TQFTs and Invariants of 3-Manifolds

Christian Kissig

February 25, 2010

Contents

1 Introduction 31.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Historical Background . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Organisation of These Notes . . . . . . . . . . . . . . . . . . . . . 4

1.3.1 Contents and Dependencies . . . . . . . . . . . . . . . . . 41.3.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Apologies and Request for Comments . . . . . . . . . . . . . . . 4

2 Preliminaries 52.1 Monoidal Categories . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Ribbon Categories . . . . . . . . . . . . . . . . . . . . . . 62.1.2 Ab-Categories . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.3 Modular Categories . . . . . . . . . . . . . . . . . . . . . 72.1.4 The Ground Ring of a Modular Category . . . . . . . . . 8

2.2 Ribbon Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.1 Ribbon Graphs, Uncoloured . . . . . . . . . . . . . . . . . 82.2.2 Ribbon Graphs, Coloured . . . . . . . . . . . . . . . . . . 8

2.3 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3.1 Collars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3.2 Surgery on Links in S3 . . . . . . . . . . . . . . . . . . . . 92.3.3 Topological Invariants of 3-Manifolds . . . . . . . . . . . . 92.3.4 Examples of Manifolds . . . . . . . . . . . . . . . . . . . . 10

3 Elements of 3D-Topological Quantum Field Theories 113.1 Cobordisms, Gluing Patterns, Cobordism Theories . . . . . . . . 113.2 2-Dimensional Modular Functors . . . . . . . . . . . . . . . . . . 123.3 3-Dimensional Topological Quantum Field Theories . . . . . . . . 12

3.3.1 Axiomatic Definition of 3D-TQFTs . . . . . . . . . . . . . 123.3.2 Homomorphisms and Isomorphisms of 3D-TQFTs, Non-

Degenerate TQFTs . . . . . . . . . . . . . . . . . . . . . . 133.4 Quantum Invariants . . . . . . . . . . . . . . . . . . . . . . . . . 14

4 A Construction of a 3-Dimensional Topological Quantum FieldTheory 164.1 3-Cobordism Theories with Decorations . . . . . . . . . . . . . . 16

4.1.1 Decorated Surfaces . . . . . . . . . . . . . . . . . . . . . . 164.1.2 Decorated Types . . . . . . . . . . . . . . . . . . . . . . . 17

4.2 A 3D-TQFT for 3-Cobordism Theories with Decorations . . . . . 17

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4.2.1 Construction of a Modular Functor from Decorations . . . 174.2.2 Operator Invariants of Decorated 3-Cobordisms . . . . . . 18

4.3 Soundness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.3.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.3.2 Representation of 3-Cobordisms by Ribbon Graphs . . . . 19

5 Anomalies 205.1 Anomaly 2-Cocycles . . . . . . . . . . . . . . . . . . . . . . . . . 205.2 Renormalisation of Cobordisms by means of 2-Cocycles . . . . . 215.3 Anomaly Cocycles of 3D-TQFTs, Anomaly-free Renormalisation 215.4 Lagrangian Spaces, Lagrangian Relations and Maslov Indices:

Preliminary Definitions . . . . . . . . . . . . . . . . . . . . . . . 225.4.1 Lagrangian Spaces and Contractions . . . . . . . . . . . . 225.4.2 Lagrangian Relations and Actions Thereof . . . . . . . . . 225.4.3 Maslov Indices . . . . . . . . . . . . . . . . . . . . . . . . 22

5.5 A Digression into Homologies of Surfaces . . . . . . . . . . . . . 235.5.1 The Lagrangian Functor . . . . . . . . . . . . . . . . . . . 23

5.6 Computing Anomalies . . . . . . . . . . . . . . . . . . . . . . . . 235.7 The Anomaly 2-Cocycle . . . . . . . . . . . . . . . . . . . . . . . 24

2

Chapter 1

Introduction

1.1 SummaryIn this tutorial we discuss invariants of 3-manifolds (with banded links embed-ded), and how to compute them by means of 3-dimensional topological quantumfield theories (3D-TQFTs). The 3D-TQFTs, we consider, shall be defined in theReshetikhin-Turaev construction[6] from a modular category V over a groundring K as modular functors from (subcategories of) the category 3Cob of 3-cobordisms into categories of projective modules over K.

The 3-cobordisms have embedded ribbon links. Modularity will use thatthese ribbon links Ω meet the boundary ∂M of the cobordisms in some compo-nent Σ, such that the bands are incident with arcs decorating surfaces. Such acobordism can be closed by gluing handlebodies with embedded bands. Mod-ularity, decoration with arcs, and gluing makes it necessary that we considerribbon graphs instead. These consist of annuli (non-M\”obius ribbon knots),bands, and coupons which are a ribbon-equivalent of nodes in graphs.

The ribbon graphs will be coloured in a modular category V. The colouringcarries over to the decoration of surfaces and to a form of gluing compatible withdecoration and colouring. It turns out that a cobordism in such a cobordismtheory with decorations is completely described by the topological propertiesof the underlying manifold and the decorations. In order to define modularfunctors from cobordism theories with decorations, it suffices to look at thedecorations in the surfaces.

The constructed 3D-TQFT is non-degenerate, but has anomalies. In factthe axiomatic characterisation of 3D-TQFT from modular functors suggest thatanomalies are introduced by gluing. An additional parameterisation in weightsin an abelian group G (basically, of the invertible elements of the ground ring)yields a standard renormalisation, which moves anomalies into gluing and pro-duces a cobordism theory with decorations and weights. The anomalies canoften be computed by 2-cocycles valued in G. In particular our constructionyields anomaly 2-cocycles, that we can compute by Maslov-indices of Lagrangiansubspaces of the homologies on the surfaces.

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1.2 Historical BackgroundModular functors arose from 2-dimensional rational conformal field theory. Segal[7],Moore and Seiberg[5] first gave an axiomatic definition. Topological quantumfield theories were inspired by the work of Witten[10, 10]. Unitary TQFTswere first axiomatised by Atiyah[1] in an extension of Segal’s modular functors.Quinn has first studied the axiomatic foundations of TQFTs in an abstractcategorical setting. Domain categories took the place of cobordism theories.

