modular invariance of minimal models from self-dual lattices

Volume 244, number 3,4 PHYSICS LETTERS B 26 July 1990

Modular invariance of minimal models from self-dual lattices

Pa t r i ck Robe r t s 1 Institute of Theoretical Physics, CTH/G U, S-412 96 G6teborg, Sweden

Received 2 April 1990

The A-D-E sequence of modular invariant partition functions for Virasoro minimal models is constructed in the Coulomb gas representation. We show that these partition functions can be written as sums and differences of even self-dual lattice partition functions.

I. Introduction

The study of conformal field theories has pro- ceeded at a considerable pace over the past few years. Not only are conformal field theories of interest in two dimensional statistical systems, but they also describe the classical vacuum of string theories. These considerations have given motivation to a program to classify all rational conformal field theories. Since constraints of modular invariance severely limit the set of such theories, the classification o f modular invariant partition functions has received much atten- tion recently [1 -7 ] . The invariants of the SU(2 ) Kac -Moody algebras, as well as their related coset models, have been shown by Cappelli et al [ 1 ] to fall into an A - D - E classification scheme, but other extended algebras have resisted attempts at such a classification. What is unfortunate is that there is no general method for constructing the modular invariant partition functions, the exceptionals being the noto- rious case.

In the study of string theories, the covariant lattice approach [ 8 ] was developed to ensure modular invariance from the outset. By using a bosonic construction of strings it was found that if the set of possible winding numbers and momenta of the bosonic fields formed an even self-dual lattice, then the model would be automatically modular invariant [ 9 ]. One of the shortcomings of this approach has been that it was limited to level-1 K a c - M o o d y and related alge-

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bras. With the development of the Coulomb gas representation [ 10-19 ] this situation has changed dra- matically. In this construction one introduces a background charge into a free bosonic field theory that affects the central charge. With such a free field representation of a conformal field theory one would expect to find similar constraints on the winding mo- mentum as in covariant lattice theories. It has been proposed [ 13 ] that all rational conformal field theories may be represented by Coulomb gas models, but in the following we restrict ourselves to the simple case of the Virasoro minimal models with c < 1.

2. Coulomb gas representation and characters of minimal models

The free field representation of minimal models has been presented in detail by many authors [ 10-12 ], so we shall only sketch the main points leaning heav- ily on the work of Caselle and Narain [ 13 ]. One be- gins by introducing a free bosonic theory with a stress tensor of the form

T(z ) = - ½ [0z~(z)] 2+ioto 0 2 ~ ( z ) , ( 1 )

where we have chosen the normalization such that

< (o(z)~o(w) > = - I n ( z - w) . (2)

The central charge is given by

c = 1 - 1 2 a 2 o (3)

and we recover the unitary series if we set

0370-2693/90/$ 03.50 © 1990 - Elsevier Science Publishers B.V. ( North-Holland ) 429


1 a o = x/(--2rn'm+ 1 ) " (4)

One represents highest weight states by vertex oper- ators of the form

Vp(z)= : e xp [ ip~ (z ) ] : . (5)

The conformal dimension of Vp(z) is given by

A B = ½ (f12- 2aofl) . (6)

In order to describe the possible values of fl it is convenient to define an abelian Lie algebra ql(x/2m(m+ 1 ) ) with "root" lattice

FR(ql (x /2rn(m+ 1 ) ) )

={kplp=x/Zm(m+ 1), ke•}. (7)

The dual of this lattice is the "weight" lattice defined by

Fw( °Y(x/Zm (m + 1 ) ) )

= {koto a o = 1 k~Z} (8) x / 2 m ( m + 1 ) ' '

which contains the root lattice as a sublattice. One may speak of conjugacy classes as cosets of the weights with respect to the root lattice:

( l) = {otol+ pkll, keY_}. (9)

When we say that fl belongs to the one dimensional lattice

fleFl = ~ ( x / 2 m ( m + l ) ) , (10)

we mean that fl belongs to the union of some set of conjugacy classes of which the root lattice (7) must be one.

