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PHYSICS LETTERS B

Physical states in quantum Liouville theory Shinobu Hosono Department of Physics, Universityof Tokyo, Bunkyo-ku, Tokyo 113, Japan Received 9 April 1992

General procedures to write down all physical states in two-dimensional (2D) gravity, coupled to a conformal matter theory with 0~

1. Introduction These three years there has been great progress in the theory of 2D gravity through the discovery of the double scaling limit in the matrix model [ 1 ]. Within this approach of the matrix model, the spectrum of the theory and their correlation functions are well understood via the string equation [ 2 - 4 ] and, surprisingly enough, the topological field theory over the moduli space of the punctured Riemann surfaces [ 57 ]. Despite these remarkable successes in the matrix model, we must say that our present status toward the complete understanding of the 2D gravity based on the continuum approach (the Liouville theory) is still unsatisfactory. In fact, there are several works [ 810] attempting to construct the quantum theory of the Liouville field following the pioneering works by Knizhnik, Polyakov and Zamolodchikov ( K P Z ) [ 11 ], David, Distler and Kawai ( D D K ) [ 12 ] as well as those done at a quite early stage [ 13 ]. In this letter we shall be concerned with the BRS cohomology of the Liouville theory which determines the spectrum of the physical states in the theory. The general properties o f the cohomology have been revealed in an excellent work by Lian and Zuckerman [14]. N o w it is known that the cohomology ~ Work supported in part by Soryuushi Shogakukai. Address after 1 April 1992: Department of Mathematics, Toyama University, Toyama 930, Japan.

explains quite well the spectrum of the matrix model. However, one of the characteristics of the BRS cohomology which we note in contrast to conventional gauge theories [ 15,16 ] is that there appear infinitely many physical states with non-vanishing ghost numbers. To study the role of these ghosts appearing in the physical states is our main topic in this letter.

2. BRS cohomology Here we summarize notations and the results by Lian and Zuckerman, and subsequent authors [ 17 ]. Let us consider the Liouville theory coupled to a conformal matter theory with the central charge and the conformal weight equal to (p_q)2 cpq= 1 - 6 - - , Pq

p

(pr-qs)2-(p-q) ZJr, S ~

(2.1)

2

4pq

1 <~rq s ,

(2.2)

respectively. Under the limit of the vanishing cosmological constant, the stress-tensor for the Liouville theory can be expressed as a free conformal theory: K ( z ) = - ½ : 0 ~ c ( z ) 0q~L(z): + i Q c 0 2 0 c ( z ) , where we chose a convention

0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.

(2.3)

(0c(z)0,(w))= 35

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PHYSICS LETTERS B

--log(z--W). The mode expansion of the stress-tensor K(z),

K.=½ ~

.",+,n"~(L),,~(L). 1 ) Q L a n(rl_}. tL)_m

,

(2.4)

m~Z

satisfies the Virasoro algebra with the central charge CL= 1 -- 12Q~. We denote the Fock space built on the vacuum state JPL) with the ground state m o m e n t u m PC by ~L (Pc)" According to the conventional analysis of the gauge theory [ 16 ], we introduce the spin (2, - 1 ) be-ghost. Then the BRS operator is defined to be

QB = Z c_.(K.+L~) - ~1 E

for what values of Pc the non-trivial cohomology elements exist. To present their results, let us denote the conformal dimensions of the singular vectors of the Virasoro Verma modulus M (Cpu, Jr.~) by

at =

( 2pqt + pr + qs)2_ ( p _ q ) 2 4pq

b,=

(2pqt_pr+qs)2_ (p_q)2 , te~, 4pq

( n - m ) : c _ , , c .... b,+,,:,

(2.5)

n,m~Z

(2.10)

and define a set E~,(p, q ) - t l ' - a , , 1 - b , lt~Z}. For each value of JeE~.~(p, q) we write

d(J)=-12t+l[,

n6; v

2 July 1992

= [2tl,

irA= l - a , , ifJ=l-b,.

