Toward classification of conformal theories

Volume 206, number 3

PHYSICS LETTERS B

26 May 1988

TOWARD CLASSIFICATION OF CONFORMAL THEORIES C u m r u n VAFA

Lyman Laboratoo, of Physics, Harvard University, Cambridge, MA 02138, USA Received 29 February 1988

By studying the representations of the mapping class groups which arise in 2D conformal theories we derive some restrictions on the value of the conformal dimension hi of operators and the central charge c of the Virasoro algebra. As a simple application we show that when there are a finite number of operators in the eonforrnal algebra, the h, and c are all rational.

Conformal theories in two d i m e n s i o n s have been under intensive investigation recently, both from the point of view of critical p h e n o m e n a in statistical mechanics, and for their role in constructing classical vacua for string theories. A systematic study of conformal theories was initiated in ref. [ 1 ], where the importance of the operator algebra of conformal theories was emphasized. Subsequently there has been a vast body of literature on the subject. An i m p o r t a n t open question is how to classify conformal theories ~ In this note we show that the operator algebra contains a lot of i n f o r m a t i o n about the possible values of the d i m e n s i o n of operators hi (the critical exponents) and c (the central charge of the Virasoro algebra). As an application we will show that when there are a finite n u m b e r of operators in the algebra, all the values of h, and c are rational (this has been shown using a different reasoning in ref. [4] ). Our m a i n tool is to study the representation of the m a p p i n g class group on the bundles over appropriate moduli space. That conformal theories give rise to bundles on ordinary moduli space of genus g surfaces, J(~, was pointed out and investigated in ref. [ 5 ]. In that work the importance of the m a p p i n g class groups in the classification of conformal theories was pointed out. Subsequently it was noted that more information can be obtained from conformal theories ~ Except for the case c< 1, where it was shown in ref. [2] that unitarity essentially determines all the possible models (the question of modular invariance in such models was settled in ref. [3]). 0 3 7 0 - 2 6 9 3 / 8 8 / $ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )

by considering the bundle structure on punctured surfaces J / ( g , n) [6,7], and the bundles on punctured surfaces together with coordinates at the punctures :3 (g, n ) [ 7 ]. The latter space is the one we will be concerned with in this paper. So far as topological questions are concerned ,~ (g, n) behaves the same way as R i e m a n n surfaces of genus g with n parametrized b o u n d a r y components, or equivalently Riem a n n surfaces with n punctures together with a nonvanishing vector at each puncture. These spaces are obtained by dividing out an appropriate contractible Teichmtiller space by m a p p i n g class group F(g, n) [8], which acts without fixed points. We therefore have

~, ( :~(g, n) ) =F(g, n ) .

(1)

Conformal theories give rise to flat projective vector bundles over ~ (g, n), which are described by (projective) representations of the f u n d a m e n t a l group of the space which is F(g, n). By studying such representations of F(g, n) we will derive restrictions on the hi and c (for an example of representations of related groups associated to conformal theories on the sphere see ref. [ 9 ] ). Throughout this paper we shall take as the definition of a conformal theory the one proposed in ref. [ 7 ]. Since the Hilbert space J{of a conformal theory admits the action of left a n d right Virasoro algebras, we can decompose the Hilbert space into a direct sum of tensor products of representations of left a n d right Virasoro algebras. So we write 421

Volume 206, number 3

~=~)0r®~r,

PHYSICS LETTERS B

(2)

i

where 0i and ~r furnish representations o f L~ and respectively. We do not require 0r ( a n d ~r ) to be irreducible representations o f the Ln ( a n d the Ln ). As is well known, in a conformal theory we must require that Lo - I 5 o have only integral eigenvalues on 0i®~r, which in turn implies that Lo and Lo have the same eigenvalue hi m o d u l o integers, when acting on 0r and 0r, respectively. Define the phases ar = exp (2ztihr). The subsector corresponding to the v a c u u m state will be labelled by 0o = 1, for which S o = 1. In this p a p e r we shall put restrictions on ~r. The m a i n i n f o r m a t i o n in a conformal theory is the assignment to each p o i n t on ~ ( g , n) o f a ray 10) in the n-fold tensor product o f the Hilbert space Y{®~. One can d e c o m p o s e the states 10) that we get from punctured surfaces, into a sum o f states