Now we introduce screening charges with vanishing conformal dimension:

Q±= f V,~+_ (z) dz, A,~_+=I. (11)

It has been shown [ 10-12 ] that one may use these screening charges to construct null states of the theory if the operator

VT(z)

= I ... f V,~_(z,)...V,+_ (z~)Vp_~,~_ (z) f i dzi i = l

(12)

is well defined. This is found to be the case for V~ + (z) and V~- (z) if

f l (p ,q )=ao[mp-(m+l)q] , p, qeZ. (13)

We may identify the null states with points on the lattice Fi by introducing the operation [ 13 ]

g: fl(p, q)~ fl(p,-q) . (14)

Given a highest weight state V~(z), we know that we have a null state V ~ ( z ) at level N=pqabove Vp(z).

Using this knowledge about null states we may write down the character [ 10-13 ] corresponding to the highest weight state Vp(z):

zp(r)

_ 1 ~ (q(fl_ao+kp)2/2 q(gfl_ao+kp)2/2)

(15)

wherep=x/2m(m+ 1 ) is a root of ~ / ( ~ 1 ) ), and

q=exp(2~rir), r / (z)=q ~/24 f i ( I - q " ) . (16) n = l

We see that the sum is over the ~ ' (x /2m(m+ 1)) conjugacy classes generated by the weights fl and gfl. With the addition of more bosons to the theory, this character formula generalizes to extended Virasoro algebras.

3. Partition functions

Given the form of the character we may construct partition functions

Z(f , r ) = ~ Nij,~i(f)Zj(z) (17) t,J

for which we will seek modular invariant solutions. Furthermore, for a sensible theory we require that the vacuum is unique, i.e. Noo= 1, and all Nij are non- negative integers.

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In the spirit of the covariant lattice approach we introduce a lorentzian lattice

F~.~ = ~ ( v / 2 m ( m + l ) ) × q l ( v / 2 m ( m + l ) ) (18)

with signature ( - , + ), and consisting of the vectors

#-~-~ ( f fL; j~R) ~ I~1,1 (19)

belonging to some subset of all pairs of q / ( x / 2 m ( m + 1)) conjugacy classes which will be specified later. Here we apply the convention of ref. [13] where flL.R=fl+Oto, SO that the conformal dimension of a highest weight state is given by

- ~ (PL,~-- O~O). ( 2 0 ) AflL. R __ 1 2 2

The partition function may be written as

Z(f , z) = ~ Z Zt~(f)Z~ ( z ) , (21) (#)eVl,l

or

1 1 (qp[/2q#~/2 B~r~.t

"~- q (gl~)2/2q (gflR)2/2 - - I~ (gflL)2/2q fl2/2

_ :l#[/2q (g~)~/2), (22)

where the sum in (21) is over conjugacy classes of F , : , and in (22) we sum over all points on the lattice. One may be worried that the lattice sum contains unphysical states. In terms of the operation g, these states are characterized by the property that they are mapped onto themselves. Thus their characters vanish and they decouple from the theory.

We assume that g is an automorphism of F~.I. Thus we may simplify (22) using

(q#[/2q#~/2)= ~ ((1(g~.)2/2q(g~)2/2), (23) f lcFLl BeFl , l

and since g2 = 1,

1 1 Z( ~, z) - 2 fl~l #~..,~. , ( qP[12qp2 /2--q(gPL)2/2qp2 /2) "

(24)

Defining the lattice A~.t containing the vectors

(g(flL);flR) e A, . , , (25)

we may write the partition function as

1 l ( ~r, -- 2 ) ~,2/2q#2/2 Z ( f , z ) = ~ # 2 . (26) ,I B~AI,I /

It is well known [9 ] that i fF~: is even, i.e. if for all f l eF t : , f12e27/, then the lattice sum over F~,~ is invariant under z--,z+ 1. Further, ifF~.~ is self-dual, that is if the volume of the unit cell is one and all scalar products among lattice vectors are integers, then the lattice sum is invariant under z--, - 1/z. Thus, the lattice partition function of an even self-dual lattice is modular invariant.