(2.11)

Finally let 0 (Pk) =- sign (i (Pc-- QL) )-

which acts on the space C~bs -- "~L(Pc) ®Vir(Cpq, Jr.,) ®A;c,

(2.6)

where L, and Vir(cpq, Ar,s) represent the matter Virasoro operator and its irreducible module, respectively, and A;~ means the ghost Fock space with the ghost number • . Under the anomaly cancellation condition C L Ji-dpq

=26,

(2.7)

the operator QB becomes nilpotent and defines the absolute BRS cohomology over the space Ca*u~. From rather technical reasons, let us consider a subcomplex

Theorem 1 (Lian-Zuckerman [14]). (a) Hr*¢L('~L(PL) ® Vir(6,q, J,...~) ® A~,¢.) # 0 iff

&(pe)~L~(p, q). (b) For the value of PL s.t. JL(PL)sE~,~(P, q), dim Hrnel(,~L(PL) @ Vir(cpq, &.,) ® Abc) = (~n,q(pL )d( d(pL ) ) "

(C) H*b~(-~L(PL) ® Vir(cpu, Jr,s) ® Abe) = H~*I(,~-L(PL) ® Vir(cpq, &,,) ® A~,.) ® coH(.-l)(,~e(Pe)®Vir(cpq, Jr.,)®Ab,.). The proof of the above theorem can be found in a recent paper by Bouwknegt et al. [ 17 ].

• --/ * C~, = tveC~u~ [b0v=0, {QB, bo}v=0},

3. G e n e r a l t h e o r y

on which the BRS operator acts effectively by

Q-B= Z c _ . ( K . + L ~ ) n#O

1

E

(n-m)c

~c .... b~+,,.

(2.8)

n,m¢O,n+m~sO

This subcomplex is known as the relative BRS complex with coefficients in .Ne(Pc)®Vir(Cpq, J,.~). We shall denote its cohomology as H*,I (gTe(Pc)®Vir(Cpq, Ar.~)®Ab~). In this way, the relative BRS cohomology is defined for each value of the Liouville momentum PL which explains the conformal dimension of the state IPL ) : d e ( P c ) -----½Pc(Pc - - 2 Q L )

.

(2.9)

The successful results by Lian and Zuckerman tell us 36

Here we will arrange a general procedure to write down the cohomology elements with non-trivial ghost numbers. Let us first replace the irreducible module Vir (Cpq,J,.,) by the zeroth cohomology of Felder's free field resolution [ 18 ], Vir(ct) q, J,.~)=H°( ~ .TM(p,(r, s)), d ) , \

/,

(3.1)

where P2, = C~r.~_2,p and P2,+ ~(r, s) = c~. . . . z,p with ~ k T ½-( 1 - k ) c ~ _ + ½ ( 1 - l ) c ~ + . . (c~+= 2 ~ p , c~ = - x/2p/q). As will be discussed later there is another choice of the resolution in which the momenta p, (r, s) are replaced by 2QM--Pt(r, s). The Virasoro operator L,, will be represented as

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PHYSICS LETTERS B

Ln= l E "t'/(M),~(M) •- , + . . . . . . . : - ( n + 1 )QMa,~M) tn~2~

(3.2)

2 July 1992

appropriate choice of Felder's resolution, and thus the spectral sequence degenerates at the Ez-term:

with Cpq= 1 - 12Q 2 and QM = 1 (O~+ +O¢_ ). Because Felder's coboundary operator ~ and our BRS operator (~B commute, we can consider the double complex ( ~ C % D) defining

Ci'J=- ['~L(PL)®.~M(pj(r, S) ) ®A},c]re,,

(3.3)

and the coboundary operator D-= ~+ ( - 1 )J(~u, D:

~.

i+j=n

C ;,j--,

y,

i+j=n+ I

C;4.

1

where p,=p,(r, s) or p,(r, s). Precisely the appropriate choice of the momenta turns out to be

p,=p,(r, s), (3.4)

Since each Fock space has natural grading (conformal dimension of states), it is easily deduced that dim Z ; + j = , C ; . J < m for a fixed PL. TO proceed further, we quote one of the fundamental results in ref. [ 17 ] setting the notation

P+(n)- ~

(3.7)

n+l (PM-T-ipL)-- ~ (QMTiQL).

(3.5)

Theorem 2 (see theorem 3.3 in ref. [17] ). If either the condition P+ (k) ~ 0 for all k~Z, k ~ 0 or the condition P_ (k) ~ 0 for all keT, k ¢ 0 are fulfilled, then

i f r / ( p L ) > 0 , ,JL(PL)~V.rs(P, q) + or r/(pL ) < 0, AL(PL) ~ F','s(P, q) - ,

=j0,(r,s),

ifrl(PL)>O, 3L(PL)eFzr~(p, q)_ or q(PL) <0, 3L (PL)EF,r,(p, q) +, (3.8)

with F-r,(P, q)+ - {1 - a , , 1 - b , _ l I teY>_,o} a n d Ers(P, q) _ -= { 1 - a , , 1 - b _ , _ l[ teY

' , rel('~*oL(PL)@Vlr(Cpq, Ars)®Ab,,)

H n ('~L (PL ) ® '~M (PM ) ® Abc, (~B) =CIPL)®[PM)®Cl 10)gh,

if n = 0 ,

= 0,

otherwise,

~nb.(nrel,(~B(n o •L(PL)® E, J~M(Pt)®mbc)) • (3.9)

where [0 ) gh represents the SL (2, C) invariant vacuum for the ghost. The cocycle IPL) ® [PM ) ® C~[0 ) ~h survives in HrOI(._~L(PL) ® ._~M(PM)NAbc, 013) if additionally the "on-shell condition" AL(PL)+ AM(PM) -- 1 = 0 is satisfied.