1.3 Organisation of These Notes

1.3.1 Contents and DependenciesIn Section 2 we introduce the foundational concepts in monoidal categories,modular categories and functors, links and graphs. These foundations allowus to define 3-dimensional topological quantum field theories in Chapter 3 ax-iomatically. Chapter 4 will be concerned with the concrete construction ofa 3-dimensional topological quantum field theory. The constructed TQFT isnon-degenerate but has anomalies. We look into the anomalies in Chapter 5,and propose methods to renormalise TQFTs. The so constructed 3-dimensionalTQFTs will be in bijection with the quantum invariants of 3-manifolds withembedded banded links.

Dependencies between the chapters are linear. The knowledgeable readermay skip any of the preliminary sections and return for reference later withoutharm.

1.3.2 BackgroundThese notes are largely based on Chapters I-V of [8], and on the lecture serieson “Integral TQFT” by Masbaum held at Copenhagen in 2007. These notes areprepared for an invited seminar at the University of Warsaw on 27 February2010. The material on homologies on surfaces stems mostly from [2].

1.4 Apologies and Request for CommentsI understand that the lack of images renders the presentation almost illegible. Iplan to add figures in later versions of these notes. My apologies for this deficite.Nevertheless, I welcome comments and suggestions at tqft@christiankissig.de.Additional material, references, and updated notes for this seminar can be foundat http://www.christiankissig.de/3dtqft.

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Chapter 2

Preliminaries

2.1 Monoidal CategoriesMonoidal categories are regular categories endowed with a monoidal structure.

Definition 1. A monoidal category V := 〈V,⊗, I〉 consist of a regular categoryV with a 2-multi-endofunctor with identity object so that there are naturalisomorphisms lX : I ⊗ X → X and rX : X ⊗ I → X, subject to identity andassociativity laws

A monoidal category may be symmetric, if it comes with a natural isomor-phism s : (−) ⊗ (+) ∼= (+) ⊗ (−), and closed if V(X,−) is right adjoint to(−)⊗X for all objects X of V.

Furthermore, monoidal categories may be braided.

Definition 2. A braiding in a monoidal category V is a natural isomorphismc : (−)⊗ (+) ⇒ (+)⊗ (−) such that

cX,Y⊗Z = (idX ⊗ cX,Z)(cX,Y ⊗ idZ) (2.1)

cX⊗Y,Z = (cX,Z ⊗ idV )(idX ⊗ cV,W ) (2.2)

A twist in a braided monoidal category V is a natural isomorphism t : (−) ⇒(−) such that

tX⊗Y = cY,XcX,Y (tX ⊗ tY ) (2.3)

Example 3. Categories of modules are symmetric monoidal closed under ⊗being the coproduct

∐(disjoint union).

Example 4. Categories of 3-cobordisms are monoidal under disjoint union∐

and the empty cobordism ∅.

Dualities generalise non-degenerate bilinear forms to monoidal categories.

Definition 5. A duality on a monoidal category V consists of an endomorphism(−)∗on the objects of V and natural transformations b : (−) ⇒ (−)⊗ (−)∗ andd : (−)∗ ⊗ (−) ⇒ (−) such that

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(idX ⊗ dX)(bX ⊗ idX) = idX (2.4)

(dX ⊗ idX∗)(idX∗ ⊗ bX) = idX∗ (2.5)

Naturality of b and d accounts for bilinearity, the rest of the analogy we leaveto the reader.

Remark 6. There are elegant graphical calculi for reasoning in monoidal cate-gories. See for instance Selinger “A survey of graphical languages for monoidalcategories” for various kinds of monoidal categories or Turaev “Quantum Invari-ants of Knots and 3-Manifolds” Section I.1.6 for ribbon categories in particular.

2.1.1 Ribbon CategoriesDefinition 7. A ribbon category is a monoidal category V with braiding c,twist t, and compatible duality ((−)∗, b, d). A ribbon category is strict if itsunderlying monoidal category is strict.

Example 8. The category Proj(K) of projective modules for a ring K is aribbon category.

Example 9. Given a multiplicative abelian group G , a commutative ring Kwith unit, a bilinear pairing c : G×G → K∗where K∗is the multiplicative groupof invertible elements of K. Bilinearity means that c(gg′, h) = c(g, h)c(g′, h)and c(g, hh′) = c(g, h)c(g, h′). We construct a ribbon category V with objectsbeing the elements of G, endomorphisms g → g being elements of K, andhomomorphisms g → h being zero. Composition is given by the multiplicationin K, the tensor by multiplication in G.

Exercise 10. Define braiding and twist for V in Example 9.

Exercise 11. Verify the axioms of ribbon categories on the category V in Ex-ample 9.

Example 12. Finite-dimensional representations of quantum groups form aribbon category. For a detailed discussion we refer the reader to [3].

Remark 13. MacLane’s coherence theorem specialises to ribbon categories.

2.1.2 Ab-CategoriesDefinition 14. A category V is an Ab-category if each homset Hom(X, Y )comes with the structure of an additive abelian group.

Exercise 15. Verify that homsets End(X) = Hom(X, X) of endomorphismsin Ab-categories are rings.

Exercise 16. Verify that homsets Hom(X, Y ) in monoidal Ab-categories havethe structure of left K-modules where K = End(I) under k · f = k ⊗ f .

Proposition 17. In ribbon Ab-categories multiplication with K is bilinear, inparticular k ⊗ idV = idV ⊗ k for all objects V of V.

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Example 18. The category Proj (K) of projective modules for a ring K is aribbon Ab-category.

Example 19. The category V of Example 9 is an Ab-category.

Definition 20. An object V of V is simple if k 7→ k ⊗ idV defines a bijectionK ∼= End(V ).

Example 21. I is simple.

Definition 22. The trace tr(f) of an endomorphism f : V → V on an objectV of V is defined as

tr(f) = dV cV,V ∗((tV f)⊗ idV ∗)bV : I → I (2.6)

Definition 23. The dimension dim(V ) of an object V is defined as the traceof the identity morphism idV , viz

dim(V ) = tr(idV ) = dV cV,V ∗(t⊗ idV ∗)bV : I → I (2.7)

Remark 24. As I is simple, dimension and trace are elements of the ground ringK.