Since the operation o fg on a lattice point changes the conformal dimension of the corresponding vertex operator by an integer, one may show that ifFl.l is even and self-dual then At,l is also even self-dual. This implies that if FI.I is even self-dual, then Z(f , z) is modular invariant.

The simplest two dimensional lorentzian lattice one may construct which is even self-dual is the one where the conjugacy classes of the two components are equivalent. This corresponds to the diagonal, or A- invariant in the classification of ref. [ 1 ], and in the following we shall call this lattice 1" A. It is worth not- ing here that the vacuum is represented by the two points

f l= (ao ;ao) , ( - O t o ; - a o ) , (27)

i.e. belongs to the conjugacy classes ( ( 1 ); ( 1 ) ) and ( ( - 1 ); ( - 1 ) ) of FA. Due to the factor of ½ in eq. (26) we have ensured that the vacuum state appears once in the partition function.

4. Shifted lattices

Given an even self-dual lattice, one may construct other even self-dual lattices by using the "shift method" presented in detail in ref. [8]. To shift a self-dual lattice F~.~ one considers a vector ~ which is not an element of F~,I, but for some smallest non-zero integer n,

n6~ F1 : . ( 2 8 )

We then define the sublattice

Fo = {flo IPo ~ F~.~ and flo-6eZ}. (29)

The new even self-dual lattice is the union of lattices

n- - I

F~= U ( F o + k 6 ) . (30) k=O

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We now shift the diagonal lattice FA by vectors belonging to the following conjugacy classes:

6 ~ ( ( - a ( m + l ) ); (a (m+ l ) ) ) , (31)

6" ~ ( ( - a m ) ; (am)) , (32)

where a~Z. Clearly these do not belong to FA, but

m6aeFA, (33)

( m + 1 )6~, ~ FA, (34)

if m and a are coprimes, otherwise the order of the shift is reduced by their common factor.

It is convenient here to introduce the standard pa- rametrization of highest weight states for minimal models:

fl(P, q; P, q)

( (rap- ( m + 1 )q); ( m p - (m+ 1 )q) )

= FA. (35)

In this notation

[P m + l 1 ( ~ m

(36)

6" . f l (p ,q;p ,q)=a - q modT1.

(37)

Thus, a shift by 6~ leaves the p-parameter unaffected, and the 6~,-shift ignores q. In the following we con- centrate on 6~ and keep in mind that the same analy- sis follows for 6a where the roles of p and q are reversed.

We now work out an interesting example with m = 12. This case has three modular invariants [ 1 ]: the diagonal (A~, A~2) which we have already constructed, the complementary (DT, A12), and the exceptional ( E6, A~2). Here we introduce the notation

1 1 m m - - I

Za -- 20q e-"-~o q--~-- m [ (p' q; p ' q) - (p' - q; p' q) ]" (38)

By shifting FA we can construct a new modular invariant partition function

20-~1 1 ( ~ r El~1"a2/2q'8~/2 Zao= p~ -- . (39) ao p~ AO~ /

I f a = 1 then the unshifted sector contains the conjugacy classes ( (mp); (rap)) and ( (mp+ ( m + 1 )m); (mp+ ( m + l ) m ) ) so that the 81 shifted lattice Fa, may be described by

Fa~ ={ ( (mp+ (m+ l )q); ( m p - ( m + l )q) )l

p = 0 .... , m; q = - m , . . . , m - 1 } . (40)

This means that Fa~ = At.t so that Za~ = - Z A which alone yields negative coefficients N o in the partition function when expressed as in eq. (17). Further- more, the sum ZA + Za~ vanishes.

I f a = 2 then the lattice Fa~ contains the conjugacy classes

Fa2={ ( (mp+ (m+ l )2r); ( m p - ( m + l )2r) )

( (mp+ ( m + 1 ) ( 6 - 2 r ) ) ;

(rap+ ( m + 1 ) ( 6 + 2 r ) ) ) 1

p = 0 , 1, ..., 12; r = - 6 , - 5 , ..., 5}. (41)

Thus,

1 1 12 5 Z62 ---~ 2 0N p=~o r=~-6 [ (p' -- 2r; p, 2r) - (p, 2r; p, 2r)