This isomorphism is the starting point of our general construction to write the cohomology elements with non-vanishing ghost numbers. We know the form of the cocycles in HaHrel,OB " o as IPL)® IP,)®c~ [0)gh. So the problem is to construct the corresponding cocy" Hr0). Let us consider, in general, a D-cocycle ~,= E;+j= n ~',,j(~';,je C";): n

Now let us consider the spectral sequence, associated with the double complex (3.3) with a choice of PL such that d L(PL) ~ F'r,s(P, q), whose El-term is

E]'J=HreLOs(/TL(PL)®'~M(Pj(r,s))®A~c) •

(3.6)

For the values of the momenta p,(r, s) used in Felder's resolution or their reversed momenta/~_,(r, s) =-2 QM--P, (r, s) in another choice of the resolution, in which the order o f the sequences of the Fock modules are reversed, we can easily verify that the conditions in theorem 2 are satisfied. Therefore we know that theorem 2 can be applied to the El-term with an

~

0

v/= qG+;,-; + ~¢-+;- I,-;+ 1 +-.. + ~,,o +... + ~'o,,, +..- + ~-k,~- -

(3.10)

Then due to the fact that both ~-cohomology and QBcohomology vanish except for n = 0, we can show that q/is D-cohomologous to both of the following states:

37

Volume 285, number 1,2

responds the following (non-trivial)

~,,.o-~,..o+ Z (-s)%+,-, k

cocycle in

n ~ O. Hrd,oBH~.

i=1

+ Z (-s)J~',,-,.,,

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PHYSICS LETTERS B

(3.11)

C~o~(pL,p,,)=(--S)n¢~(pL, p,),

if n > 0 ,

j=l

=(-S)

n+l i=l k--n

+ ~ (-S)J~/_/.,,+/, j=l

(3.12)

where we define S - ( - 1 )J0B6-i and S--_ ( - 1 )J ×

~0~'. Case II (when n < 0). In this case, we must be a little careful in applying the above reasoning. What we can say is that the ~ is D-cohomologous to ~, ~[=~[O,n'~-~[_l.n+

"[- .., -[- ~ben+ 1. _ 1 "~- ~n,0 ,

(3.13)

(~(PL,Pn),

ifn~<0,

(3.16)

with S = ( - 1 )/QB6- ~, S ' = ( - 1 ) J O g ~6 and with the choice 3( - S ' )-"g~(PL, P,) = 0 for n~<0. When we inspect the way of the operations ( - S ) n or ( - S ' ) - " on C:~(pL, p,,), the following can be deduced.

Proposition 2. The cocycles (:0~ (PL, Pn) in HrekOsHan o are written in the form of (o~(PL, P , , ) = P ( { L .... }, {C.... }) IPL)

with definitions @ Ip0)@Cl

I 0)gh,

if n > 0 ,

k

= P ( { K .... },{L .... }, {b_,,}) [pL)

(U..o=~U..o+ /=l Z -Sy~u,,_,.,,

(~) I p 0 ) @ C 1 [ O ) g h ,

n+l

(Uo,,-~Uo.,+ Z

i=1

-S)~g6 ......

(3.14)

It can be shown easily that the map q:~q),,o

(~U~o,,) constitute the isomorphism HreLD H~L0~Ha (Hrel,D~-H~H~.O_B). Case I is the standard isomorphism we see, for example, in ref. [ 19]. In contrast to this, case II belongs to one of the characteristic properties in the semi-infinite cohomology theory [ 20 ]. Now we are ready to state our construction. Let us consider the cocycle (:a(Pe, P,,)~ I P L ) ® IP,,)® C[ } 0 ) g h in HaHr~LOB. , o . If n > 0 , then this cocycle can be a cocycle in H~LD at the same time. Therefore we can apply case I above to obtain the corresponding cocycle C:O~(PL, P,,) in " H~I,o.H~. , , o On the other hand, if n ~<0 then we can construct the corresponding Dcocycle C'o(PL, P,) as

(:'D(pL, p,,)= [ I + ( - S ' ) + ( - S ' ) z + . , . + ( - S ' )-"1 (:~(PL, P,,),

(3.15)

with S ' -= ( - 1 ) : 0 ~ ~~ and 3 ( - S ' ) - " ~ (PL, P , ) = O. Summarizing, we can state the following.

where P({ }, ) means some polynomial of the arguments { } andpo=Po(r, s) or 2QM--po(r, s) depending on the value PL. The above two propositions are among our main results in this letter. In the next section, we will apply our general construction to the case of c = 0, the pure Liouville theory.