L,,

10)=~

/

I~u,)l~,),

(3)

by d e c o m p o s i n g it to its left and right components, in such a way that I~ur) ~0z, ®...®0r,,- C o n s i d e r going a r o u n d non-trivial loops in ~(g, n). In a well-defined conformal theory the state 10) should come back to itself ( u p to n o r m a l i z a t i o n ) u n d e r such an operation. Let us see what happens to I ~0'r) under such an operation. Since parallel t r a n s p o r t on ~ (g, n) is accomplished by the Virasoro algebra operating on the state and since the 0~ form representations o f Virasoro algebra, u n d e r parallel t r a n s p o r t I ~ur) will still be an element of0r, ®...®0r,,, but it does not have to come back to itself: I f there are other indices l' for which [~ur )~0r, ®...®0r,, appears in ( 3 ) , then as we go a r o u n d non-trivial loops o f ~ ( g , n), 1~9) could mix with them. Let Nr,...~,, denote the n u m b e r o f / ' s such that Igtr)e0a ®..-®0~,,. In this way we see that we end up with a projectively flat vector bundle over ~ ( g , n) o f d i m e n s i o n Nr, r,,. By the p e r m u t a t i o n property o f the state I 0 ) , N is totally symmetric in its indices ( a n d it clearly d e p e n d s on the surface only through g). I f it weren't for the fact that the bundles we get are projective, we would have o b t a i n e d in this way a representation o f F(g, n) o f d i m e n s i o n Nr, .to telling us how we mix the I q/r) as we go a r o u n d nontrivial loops (recall eq. ( 1 ) ). The projectivity o f the vector bundle prevents us from m a k i n g such a gen422

26 May 1988

eral statement. However, for the cases o f the sphere and the torus we can actually obtain representations o f the m a p p i n g class groups ~2 (see ref. [ 5 ] ). By sewing operation one can get all the rays starting from the ray corresponding to ~ ( 0 , 3 ) , i.e., a sphere with three punctures. So all the i n f o r m a t i o n o f a conformal theory is s u m m a r i z e d in the rays corresponding to ~ (0,3). Using the rays corresponding to 2 ( 0 , 3) we obtain the fusion rules [1 ] o f a conformal theory which is defined by ~3 Oi X Oj = NijkOk •

(4)

In ref. [ 1 ] this is m e a n t m o r e as a symbolic representation and the i n f o r m a t i o n is in whether the Nrjk vanish or not. However recently, in a beautiful work [ 10 ], Verlinde has taken Nrjk to c o r r e s p o n d to the d i m e n sions o f bundles (as described a b o v e ) ~4 and has d e m o n s t r a t e d in m a n y examples that the m o d u l a r t r a n s f o r m a t i o n matrix at one loop, corresponding to z ~ - 1 / r diagonalizes Nak (viewed as a matrix whose rows a n d columns are labelled by j and k). Furtherm o r e he has conjectured this to be true in general a n d has used this conjecture to derive some restrictions on the values o f the hi and c entirely from the fusion rules. In our work the actual values o f Nrik will also play a f u n d a m e n t a l role, though our m e t h o d o f derivation, as well as our results, differ from ref. [ 10 ]. In this p a p e r by focusing on the sphere with four punctures ( a n d coordinates) and the representation of the m a p p i n g class group F ( 0 , 4) it gives rise to, we derive restrictions on the hr. We then use the representation o f the m o d u l a r group for the torus with no punctures F ( 1 , 0 ) = SL ( 2 7 ) , to put restrictions on c. F(g, n) is generated by D e h n twists about curves on the surface [ 11 ], just as is the case for the ordinary m a p p i n g class group, except that now we have m o r e curves. F o r example, performing Dehn twists about curves that are contractible but include punctures in their interiors are non-trivial. This is true, even if the curve encircles only one puncture, in which case the Dehn twisting will be equivalent to changing ,2 At higher loops we may supplement the conformal system by some other system such that ctota~=0and for such a combined system we obtain representations ofF(g, n). ~3 I f ~ is not self-conjugatethen its conjugate appears on the righthand side ofeq. (4). ,4 We shall assume in this paper that N,jk is finite which we believe may be true for an arbitrary conformal theory.