2.1.3 Modular CategoriesA decomposition in direct sums of indecomposables is known to exist for ar-tinian, noetherian or semi-simple modules. General ribbon categories do notadmit direct sums of objects, but we obtain decomposition of their identitymorphisms via the notion of domination:

Definition 25. An object V of V is said to be dominated by a family Vii∈I ofobjects of V if idV decomposes into the sum

∑j∈J fw(j)gw(j) of a finite zig-zag

of morphisms fw(j) : Vw(j) → V and gw(j) : V → Vw(j) indexed by a finitewordw ∈ I∗.

Exercise 26. Show that if V admits direct sums, then an object V of V isdominated by a family of objects of V iff there is an object W of V such thatV ⊕W decomposes into a finite direct sum of objects of this category.

Definition 27. A modular category is a ribbon Ab-category V with a finitefamily V =Vii∈I of objects of V satisfying

1. (normalisation) I ∈ V

2. (duality) V is closed under (−)∗.

3. (domination) Each object of V is dominated by V.

4. (non-degeneracy) The square matrix [Sij ]i,j∈I is invertible over the groundring K of V.

Sij = tr(cVj ,Vi cVi,Vj

) (2.8)

Exercise 28. Using the (non-degeneracy) axiom show that I appears preciselyonce in V.

Remark 29. Semi-simple ribbon categories generalise modular categories in thatthey allow arbitrary families V = Vii∈I . Schur’s axiom assertsV to be discretein the sense that Hom(Vi, Vj) = 0 for all i, j ∈ I with i 6= j. Semi-simple ribboncategories give rise to topological invariants of links and ribbon graphs in R3,but not 3-manifolds.

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2.1.4 The Ground Ring of a Modular CategoryLet V be a modular category.

Definition 30. The rank of V is an element rank(V) ∈ K of the ground ringK = End(I) of V, such that

rank(V) =∑i∈I

(dim(i))2 (2.9)

The rank is not unique, in particular −rank(V) is a rank of V.

Definition 31. ∆ ∈ K is uniquely determined by

∆ =∑i∈I

v−1i (dim(i))2 (2.10)

2.2 Ribbon Graphs

2.2.1 Ribbon Graphs, UncolouredDefinition 32. A ribbon graph Ω embedded in a manifold M is a compactoriented surface composite of a finite set of

1. bands, that are homeomorphic images of [0, 1]×[0, 1] where we call [0, 1]×0and [0, 1]× 1 the bases of the band

2. coupons, that are bands with one base distinguished as their top base, andone as their bottom base

3. and annuli, that are homeomorphic images of cylinders S1 × [0, 1] (Infor-mally, these are knots that are not Moebius strips.)

Remark 33. Choosing an orientation of a ribbon graph is equivalent to choosinga preferred side.

2.2.2 Ribbon Graphs, ColouredDefinition 34. In this text we consider ribbon graphs that are coloured in amodular category V with duality. Bands and annuli are coloured with objectsof V, and coupons are coloured with morphisms

f : V d01 ⊗ ...⊗ V dm

m → W e11 ⊗ ...⊗W en

n (2.11)

where V1, ..., Vm are the colours and d1, ..., dm are the directions (in- or out-going) of the bands incident to the top edge, W1, ...,Wn are the colours ande1, ..., enare the directions of the bands incident to the bottom edge. WhereV +1 = V and V −1 = V ∗ and similar for W .

Example 35. The category RibV of V-coloured ribbon graphs has finite squences〈〈V1, v1〉, ..., 〈Vm, vm〉〉 of pairs of coloursVi and signs vi ∈ −1,+1 as objects, andisotopy types of V-coloured ribbon graphs as morphisms with colours and di-rections at top and bottom edge incident with the respective object.

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The following is a fundamental observation for the definition of isotopy in-variants of coloured ribbon graphs, topological invariants of 3-manifolds withembedded ribbon graphs, and later on for the definition of quantum invariantsof 3-cobordisms. F from the its action on the basic directed components ofribbon graphs, overcrossing X, loop φ, cup

⋃, and cap

⋂.

Definition 36. There is a unique functor F : RibV → V taking

• objects 〈V,+1〉 to V , and 〈V,−1〉to V ∗

• for colours V and W , F (X+V,W ) = cV,W , F (φV ) = θV , F (

⋃V ) = bV , and

F (⋂

V ) = dV where X+is a positive overcrossing, φis a positive loop,⋂

isa cap directed left-to-right,

⋃is a cup directed left-to-right

• a V-coloured graph with one coupon to the colour of the coupon

2.3 ManifoldsBy manifolds we mean compact oriented manifold M with boundary ∂M . Amanifolds with ∂M = ∅ is called closed.

2.3.1 CollarsA manifold M is said to have collars if it is homeomorphic to a copy of itselfwith a cylinder B1 × [0, 1] glued to each component Σ of the boundary ∂M .

2.3.2 Surgery on Links in S3

Let L be a framed link in S3 with m components, L1, ..., Lm. A closed regularneighbourhood U of L consists of m disjoint solid tori, U1,..., Um, whose coresare the components of L. Removing the solid tori from S3 yields a remainderR, one can define a bijection f between ∂R and S3and glues R and S3backtogether along f .

Remark 37. Note that in the context of differentiable manifolds surgery alongembedded links in S3 creates Wilson loops.

Theorem 38. Every closed connected oriented 3-manifold M is obtained bysurgery on S3along a framed link L.

This theorem is due to Lickorish[4] and Wallace[9]. The proof uses thetheorem of Rokhlin that every closed oriented 3-manifold bounds a compactoriented piecewise-linear 4-manifold.

2.3.3 Topological Invariants of 3-ManifoldsTopological invariants of 3-manifolds are topological properties invariant un-der homeomorphisms. Here we consider a construction such invariants for 3-manifolds with embedded ribbon graphs coloured in modular category V, suchthat braiding, twist, and duality reflect the basic components of ribbon graphs.