4. Physical states in the Liouville theory

Here we restrict our attention to the (pure) Liouville theory which couples to the c = 0 matter. To begin with, let us recall some results on Felder's free field resolution and the structure of the Fock space ffL(PL) as a Virasoro module. In our case of c = 0 , whose detailed data are (p, q) = (2, 3), (r, s) = ( 1, 1 ) and QM = 1/2x/3 , Felder's resolution has a quite simple form; its complicated coboundary operator can be expressed via single contour integration: 6 = f dz :exp(ix/3 ¢~M): ,

Proposition 1. To each (non-trivial) cocycle ~:6(PL, p,,)=[pL)@lp,,)@cl[O)gh i n H ~,, H 0r~l.O., , there cor38

ifn~<0,

(4.1)

and the conformal weights of the singular vectors are

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PHYSICS LETTERS B

given by the so-called Euler pentagon numbers,

a,=(2t+l)(3t+l),

b,=t(6t+l),

t~Z.

2 July 1992

~O~(PL,P,) =P( {C-m} ) IPL ) (4.2)

The Liouville Fock space belongs to type III+ ( + ) or III+ ( - ), according to the classification in ref. [ 21 ], depending on q(PL)<0 and q(PL)> 0, respectively. Thus the Seiberg condition [22 ], which is equivalent to q(PL) < 0, determines the type of the Fock module. The embedding structures of the Virasoro module III+ ( _+ ) are drawn in fig. 1, where the degrees of the (co-)singular vectors are given by l - a , or l - b ,

(t~2~). According to theorem 1, the physical states exist for each of the values of the Liouville momentum PL satisfying the on-shell condition AL(PL)= 1--at or 1 -- b,. Solving this equation, we determine the values of PL(t) as

@ l P o ) ® q IO)gh,

for q(pL) >0,

=P({K-m}, {b_m })IPL ) ® ]Po)®Cl [0)gh,

for q(pL) <0, (4.4)

with Po = 0 or 2QM depending on the value ofpL. The case of q(pz)> 0 "anti-Seiberg states". In this case all physical states are described by the polynomials P({c_ m} ). Therefore the physical states live in the subspace IPL) ® IPo) ® Ac where A c E{........ k} CO-hi ...C-nk (el [ 0 ) g h ) . Since the BRS operator turns out to act in this subspace effectively as an operator

d=-~

~

(n-m)c_,c_,,b,+m,

(4.5)

n,m>O

pL(2t+ 1)~ = - - i

5-T- 112t+51 2X/~ ,

for the choice of at, pL(2t) z = --i

5-T- 112t+ 11 2X/~ ,

for the choice of b_,,

(4.3)

with the - sign for r/(pL)< 0 and the + sign for q (PL) > 0. The cocycles 60. (PL, P,) in proposition 2 further simplify their form as

the cohomology is naturally embedded into a simpler cohomology H*(A,, d). One should note that the cohomology H*(A,., d) is nothing but the standard Lie algebra cohomology [ 23 ] H* (A~+, C) with trivial coefficient C. Here 50+ means a subalgebra, spanned by the generators with positive conformal dimensions of the Virasoro algebra Y = ~+ ~ % ~ e _ . The embedding is in fact an isomorphism.

Proposition 3. All physical states with Y/(PL) > 0 (antiSeiberg states), are described by H*(A~+, C), i.e., H*(~L(PL(t) + ) ®C®Ah,, (~B)

III+ (-)

III+ (+)

/1 TxT

T",, lxl

TxT \1

lxl T/

Fig. 1. Structure of the Fock module for the Liouville part.

t~2 ~

-~ H*(ALP+, C ) .