Volume 206, number 3

PHYSICS LETTERS B

the coordinate at the puncture from z~exp(2niO)z as 0 goes from 0 to 1 (at the Hilbert space of leftmovers this is represented by exp(2niOLo)). This would have been a trivial twist if we had not chosen coordinates at the punctures ~5. Now consider the sphere with four punctures (with coordinates) labelled from 1 to 4 (see fig. 1 ). Consider the Dehn twists about curves shown in fig. 1. If we denote Dehn twist about the curve C, by 2"~,it is possible to show that the relation 2"12"22"32"4 = 2-122-132-23

(5)

holds. In fact this relation is the one which was used in ref. [ 8 ] to show that HI ( ~ (g, n) ) = 0 (for g>~ 3 ) and this fact was used in ref. [ 7 ] to show that once the local conditions required for a well-defined conformal theory are satisfied there are no global conformal anomalies. It is curious that this very same relation is the source o f our restrictions on the hi. We fix the punctures at 1, ..., 4, to correspond to i, j, k, l Hilbert spaces. As discussed above we obtain a ~5 In particular this shows why the mapping class group we are considering could be thought of as a refinement of the braid group, for which this operation would have been trivial.

.4

26 May 1988

vector bundle of dimension N,jkt. Therefore each Dehn twist in eq. (5) is represented by an Nijkt X Nijkl matrix. It is easy to see how the left-hand side of (5) is represented: Each Dehn twist is equivalent to a rotation by 2n, which is represented by exp(2niLo) acting on the Hilbert space o f the puncture. This implies that rl, 2"2, 2"3,2"4 are represented, respectively, by ai, aj, o~k, oq times the NoktXNijkt identity matrix. The right-hand side o f ( 5 ) is not as easily obtainable; this is because it is not in general possible to find a basis for the bundle for which 2"12, 2"23and 2"13are simultaneously diagonal. For example, if we represent the bundles as obtained by sewing in the 1-2 channel (see fig. 2 ), and choose a basis for the bundle one for each possible intermediate state configuration, then in this basis 2"12is a diagonal matrix. This is so, because by the assumption of what constitutes a well-defined conformal theory, we can construct the sphere with four punctures as sewing two spheres, each with three punctures. Then 2"~2is represented in this basis by exp (2niLo) acting on the Hilbert space which is sewn. Therefore if the intermediate state is in the Or sub-

1-2 Channel

~

t

k

2-3 Channel I

i=1,j=2, k=3,1=4 Fig. I. A Riemann sphere with four punctures at 1, ..., 4 (with a choice of coordinate at each puncture). The Dehn twists about C~ satisfy the non-trivial identity given in eq. ( 5 ).

Fig. 2. If we choose a basis for the bundle suggested by the 1-2 channel intermediate states, r~2 acts diagonally. In the 2-3 basis z23 is diagonal, but r~2 may be non-diagonal.

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space, this gives ce~ as a diagonal element o f z~2. So we see that z~2 is represented by a diagonal matrix in which the a r appears as the diagonal element Nii~rkZ times (note that since Z~N,j~Nm=Nijkl, we obtain an Nijkt × N~jktm a t r i x ) . Similarly, if we had chosen a basis for the bundle suggested by the 2 - 3 channel intermediate states (see fig. 2), then "c23 would have been diagonal, and in this basis q 2 - - , U z ~ 2 U -~, where U represents the matrix for the change o f basis. N o w we take the d e t e r m i n a n t o f both sides o f ( 5 ) to obtain an equality which does not d e p e n d on the U's. Define Nljkl, r = N i j r N k l r ~- NjkrNilr

~l- NtkrNjl r ,

(6)

Then taking the d e t e r m i n a n t o f ( 5 ) we obtain the following relation: ( O L l a j a k O ( i )N#~l = I ~ 0(. Ndl'l'" . r

(7)