Roughly we obtain topological invarians of 3-manifolds as follows. Let Mbe a closed connected oriented 3-manifold. By Theorem 38 M is obtained by

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surgery on S3along a framed link L. Let L have components L1, ..., Lm. Fix anorientation of L, and let col(L) denote the finite set of all possible colourings ofthe components of L in a set I. Fixing a colouring λ, we obtain a ribbon graphΓ(L, λ) without loose ends. Define

Lλ = dim(λ)F (Γ(L, λ)) where dim(λ) =m∏

i=1

dim(λ(Li)) (2.12)

and L =∑

λ∈col(L)

Lλ (2.13)

Remark 39. L does not depend on the choice of orientation of L.

Then defineτ(M) = ∆σ(L)(rankL)−σ(L)−m−1 (2.14)

Theorem 40. Let M be a 3-manifold, then τ(M) is a topological invariant ofM .

2.3.4 Examples of Manifolds1. S1 is the unit circle. S1 bounds the unit disc B2, S1 = ∂B2

2. The 3-sphere S3 is 4-dimensional hypersphere. S3 bounds the closed 4-ballB4, S3 = ∂B4.

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Chapter 3

Elements of 3D-TopologicalQuantum Field Theories

3.1 Cobordisms, Gluing Patterns, Cobordism The-ories

Cobordisms are 3-manifolds in which we distinguish in- and out-boundary.

Definition 41. A 3-cobordism is a 3-manifold M whose boundary ∂M is par-titioned into ∂−M (in-boundary) and ∂+M (out-boundary).

Example 42. Let X be a closed oriented surface, the cylinder X × [0, 1] is a3-cobordism with boundaries ∂−(X× [0, 1]) = −X and ∂+(X× [0, 1]) = X. −Xis X with reversed orientation.

Cobordisms may be glued along a parameterising homeomorphism. Pointsfrom M and N coincide in the resulting cobordism iff they are mapped ontoeach other by the parameterising morphism. We refer to the standard literatureto a more detailed account of gluing.

Definition 43. Let 〈M,∂−M,∂+M〉and 〈N, ∂−N, ∂+N〉 be 3-cobordisms, andlet f : ∂+M → ∂−N be a homeomorphism. The result of gluing M and Nalong f is a 3-cobordism 〈M ∪f N, ∂−M,∂+N〉 in which points m ∈ M andn ∈ N coincide iff n = f(m). The triple 〈M,N, f〉is called a gluing pat-tern. A homeomorphism of gluing patterns 〈M,N, f〉 → 〈M ′, N ′, f ′〉consistsof homeomorphisms h : M → M ′and g : N → N ′ preserving bases such thatf ′g |∂+M= hf : ∂+M → ∂−N ′.

Definition 44. 3-cobordisms and their surfaces form a (monoidal) category,3Cob, such that gluing as composition and cylinders as identity morphisms.Disjoint union,

∐, and empty cobordism and surface, ∅, provide the monoidal

structure. By surface we mean surfaces of 3-cobordisms.

Definition 45. The cobordism theories C, we regard in these notes, are sub-categories of 3Cob closed under cylinders, gluing, disjoint union, and emptycobordisms and surfaces.

Example 46. In particular, 3Cob is a cobordism theory.

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3.2 2-Dimensional Modular FunctorsDefinition 47. A 2-dimensional modular functor for a ground ring K is amonoidal functor from the category of 2-dimensional surfaces with homeomor-phisms to the category of projective K-modules and K-modules isomorphisms.

3.3 3-Dimensional Topological Quantum Field The-ories

TQFTs extend 2-dimensional modular functors by maps assigning to 3-cobordismsoperator invariants.

3.3.1 Axiomatic Definition of 3D-TQFTsWe introduce 3D-TQFTs axiomatically.

Definition 48. A 3-dimensional topological quantum field theory (3D-TQFT)on a 3-cobordism theory C consists of a modular functor T with ground ringK and a map τ assigning to every cobordism M a K-module homomorphismτ(M) : T ∂−M → T ∂+M , the operator invariant of M .

1. (naturality) Let M1and M2 be cobordisms of C and f : M1 → M2bea homeomorphism preserving their bases, then T (f |∂+(M1)) τ(M1) =τ(M2) T (f |∂−(M1)) commutes.

2. (multiplicativity) Let M be the disjoint union of cobordisms M1 and M2

of C then τ(M) = τ(M1)⊗ τ(M2).

3. (functoriality) Let M be obtained by gluing cobordisms M1 and M2 of Calong a homeomorphism f : ∂+(M1) → ∂−(M2) then there is an invertiblek ∈ K, such that

τ(M) = kτ(M2) (T f) τ(M1) (3.1)

4. (normalisation) For any closed oriented surface X,

τ(X × [0, 1], X × 0, X × 1) = idT (X) (3.2)

Remark 49. The weight-parameter k ∈ K spoils τ being a monoidal functor.We may recover a functorial behaviour by renormalising the cobordism theoryin Chapter 5. This situation indicates anomalies.

Example 50. Let T be the modular functor restricted to finite cell spaces,and fix an invertible element k ∈ K. For any finite cell triple 〈M,X, Y 〉, definethe operator invariant τ〈M,X, Y 〉 := kχ(M,X) where χ computes the Eulercharacteristic. Then 〈T , τ〉 is an anomaly-free TQFT.

Example 51. Fix an integer i ≥ 0 and a finite abelian group G whose orderis invertible in K. For any finite cell-space X, set T (X) = K[Hi(X;G)]. ThusT (X) is a module of formal linear combinations of elements of the homologyHi(X;G) with coefficients in K. Note that T (∅) = K[Hi(X;G)] = K[0] =

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K. The action of cell homeomorphisms is induced by their action in Hi. Theoperator invariant τ〈M,X, Y 〉of a finite cell-triple carries any g ∈ Hi(X;G) intothe formal sum of all those h ∈ Hi(X;G) homological to g in M . This yields aTQFT with non-trivial anomalies, that we inspect in Chapter 5.

Exercise 52. Verify the axioms of TQFTs in Examples 50 and 51.

Definition 53. A cobordism theory C is said to be involutive if we have an in-volution on the cobordisms M in C, and the boundaries of cobordisms commutewith negation, −∂M = ∂(−M). An involutive cobordism theory gives rise to adual TQFT such that T ∗(M) = (T (M))∗and τ∗(M) = (τ(M))∗.

Exercise 54. Show that the dual of a TQFT is a TQFT, in particular thatτ∗(M) = τ(−M) where (T (∂M))∗ and T (−∂M) are identified via dτ

∂M .