+ (p, 6 - 2 r ; p , 6 + 2 r ) - (p , -6+2r;p , 6 + 2 r ) ] . (42)

This contains no vacuum state, so as it stands it does not constitute a partition function for a conformal field theory. On the other hand, the sum ZA + Za2 does contain the vacuum state, and the first two terms in the brackets of (42) exactly cancel the q=even terms in ZA. The last two terms may be rewritten such that

z.+za~ i 1 1 2 ( II

- 2 ~/r~O \q=~odd [ (p ' q;P' q ) - ( P ' - q ; P ' q)]

1o ) + ~ [(p,q;p, 1 2 - q ) - ( p , - q ; p , 12--q)] .

q=even (43)

In terms of non-vanishing characters,

,o ) ZA..I_Za2~_½ ~ [ ~p,q [ 2 .~_ E ~p,q~p, 12--q •

p=O \q=odd q=even (44)

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which we recognise as the complementary invariant

(D7, AI2) . I f we let a = 3, then

F~ 3 = { ( ( r a p + ( m + l)3s) ; (rap- ( m + 1)3s))

( (mp- ( m + 1 ) ( 4 - 3s) );

(mp- ( m + 1 ) ( 4 + 3 s ) ) )

( (mp- ( m + 1 ) ( 8 - 3s) );

(mp- ( m + 1 ) ( 8 + 3 s ) ) ) 1

p = 0 , 1, ..., 12; s = - 4 , - 3 , ..., 3}. (45)

As above, we find here that Z~ 3 contains no vacuum state, and the only combination which can be expressed in terms of characters as in eq. (17) with non- negative N o coefficients and a non-degenerate vacuum

12 ZA+Z~+Z63=½ ~ (IXp,,+Z.,712

p=O

"~ [)~p,4dl-)~p,812+ Ix,,5 +Z,,t~ 12), (46)

which is the exceptional invariant (E6, A~2). All other values of a in ~a yield partition functions

equivalent to one of the above so we have exhausted all possible invariants for m--- 12 using shifts of this form.

This method may be applied to all other values of m and we summarize the results in relation to the classification of ref. [ 1 ] as follows:

Complementary invariants:

m=2j, j~Z +, ZD=ZA"FZ~2.

Exceptional invariants:

m=2.2 .3 , ZE6=ZA +Z62+Za3,

m=2.3 .3 , ZE7=ZA +Za2+Za3,

m=2.3.5, ZEs=ZAdI-Zt$2-~Z63"~Z~5. (47)

By replacing the values of m with m + 1, one finds the same invariants where 6" replaces ~,.

rice partition functions. As in the covariant lattice approach to string theory, this places the study of modular invariances into the study of self-dual lattices. A major advantages of this approach is that ex- ceptionalinvariants may be studied on the same foot- ing as the diagonal and complementary invariants.

It would be interesting to know the relation be- tween these shift vectors and the action of the Weyl group [20] on the lattice FI,~= [ ~ ( x / 2 m ( m + 1 ) ) ]2. One would then expect to find some group theoretic reason for this particular set of modular invariants, and perhaps shed some light on the mysterious A - D - E classification.

One may apply these methods to any other Coulomb gas representation of conformal field theories with more bosons as long as the screening charges can be expressed as

Qa= f :exp[iata.q~(z)]: dz, A,,.=l . (48)

In the important case of the Kac-Moody algebras, the holomorphic winding momenta live on a lorentzian lattice [14-16]. This can be inconvenient for the construction of lattice partition functions because one must limit the sum to ensure that all states have non- negative conformal dimension. A possible remedy to this difficulty would be to euclideanizethe lattice [ 8 ] and study its modular properties as in the even lattice formulation of string theories. One may hope that the study of these and other lattices will lead to a general classification of the modular invariant partition functions for Kac-Moody algebras.

Acknowledgement

I wish to thank Per Salomonson and Bengt Nilsson for discussions and many helpful suggestions on the manuscript.

References

5. Conclusion

We have shown how to write modular invariant partition functions of the Virasoro minimal models in terms of sums and differences of even self-dual lat-

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modular invariance of minimal models from self-dual lattices

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