This relation is the source o f our restrictions on hi. Before discussing some examples let us make a few remarks. Note that eq. ( 7 ) is symmetric in its four indices, and so we obtain one relation, for each four choices o f fields ( n o t necessarily distinct) in the algebra. Also note that we never had to assume that we have a finite n u m b e r of operators Oi in our theory: we only have to assume that Nij~.z is finite. F o r the rest o f this paper, we shall suppose we have only a finite n u m b e r o f subsectors ~z where i = 0 , 1. . . . , N. Such theories are called [ 12 ] rational conformal field theories R C F T (see also refs. [ 13,4] ). In such cases we seem to get an overconstrained system. There are ( N + 1 ) ( N + 2 ) ( N + 3) ( N + 4 ) / 4 ! relations, but only N unknowns namely the c~ (note that ao = 1 ). So one would expect no solutions (apart from the trivial one ozi= 1 ), unless the coefficients Nijk are very special, as is the case for conformal theories. To get a feeling for the relations ( 7 ) we shall now consider some examples. Consider a conformal theory which satisfies the algebra o f the Ising model: q/X q/= 1, aX

q/=a

.

aXa= 1+~/,

(8)

Let oq, a2 be the corresponding eigenvalues o f exp (2niLo) for ~ua n d a, respectively. There are three non-trivial relations following from ( 7 ) , correspond424

26 May 1988

ing to the cases ~/~u~,, ~,~aa and aaaa. These give the relations ~6 WW~,~-~a2 = 1, ~ a a - - - , (oq % ) 2 = a 2 , cwaG-~ ( ~ 4 ) 2 = ~ .

F r o m these relations we find that a ,~ = 1 ,

~=a,.

So if we look for solutions with a ~ # 1, we have a~ = - 1 and % = e x p [ 2 7 r i ( 2 p + 1 ) / 1 6 ] for some integer p, in perfect agreement with what one expects for the Ising model, where p = 0 (the case p = 1 corresponds to the SU ( 2 ) K a c - M o o d y algebra at level 2). We see that it is quite easy to derive restrictions for the h, which are indeed saturated. As another exa m p l e consider SU ( 2 ) level 3. The o p e r a t o r algebra of SU ( 2 ) level 3 has three non-trivial blocks. Eqs. ( 7 ) give a~ = 1, c¢25= 1 and a l = a2a3. These restrictions are to be c o m p a r e d with the values h 3 = 3 / 4 , h 2 = 2 / 5 , h~ = 3/20. We leave writing down m o r e examples to the reader, as it is quite easy a n d fun to check restrictions on conformal dimensions of known models. As an application we now wish to prove quite generally that in a R C F T the hi are all rational. Consider eqs. ( 7 ) setting i = j = k = l, and consider N possibilities, as i takes values from 1 to N. We get N equations with N unknowns ~7. So in principle there could be a solution. The ith equation reads ( a i )4N ..... Ntii,,, U

O L r N ...... =

1.

(9)

r=/-i

Define the matrix Mir = ~ri ( 4 N , ii -- N , ii,i ) + ( 1 - 5~i ) ( -- N , ii,~ ) • Then eq. ( 9 ) can be rewritten as M h = O m o d 1,

(10)

where h represents the column vector with hi as its coefficients (as i runs from 1 to N). We will now show r46The relation a~ = 1 was also derived in ref. [ 14] using braid group relations. ~7 Here we are assuming that ~i is self-conjugate,i.e., ~,X 0i= I + .... The argument we are presenting could be easily modified for the case when some of the fields are not self-conjugate, by taking i=j, k= l= {and noting that a,= at.

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that M is invertible. This is not to difficult as one notices that M is a matrix with positive ( n o n - z e r o ) diagonal entries and negative off-diagonal elements, such that the diagonal element is bigger in absolute value than the sum o f all the off-diagonal elements in the same row (note that Niiii ~- "~Zr=oNiiii,r ~ "~ and Nuu >/1 ). Then it is easy to see that such a m a t r i x cannot have a non-trivial eigenvector with eigenvalue 0, so det M S 0. Set k = det M . Since M is an integral matrix, it follows that k ( M ) is also an integral matrix. Multiplying eq. (10) by k ( M ) - 1 we see that kh = 0 mod 1 .