3.3.2 Homomorphisms and Isomorphisms of 3D-TQFTs,Non-Degenerate TQFTs

Definition 55. A homomorphism〈T1, τ1〉 ⇒ 〈T2, τ2〉 between TQFTs on thesame 3-cobordism theory C and over the same ground ring K is a natural trans-formation between the underlying modular functors. Isomorphisms 〈T1, τ1〉 ∼=〈T2, τ2〉are respectively natural isomorphisms T1

∼= T2.

Example 56. The identity natural transformation T ⇒ T extends to an iso-morphism 〈T , τ〉 ∼= 〈T , τ〉 .

The following theorem implies that operator invariants of isomorphic 3D-TQFTs coincide.

Lemma 57. Let g : 〈T1, τ1〉 → 〈T2, τ2〉be an isomorphism of 3D-TQFTs , theng(∅) = idK and for any 3-cobordism M we have τ2(M) = g(∂M)(τ1(M)) ∈T2(∂M).

Definition 58. A TQFT 〈T , τ〉 is called non-degenerate if for any closed ori-ented surface X, τ〈M,f〉 ∈ T (X)〈M,f〉 generate the module T (X) over theground ring K, where 〈M,f〉runs over all 3-cobordisms M with X-parameterisationf .

Corollary 59. It follows from Lemma 57 that non-degeneracy is invariant un-der isomorphisms.

The following theorem reduces the isomorphism problem of TQFTs to asimpler problem of K-valued functions - of the operator invariants.

Theorem 60. Anomaly-free non-degenerate TQFTs 〈T1, τ1〉 and 〈T2, τ2〉basedon the same ground ring K and the same cobordism theory C are isomorphic ifτ1(M) = τ2(M) for all cobordisms M .

The proof of this theorem uses two constructions of modules for surfacesX, ατ (X) and βτ (X). ατ (X) is the submodule of T (X) generated by theset τ(M,f)(M,f) over all cobordisms M with X-parameterised (f) boundary.Then βτ yields the quotient module

βτ (X) = ατ (X)/(ατ (X) ∩Ann(ατ (−X))) (3.3)

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where Ann(−) computes the set of annihilators with respect to the pairingdτ

X : T (X)⊗K T (−X) → K.If K has zero-characteristic we refine the above theorem, such that only one

of the TQFTs must be non-degenerate.

Theorem 61. Anomaly-free 3D-TQFTs 〈T1, τ1〉 and 〈T2, τ2〉 (one of them, non-degenerate) with ground ring K of zero characteristic (non-zero in the additivepart) are isomorphic if τ1(M) = τ2(M) for all 3-cobordisms M .

Proof.

Exercise 62. Verify that the dual of a non-degenerate TQFT for an involutivecobordism theory C is non-degenerate.

3.4 Quantum InvariantsWe can glue 3-cobordisms M and N along a surface X, if M and N come withparametrising maps, that are homeomorphisms f : X → ∂M and g : X → ∂N .We call 〈M,f〉 and 〈N, g〉 cobordisms with X-parametrized boundary. Then set

τX〈M,f〉 := (T f)−1 (τ(M)) ∈ T (X) (3.4)

Splitting systems allow us to look at 3-cobordisms as composites of finitefamilies of 3-cobordisms. A quantum invariant τ0 is said to have a splittingsystem, if it commutes suitably with the composite structure.

Definition 63. A splitting system for a 2-manifold X is a finite family of triples〈ki,Mi, Ni〉 | i ∈ I consisting of

• an elementki ∈ K of the ground ring,

• a 3-cobordism Mi with X-parameterised boundary and collar, and

• a 3-cobordism Ni with (−X)-parameterised boundary and collar,

such that for all cobordisms M and N of respectively X- and (−X)-parameterisedboundary,

τ0(M ∪X N) =∑i∈I

kiτ0(M ∪X Ni)τ0(Mi ∪X N) (3.5)

A quantum invariant τ0 is said to preserve splitting systems, if the splittingstructure is compatible with the monoidal composition in 3Cob. This is part ofthe definition of quantum invariants:

Definition 64. A quantum invariant is a K-valued homeomorphism invariantτ0of closed 3-manifolds, satisfying

1. (normalisation) τ0(∅) = I

2. (multiplicativity) τ0(M∐

N) = τ0(M)τ0(N)

3. (splitting) τ0preserves splitting systems

The following is immediate from the definition of TQFT isomorphisms above.

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Proposition 65. The quantum invariants of 3Cob classify TQFTs up to iso-morphism.

Lemma 66. Let 〈T , τ〉 be a TQFT based on 3Cob, the function τ on the classof closed 3-cobordisms is a quantum invariant.

Proof. Axioms (normalisation) and (multiplicativity) of Definition 64 are im-mediate from the definition of TQFTs. It remains to verify the (splitting) ax-iom.

Theorem 67. Every quantum invariant τ0 of closed 3-cobordisms extends to anon-degenerate anomaly-free TQFT based on 3Cob.

Proof. Let T assign to a closed oriented surface X the free K-module generatedby 3-cobordisms with X-parameterised boundary quotiented by AnnX , the leftannihilators of the bilinear pairing w(M,N) = τ0(M ∪X N). τ extends τ0suchthat for any 3-cobordism M with X-parameterised boundary, τ(M) takes agenerator N of T (X) to result of gluing N ∪X M .

Theorem 68. There is a bijective correspondence between isomorphism classesof non-degenerate anomaly-free TQFTs (based on 3Cob) and quantum invari-ants of closed 3-cobordisms.

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Chapter 4

A Construction of a3-Dimensional TopologicalQuantum Field Theory

In this section we construct a preliminary 3D-TQFT for cobordisms with embed-ded ribbon graphs. The construction uses that the ribbon graphs are colouredin a modular category V and meet the boundary of the cobordisms in distin-guished arcs. These lie in the components of the boundary and are themselvesV−coloured (compatibly with the incident bands of the embedded ribbon graph)and signed. The disinguished arcs are joined in a V-coloured coupon. The deco-ration gives rise to a form of gluing that is parameterised in decorated surfaces.

