras J Pure Appl Alg 157 121ndash133 (2001)[BK] Bakalov B Kirillov AA Lectures on Tensor Categories and Modular Functors Providence

RI Amer Math Soc 2001[C1] Cardy JL Operator content of two-dimensional conformal invariant theories Nucl Phys

B 270 186ndash204 (1986)[C2] Cardy JL Boundary conditions fusion rules and the verlinde formula Nucl Phys B 324

581ndash596 (1989)[CL] Cardy JL Lewellen DC Bulk and boundary operators in conformal field theory Phys Lett

B 259 274ndash278 (1991)[DMZ] Dong C-Y Mason G Zhu Y-C Discrete series of the Virasoro algebra and the moonshine

module In Algebraic Groups and Their Generalizations Quantum and infinite-dimensional Meth-ods Proc Symp Pure Math 56 Part 2 Providence RI Amer MathSoc 1994 pp 295ndash316

[DP] Day B Pastro C Note on frobenius monoidal functors New York J Math 14 733ndash742 (2008)[ENO1] Etingof PI Nikshych D Ostrik V On fusion categories Ann Math 162 581ndash642 (2005)[ENO2] Etingof PI Nikshych D Ostrik V An analogue of radfordrsquos s4 formula for finite tensor

categories Int Math Research Notices 54 2915ndash2933 (2004)[Fe] Felder G Froumlhlich J Fuchs J Schweigert C Correlation functions and boundary conditions in

rational conformal field theory and three-dimensional topology Comp Math 131 189ndash237 (2002)[FHL] Frenkel IB Huang Y-Z Lepowsky J On axiomatic approaches to vertex operator algebras

and modules Mem Amer Math Soc 104 (1993)[Fj] Fjelstad J Fuchs J Runkel I Schweigert C Uniqueness of openclosed rational cft with

given algebra of open states Adv Theor Math Phys 12 1283ndash1375 (2008)[FK] Froumlhlich J King C The chern-simons theory and knot polynomials Commun Math

Phys 126 167ndash199 (1989)[Fr] Froumlhlich J Fuchs J Runkel I Schweigert C Correspondences of ribbon categories Adv

Math 199 192ndash329 (2006)[FRS] Fuchs J Runkel I Schweigert C Tft construction of rcft correlators I partition functions Nucl

Phys B 646 353ndash497 (2002)[FS] Fuchs J Schweigert C Category theory for conformal boundary conditions Fields Inst Com-

mun 39 25ndash71 (2003)[H1] Huang Y-Z Two-dimensional conformal geometry and vertex operator algebras Progress in

Mathematics Vol 148 Boston Birkhaumluser 1997[H2] Huang Y-Z Rigidity and modularity of vertex tensor categories Comm Contemp Math 10

871ndash911 (2008)[HK1] Huang Y-Z Kong L Open-string vertex algebra category and operads Commun Math

Phys 250 433ndash471 (2004)[HK2] Huang Y-Z Kong L Full field algebras Commun Math Phys 272 345ndash396 (2007)[HK3] Huang Y-Z Kong L Modular invariance for conformal full field algebras httparxivorgabs

math0609570v2[mathQA] 2006[HL] Huang Y-Z Lepowsky J Tensor products of modules for a vertex operator algebra and vertex

tensor categories In Lie Theory and Geometry in honor of Bertram Kostant ed R BrylinskiJ-L Brylinski V Guillemin V Kac Boston Birkhaumluser 1994 pp 349ndash383

[JS] Joyal A Street R Braided tensor categories Adv Math 102 20ndash78 (1993)[K1] Kong L Full field algebras operads and tensor categories Adv Math 213 271ndash340 (2007)[K2] Kong L Open-closed field algebras Commun Math Phys 280 207ndash261 (2008)[K3] Kong L Cardy condition for open-closed field algebras Commun Math Phys 283 25ndash92 (2008)[Ki] Kitaev A Private communication[KO] Kirillov AA Ostrik V On q-analog of mckay correspondence and ade classification of sl(2)

conformal field theories Adv Math 171 183ndash227 (2002)[KR] Kong L Runkel I Morita classes of algebras in modular tensor categories Adv Math 219