This implies that all h~ are multiples o f 1 / k as was to be shown. The k that we found here m a y not be the smallest such integer, as we have used only a small subclass o f constraints coming from (7). F o r example in the Ising case, we get k = 32, whereas we know that if we had utilized all the constraints coming from (7) we could show that the h~ are all multiples o f 1/ 16. Finally we come to putting restrictions on c. But that is quite easy: At one loop the generators for the m a p p i n g class group which is SL(2, 72) are given by S:

z~-

T:

z--+z+ l ,

N o Lc( N + I ) ~ 1 ~ r= 1

26 May 1988 N

°t6

or

OL2(N+I)=I~

OZ) 2

(13)

r= 1

where c~c = exp ( 2 n i c / 4 ) . Using the rationality o f the h,, eq. ( 1 3 ) implies, in particular, that c is rational for a R C F T . Just as an example, if we consider the case o f conformal theories with the algebra o f the Ising model, the above restriction, together with what we found before, implies that c=p+½ mod 4.

It seems somewhat surprising that by some very simple considerations on the representations o f m a p ping class groups arising in conformal theories we have o b t a i n e d relatively strong restrictions on the h~ and c. It is clear that there is m o r e i n f o r m a t i o n in eqs. ( 5 ) a n d ( 11 ) than their d e t e r m i n a n t s which we have used. It would be o f interest to try to get m o r e inform a t i o n from them. Also, we have considered here only some o f the simplest cases o f m a p p i n g class groups. It would be o f interest to extend this analysis to surfaces with m o r e punctures, and also to surfaces with higher genus.

1/z,

and they satisfy (ST)3=I .

(11)

Acting on the bundles corresponding to ~ (1, 0), S and T will be represented by ( N + 1 ) × ( N + 1 ) matrices. D e p e n d i n g on whether or not the fields are self conjugate or not, $ 2 = 1 or $ 4 = 1. Since we do not know the explicit form o f S, we can eliminate it from ( 11 ) by taking d e t e r m i n a n t s o f both sides a n d raising it to the power o f 2 or 4, d e p e n d i n g on which case we have at hand. We see that det(T)6=l

or

det(T)~2=l.

Since N

det T = 1-[ exp( - 2 z r i c / 2 4 ) t ~ , , r=O

we see that

(12)

It is a pleasure to thank S. G i d d i n g s a n d G. M o o r e for discussions. We also wish to thank E. Verlinde for discussions on his work. This research was s u p p o r t e d in part by N S F contract PHY-82-15249, and by a fellowship from the H a r v a r d Society o f Fellows.

References [ 1] A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Nucl. Phys. B 241 (1984) 333. [2] D. Friedan, Z. Qiu and S.H. Shenker, Phys. Rev. Lett. 52 (1984) 1575. [3] J.L Cardy, Nucl. Phys. B 270 [FS16] (1986) 186; A. Capelli, C. Itzykson and J.-B. Zuber, Nucl. Phys. B 280 [FS 18] (1987) 445; Commun. Math. Phys. 113 (1987) 1; D. Gepner, Nucl. Phys. B 287 (1987) 111. [4] G. Anderson and G. Moore, IAS preprint IASSNS-HEP87/69. [5] D. Friedan and S. Shenker, Phys. Lett. B 175 (1986) 287; Nucl. Phys. B 281 (1987) 509. [ 6 ] E. Martinec and S. Shenker, unpublished. [7] C. Vafa, Phys. Lett. B 199 (1987) 195. 425

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[8]J. Harer, Inv. Math. 72 (1983) 221. [ 9 ] A. Tsuchiya and Y. Kanie, Lett. Math. Phys. 13 ( 1987 ) 303. [ 10] E. Verlinde, to be published. [ 11 ] M. Dehn, Acta Math. 69 (1938) 135. [ 12 ] D. Friedan and S. Shenker, unpublished.

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[ 13] J. Harvey, G. Moore, and C. Vafa, Nucl. Phys. B, to be published; D. Kastor, E. Martinec and Z. Qiu, E. Fermi Institute preprint EFI-87-58. [ 14] G. Moore and N. Seiberg, unpublished.