4.1 3-Cobordism Theories with DecorationsWe consider decorated cobordisms, where the arcs in the boundary are colouredwith objects of V. The colouring of the embedded ribbon graphs shall be com-patible with the colouring of the arcs in the boundary. This will allow us todescribe the properties of the so decorated cobordisms entirely by the topologicalproperties of the underlying spaces and the colouring of the decorations.

4.1.1 Decorated SurfacesDefinition 69. A marked arc is a simple arc α with an object V of V anda sign v ∈ −1,+1. A closed connected orientable surface is called decoratedif it is oriented and comes with a countable set of distinguished marked arcs.A non-connected surface is decorated, if all of its components are. We calldecorated surfaces also d-surfaces. Homeomorphisms between d-surfaces thatin addition preserve the decoration are called decorated homeomorphisms, ord-homeomorphisms for short.

Definition 70. The inverse−Σ of a connected d-surface Σ is obtained by re-versing the orientation of Σ and inverting the signs of all marked arcs lying onon Σ. Inversion extends to non-connected surfaces in the obvious way.

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4.1.2 Decorated TypesDecorated types abstract decorated surfaces.

Definition 71. We define a decorated type t to consist of a natural number g,the genus, and an m-tuple of marked arcs 〈Wi, vi〉i≤m. We briefly call decoratedtypes d-types.

Definition 72. The d-type of a d-surfaces of genus g and with m marked arcs〈Wi, vi〉i≤m is a tuple 〈g; 〈W1, v1〉, ..., 〈Wm, vm〉〉.

Definition 73. The inverse −t of a decorated type t is obtained by invertingall signs.

Each d-type t gives rise to a canonical d-surfaces Σt as follows.

Definition 74. The canonical surface Σt of a d-type t = 〈g; 〈V1, v1〉, ..., 〈Vm, vm〉〉is constructed as follows. Let Rt be the ribbon graph with one coupon and m+gbands. The firstm bands are untwisted and unlinked. For i ≤ m, the i-th bandis coloured in the respective label Vi. The sign determines the orientation of theband. In addition there are g bands that with the coupon form unknots.

We then fix a closed regular neighbourhood Ut of Rt. Then Ut is a handle-body of genus g. Rt lies in the interior of Ut, except for the m bands that meetthe boundary ∂Ut. The desired d-surface is then Σt := ∂Ut.

The reader should not have problems verifying the soundness of the con-struction above.

Exercise 75. Show that Σt is of d-type t.

The following is easy to see, given that t and −t differ only in the signs. Weleave the proof to the reader.

Exercise 76. Prove that Σ−t = −Σt are homeomorphic.

Definition 77. A connected d-surface Σ of d-type t is said to be parameterisedif it is homeomorphic to Σt. The homeomorphism Σt → Σ is called the param-eterisation of Σ.

4.2 A 3D-TQFT for 3-Cobordism Theories withDecorations

4.2.1 Construction of a Modular Functor from Decora-tions

Each d-type t = 〈g; 〈W1, v1〉, ..., 〈Wm, vm〉〉 defines a projective K-module Ψt asfollows:

Φ(t; i) = W v11 ⊗ ...⊗W vm

m ⊗g⊗

r=1

(Vir ⊗ V ∗ir

) (4.1)

where i = (i1, ..., ig) ∈ Ig, and then set

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Ψt =⊕i∈Ig

Hom(I,Φ(t; i)) (4.2)

Then define T (Σ) to be the tensor product of Ψt for types t of the componentsof Σ, thus for connected Σ of typet we get T (Σ) = Ψt. The action T on d-homeomorphisms f is defined by the parameterisations. For connected Σ andΣ′, we thus obtain T (f) to be the identity morphism. Otherwise T (f) is thetensor product of identity morphisms by restriction to connected components.The proof of the following is immediate from the above construction.

Lemma 78. T is a modular functor.

Below we need an endomorphism η(Σ) : T (Σ) → T (Σ) for each closedoriented surface Σ. η is defined on the summands Hom(I,Φ(t; i)) by multiplica-tion with (rank(V))1−g

∏gn=1 dim(ig), and on non-connected surfaces such that

η(Σ1

∐Σ2) = η(Σ1)⊗ η(Σ2) and η(∅) = idK .

4.2.2 Operator Invariants of Decorated 3-CobordismsWe define an operator invariant for decorated 3-cobordisms M roughly as fol-lows. First we glue standard handlebodies to the boundary of M , which yieldsa closed oriented 3-manifolds M with embedded ribbon graph Ω. The colouringof the extension of Ω over Ω is not unique. Fixing a colouring we can apply thetopological invariant of ribbon graphs developed in Section ?? to (M, Ω) . Thisyields an element of K. This assignment is polylinear in the coupons on theboundary of M , and thus yields a K-homomorphism:

T (∂−M)⊗K (T (∂+M))∗ → K (4.3)

The action of τ(M) is then defined as the composition of the adjoint trans-pose T (∂−M) → T (∂+M) with the endomorphism η(∂+M) : T (∂+M) →T (∂+M) constructed above.

4.3 SoundnessIn the remainder of this section we outline cornerstones of the proof that theconstruction yields a non-degenerate TQFT.

Theorem 79. 〈T , τ〉 is a non-degenerate 3D-TQFT.

4.3.1 OutlineIn Lemma... we have shown T to be a modular functor, so that it remains toverify the following axioms of Definition 48.

• (naturality) follows from τ being a topological invariant of 〈M, Ω〉.

• (multiplicativity) follows from the multiplicativity of τ with respect toclosed 3-manifolds and ultimately from the multiplicativity of the functorF : RibV → V.

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• We prove (functoriality) and (normalisation) below. The proofs require arepresentation of 3-cobordisms by ribbon graphs.

Lemma 80. (normalisation) τ(Σ× [0, 1]) = idT (Σ) for any d-surface Σ.

Lemma 81. (functoriality) Let a decorated 3-cobordism M be obtained by glue-ing decorated 3-cobordisms M1and M2along a d-homeomorphism f : ∂+M1 →∂−M2, then τ(M) = kτ(M2)T fτ(M2).