1548ndash1576 (2008)[Ld] Lauda AD Frobenius algebras and ambidextrous adjunctions Theo Appl Cat 16 84ndash

122 (2006)[Li1] Li H-S Regular representations of vertex operator algebras Commun Contemp Math 4

639ndash683 (2002)[Li2] Li H-S Regular representations and huang-lepowsky tensor functors for vertex operator

algebras J Alge 255 423ndash462 (2002)

912 L Kong I Runkel

[Ln] Leinster T Higher operads higher categories London Mathematical Society Lecture Note Series298 Cambridge Cambridge University Press 2004

[LP] Lauda A Pfeiffer H Open-closed strings two-dimensional extended tqfts and frobeniusalgebras Topology Appl 155 623ndash666 (2008)

[LR] Longo R Rehren KH Local fields in boundary conformal qft Rev Math Phys 16 909 (2004)[Lw] Lewellen DC Sewing constraints for conformal field theories on surfaces with boundaries Nucl

Phys B 372 654ndash682 (1992)[Lz] Lazaroiu CI On the structure of open-closed topological field theory in two dimensions Nucl

Phys B 603 497ndash530 (2001)[Ma] Mac Lane S Categories for the working mathematician Brelin-Heidelberg-NewYork Springer

1998[Mo] Moore G Some comments on branes G-flux and K -theory Int J Mod Phys A16 936ndash

944 (2001)[MS] Moore G Segal G D-branes and K-theory in 2D topological field theory httparxivorgabs

hep-th0609042v1 2006[Muuml1] Muumlger M From subfactors to categories and topology II The Quantum Double of Tensor Cate-

gories and Subfactors J Pure Appl Alg 180 159ndash219 (2003)[Muuml2] Muumlger M Talk at workshop lsquoQuantum Structuresrsquo (Leipzig 28 June 2007) Preprint in prepa-

ration[O] Ostrik V Module categories weak hopf algebras and modular invariants Transform Groups 8

177ndash206 (2003)[P] Pfeiffer H Finitely semisimple spherical categories and modular categories are self-dual Adv

Math 221 1608ndash1652 (2009)[RT] Reshetikhin N Turaev VG Invariants of 3-manifolds via link polynomials and quantum

groups Inv Math 103 547ndash597 (1991)[Se] Segal G The definition of conformal field theory Preprint 1988 also in U Tillmann (ed)

Topology geometry and quantum field theory London Math Soc Lect Note Ser 308 Cam-bridge Cambridge Univ Press 2004 pp 421ndash577

[So] Sonoda H Sewing conformal field theories II Nucl Phys B 311 417ndash432 (1988)[Sz] Szlachaacutenyi K Adjointable monoidal functors and quantum groupoids In Hopf algebras in non-

commutative geometry and physics Caenepeel S Oystaeyen FV (eds) Lecture Notes in Pureand Applied Mathematics 239 Boca Raton FL CRC Press 2004 pp 297ndash307

[T] Turaev VG Quantum Invariants of Knots and 3-Manifolds New York de Gruyter 1994[V] Vafa C Conformal theories and punctured surfaces Phys Lett B 199 195ndash202 (1987)[VZ] Van Oystaeyen F Zhang YH The brauer group of a braided monoidal category J Algebra 202

96ndash128 (1998)[W] Witten E Quantum field theory and the jones polynomial Commun Math Phys 121 351ndash399

(1989)[Y] Yetter DN Functorial knot theory Categories of tangles coherence categorical deformations

and topological invariants Series on Knots and Everything 26 River Edge NJ World Scientific2001

Communicated by Y Kawahigashi