To show non-degeneracy, we compute τ on the decorated handlebody H(Ut, Rt, i, x)for x ∈ Hom(I,Φ(t; i)), as a cobordism between the empty surface and thecanonical surface ∂H(Ut, Rt, i, x) = Σtof the d-type t. We see that τ(H(Ut, Rt, i, x)) ∈T (Σt) = Ψt.

Lemma 82. For any d-type t of genus g and any x ∈ Hom(I,Φ(t; i)) withi ∈ Ig, we have τ(H(Ut, Rt, i, x)) = x.

4.3.2 Representation of 3-Cobordisms by Ribbon GraphsTo show (normalisation) and (functoriality) one needs to represent cobordismsM with embedded ribbon graphs and by special ribbon graphs Ω. Then oneperforms surgery along Ω in S3and obtains a cobordism of the same d-typesand genus. The latter can be shown by first attaching coupons to the free endsof the embedded graph and then using Theorem 38 . The preserved propertiessuffice for the presentation of 3-cobordisms by ribbon graphs.

One then uses that τ and F coincide on M and Ω, respectively. The prop-erties (normalisation) and (functoriality) then follow from the functoriality ofF .

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Chapter 5

Anomalies

5.1 Anomaly 2-CocyclesThe (functoriality) axioms of Definition 48 tells us that each pair of cobordismsM1 and M2, and homeomorphism f : ∂+(M1) → ∂−(M2) give rise to gluinganomalies. We study cases where anomalies yield so-called 2-cocycles of glu-ing patterns in abelian groups. In these cases we may replace the underlyingcobordism theory by one determined by the cocycle.

Definition 83. Let G be an abelian group. A G-valued 2-cocycle g assigns toeach gluing pattern an element of G, such that

1. (naturality) g identifies homeomorphic gluing patterns.

2. (multiplicativity) Let P be the disjoint union of gluing patterns P1 andP2, then g(P ) = g(P1)g(P2).

3. (normalisation) g takes gluing with a cylinder to 1, the unit of G.

4. (compatibility) with disjoint union:

g(M∐

N,W, e∐

f) = g(M, (W,WM , ∂+W∐

−WN ), e)g(N, (M∪eW,WN ,−∂−M∐

∂+W ), f)(5.1)

where W = WM

∐WN

Exchanging the roles of M and N in (compatibility) one finds

g(M, 〈W,WM , ∂+W∐

−WN ), e〉g〈N, 〈M∪eW,WN ,−∂−M∐

∂+W 〉, f〉 = g〈N, 〈W,WN , ∂+W∐

−WM 〉, f〉g〈M, (N∪fW,WN ,−∂−N∐

∂+W 〉, e)

The latter necessitates the notion of symmetric gluing patterns, which identifiesa gluing patter 〈M,N, f〉 with its opposite

〈〈N,−∂+N,−∂−N〉, 〈M,−∂+M,−∂−M〉,−f−1 : −∂−N → −∂+M〉 (5.2)

Symmetric 2-cocycles satisfy the following variant (compatibility)’ of the aboveaxiom.

g〈W,M∐

N, e∐

f〉 = g〈〈W,∂−W∐

−Y, X〉,M, e〉g〈〈W∪eM,∂−W∐

−∂+M,Y 〉, N, f〉(5.3)

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5.2 Renormalisation of Cobordisms by means of2-Cocycles

In the following let G be an abelian group and g a G-valued 2-cocycle.

Definition 84. The cobordism M weighted inG is a pair 〈M,k〉 where k ∈ Gis called the multiplicative weight of M .

1. Homeomorphisms between G-weighted spaces are homeomorphisms be-tween the underlying spaces that preserve the weight.

2. The weights are compatible with disjoint union, such that〈M,k〉∐〈N, l〉 =

〈M∐

N, kl〉.

3. The boundaries of weighted cobordisms are the boundaries of the under-lying cobordisms.

Gluing of weighted cobordisms is defined by gluing cylinders to weighted cobor-disms.

Definition 85. Consider a cobordism M whose boundary is a disjoint unionof surfaces X, Y , and Z, and let f : X → −Y be a parameterising homeo-morphism. Then we define the associated gluing pattern P (M,X, Y, Z, f) =〈〈X × [0, 1]〉, ∅, (−X × 0)

∐(X × 1)〉, 〈M,−X

∐−Y, Z〉, id−X

∐f〉. The gluing

pattern evaluates to 〈M ′, kg(P (M,X, Y, Z, f)〉 where M ′ is the result of gluingX to Y along f .

Proposition 86. Weighted cobordisms with weighted gluing form a cobordismtheory.

5.3 Anomaly Cocycles of 3D-TQFTs, Anomaly-free Renormalisation

Let 〈T , τ〉 be a 3D-TQFT with ground ring K. Let G be the abelian group ifinvertible elements of K. An anomaly cocycle of 〈T , τ〉 is a symmetric G-valued2-cocycle g of the underlying cobordism theory.

We shall assume that all cobordisms at hand have collars, or otherwise theweight is stable under gluing cylinders to weighted cobordisms.

Example 87. The trivial cocycle is an anomaly cocycle in any anomaly-freeTQFT.

Let g be an anomaly cocycle for〈T , τ〉, we define a renormalised anomaly-freeTQFT 〈T , τg〉 where

τg〈M,k〉 := k−1τ(M) : T (∂−M) → T (∂+M)). (5.4)

The so obtained TQFT is anomaly free.

Theorem 88. Let 〈T , τ〉be a TQFT and g an anomaly cocycle of 〈T , τ〉thenthe renormalisation 〈T , τg〉 is an anomaly free TQFT.

Example 89. The TQFT of Example 51 has anomaly cocycles.

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Proposition 90. Renormalising TQFTs preserves non-degeneracy.

Anomaly cocycles for TQFTs based on involutive cobordism theories areanomaly cocycles for the dual TQFT.

5.4 Lagrangian Spaces, Lagrangian Relations andMaslov Indices: Preliminary Definitions

5.4.1 Lagrangian Spaces and ContractionsDefinition 91. A symplectic vector space is a finite dimensional real vector-space H with an antisymmetric bilinear form w : H×H → R. The same vectorspace with the opposite form will be denoted −H. Let A ⊆ H be linear subspaceof H, an annihilator of A is an element h ∈ H such that w(h, a) = 0. Denoteby Ann(A) the set of annihilators of A. A lagrangian subspace is the maximallinear subspace A ⊆ H such that A ⊆ Ann(A). The latter condition is calledisotropy. The set of lagrangian subspaces of H is denoted as Λ(H).

Definition 92. Let A ⊆ H be an isotropic subspace, we call λ | A the la-grangrian contraction of a linear subspace λ ⊆ H along A

λ | A = (λ + A) ∩Ann(A)/A ⊆ H | A (5.5)

5.4.2 Lagrangian Relations and Actions ThereofDefinition 93. A lagrangian relation between non-degenerate symplectic vectorspaces H1and H2is a lagangian subspace N of −H1⊕H2. We denote lagrangianrelations like this one as N : H1 ⇒ H2.

Lagrangian relations generalise symplectic relations.

Exercise 94. Show that lagrangian relations are

1. reflexive, such that diag(H) is a lagrangian relation

2. symmetric, such that whenever N : H1 ⇒ H2, then Ns := (h2, h1) | (h1, h2) ∈ Nis a lagrangian relation H2 ⇒ H1

3. and transitive, such that whenever N12 : H1 ⇒ H2and N23 : H2 ⇒ H3

are lagrangian relations, then so is N12 ⊕N23 : H1 ⇒ H3

5.4.3 Maslov IndicesSubsequently let λ1, λ2, λ3be isotropic subspaces of a symplectic vector space〈H,w〉. We define a bilinear form 〈_,_〉 on (λ1 ⊕ λ2) ∩ λ3such that for alla, b ∈ (λ1 ⊕ λ2) ∩ λ3with a = a1 + a2.

〈a, b〉 = w(a2, b) (5.6)

Note that a2 is determined by a up to the addition by a1. 〈_,_〉 is well-definedbecause the possible a1annihilate b.

Exercise 95. Verify that 〈_,_〉 is symmetric.

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Note that 〈_,_〉may be degenerate as its annihilator contains (λ1 ∩ λ2) +(λ2 ∩ λ3).

Definition 96. Define the Maslov index µ(λ1, λ2, λ3) to be the signature of〈_,_〉, that is the number of positive and negative entries.

Proposition 97. The Maslov index is antisymmetric, in particular we have forany isotropic spaces

µ(λ1, λ2, λ3) = −µ(λ2, λ1, λ3) = −µ(λ1, λ3, λ2) (5.7)

5.5 A Digression into Homologies of SurfacesLet Σ be an oriented surface, we define the real vector space H1(Σ; R). Theexistence of this homology is a corollary of the Poincare Duality Theorem (see[2] for details).

5.5.1 The Lagrangian FunctorWe define a covariant functor from the category of decorated surfaces and 3-cobordisms to the category of non-degenerate symplectic vector spaces withfixed Lagrangian subspaces and Lagrangian relations. This functor assigns toeach d-surfaces Σ a lagrangian subspace of its homologies, H1(Σ; R), and to eachdecorated 3-cobordism a lagrangian relation between homologies of its bases.

Let Σ be a connected parameterised d-surface, so that there is a parameteris-ing d-homeomorphism Σ → Σt. We define a distinguished Lagrangian subspaceλ(Σ) as the kernel of the inclusion map H1(Σt; R) → H1(Ut; R). Then for Σset λ(Σ) = f∗[λ(Σt)]. The distinguished Lagrangian space for disconnected sur-faces Σ is defined as the Lagrangian subspace of H1(Σ, R) generated from thedistinguished Lagrangian subspaces of its components.

For any decorated 3-cobordism M we have

H1(∂M ; R) = −H1(∂−M ; R)⊕H1(∂+M ; R) (5.8)

Taking the kernel of the inclusion morphism H1(∂M ; R) → H1(M ; R) yields aLagrangian relation N(M) : H1(∂−M ; R) ⇒ H1(∂+M ; R).Remark 98. The definition of N(M) is independent from the decoration.

Exercise 99. Verify covariance, in particular for M being the result of gluingcobordisms M1 and M2 along a homeomorphism p : ∂+M1 → ∂−M2, thenN(M) = N(M2)p∗N(M1).

5.6 Computing AnomaliesTheorem 100. Let M be obtained by gluing decorated 3-cobordisms M1and M2

along a d-homeomorphism p : ∂+M1 → ∂−M2 commuting with the parameteri-sations, then

τ(M) = (rank(V)∆−1)µ(p∗(N1)∗(λ−(M1)),λ−(M2),N∗2 (λ+(M2))τ(M2)(T p)τ(M1)

(5.9)and for both i ∈ 0, 1

Ni = N(Mi) : H1(∂−Mi; R) ⇒ H1(∂+(Mi); R) (5.10)

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Remark 101. The gluing anomalies are completely determined by topology ofthe underling manifold under gluing and by the parameterisations of their bases.

5.7 The Anomaly 2-CocycleWe define actions of a modular groupoid which determine the anomaly 2-cocycle.

Definition 102. The mapping class group Modg is the group of isotopy classesof (degree-1) homeomorphisms on a closed oriented surface of genus g. Thisdefinition extends to decorated types t, yielding Modt.

A weak homeomorphisms as degree-1 homeomorphisms preserving the dec-oration up to a permutation of arcs. Then the definition of the mapping classgroup extends to the groupoid of weak homeomorphisms. A weak homeomor-phism g : Σ → Σ′ defines a cobordism M(g) as the cylinder Σ′ × [0, 1] with thetop base decorated by idΣ′ and the bottom base is decorated and parameterisedby g. We obtain an action τ(M(g)) for g of the mapping class groupoid withthe following value.

Theorem 103. Let Σ,Σ′,Σ′′ be closed oriented surfaces and idΣ : Σ → Σ,g : Σ → Σ′, and h : Σ′ → Σ′′ be weak homeomorphisms, then we obtain byTheorem 100

τ(M(idΣ)) = idT (Σ)τ(M(gh)) = (rank(V)∆−1)µ(h∗(λ(Σ)),λ(Σ′),g−1∗ (λ(Σ′′)))τ(M(g))τ(M(h))

(5.11)where we use that µ((gh)∗(λt), g∗(λt), λt) = µ(h∗(λt), λt, g

−1(λt)).

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