(4.6)

The problem to calculate the cohomology H*(A~+, C) was first posed by I.M. Gel'fand at the 1970 mathematical congress, at the early stage of the representation theory of the Virasoro algebra, and subsequently it was solved by Goncharova [24] and Gel'fand, Feigin and Fuchs [25]. To describe their results, let us define the coherent state representation for a state ] ~ ) =Ea(nt, ..., n~)c_,,...c_,s(q [0)gh) with a fixed ghost number s by

~(z~ .... , Zs) =- ~0'( ~o ~ 7~b,, ) ...( ~~ z~"~b,s ), ~) , t/ ns=0

(4.7)

39

Volume 285, n u m b e r 1,2

P H Y S I C S LETTERS B

where (0l means (cllO)gh)*=(O]b-I. It isan easy task to recover the form of I 5v) from the totally antisymmetric function ~(zL ..... z~). Within this representation of states Gel'land et al. obtained all the harmonic cocycles in H*(AS+, C) as Zl...Zq//3 and z~...zqH 3 where H q - H , ~ , < j ~ q ( z , - z j ) and q~2~>o. Using their results, we can express all physical states by ~ = [Pc(S) + >®z, ...z,H 3, =IPL(S)+>®z~...z2-~H[,,

where X3 and 24 = 19Z4-- 9 K tZ3 are determined by 3+½K_2K_I+TsK~I

z4=K

4 +T~K_3K_I 13

3

,

+ 3 K 2 - _,

2K ~--~+ 3 K 4 _ , .

(4.10)

The states Z3 [Pc( - 2) ) and Z4 IPL( -- 2) ) would constitute the first two singular vectors which exist in the Virasoro Verma module M ( c = 2 6 , A = - 4 ) . However, in our Fock module these expressions vanish identically due to the existence of the co-singular vectors as indicated in fig. 1 (III+ ( + ) ).

(4.8)

In table 1, we list some of the observables in the form of operators using the state and the operator correspondence. The case of q(pL) < 0 (Seiberg states). In this case, the physical states can exist as local operators and have a direct relation to the operators in the matrix model. So this case is most important for establishing the relation. In table 1, we present the first few observables obtained through the general formula (3.16). It seems that there exists a combinatorical representation for these cocycles like the case of the singular vectors (see ref. [26] ). Here it may be instructive to see how these physical states satisfy the cocycle condition; let us present the results of the action 0B on the state, as an example, I ( ~ ) = [b-2b-l(K-2 + LK2 6 , ) _ ~b-3b 1K l "~- ~ b - 4 b - l b - 3 b - 2 ] ] P L ( - 2 ) ) ® l ® c l ] 0)g~,,

5. Discussions Here we discuss some physical implications of our results. Let us consider the Hartle-Hawking-like construction of the states through the zero mode approximation of the Liouville theory. In ref. [22], Seiberg showed that the anti-Seiberg states cannot be a local operator with respect to the quantum metric g~,-exp(0c)~/, and describe the macroscopic loops, which constitute a normalizable wavefunction. In contrast, the Seiberg states can be a local operator and describe the microscopic loops, which constitute a non-normalizable wavefunction. Our BRS analysis (proposition 2) tells us that the c-ghosts and the bghosts can be attached only to the macroscopic loops and microscopic loops, respectively. This result might arise from some fundamental difference in nature be-

O~Bl (q, ) = (~3b_,24- 2 b _ 2 z 3 ) [ P c ( - 2 ) ) ® l ® c , [0)gh ,

1

z3=K

+~K

s>_.O, S<0.

2 July 1992

(4.9)

Table l List of the observables. (rt and (~ correspond to the anti-Seiberg states and Seiberg states, respectively• The subscripts indicate the KdV flow which these operators are assumed to generate under the correspondence to the matrix model. Anti-Seiberg states ~71= c e x p ( \ / ~ 3 0e) ~

Seiberg states

2 -cexp --7=~OL/"~ -- (.V/3

5

(s =exp(~/.-g~ OLX~c ~)2c

\\,' 3 } ~'v = e x p ( 2 , j 3 0c) c 03C ~

8

(7is = e x p ( 3,,//3 0C ) (C 04C i?C-- -~C03C 06C)

(v=:(bc+~.Y:l)

~v"3~L): i ) - g d b . ) f I +~02b-Obbc]exp( -.~/3 0L):

exp( -

" [ 2 (( it = . . [b(.h/_2 + g.)f

1"

• • " ')f ' 1 + 4136 51. 4 ) ('13='[b('kf~4q-t~4')f-3 ~ 'fir2-2 q_ 6~4 .)~2 2j{ 2 i -1-5~T5'~-1 17 Io6 - 2.W-I+~.)~_t • 3 '3 ) + ~ 3 2b ( . W . _ 2 + ~iv# [,-~ ~ ) --~0b(,)f-3+~ffTJ(( 136

"

3

"2

-~Obbc(.h/ 2+~.x_l)-7~fOZbbc.)f_ t - ~ O 3 b b c + ~O-bOtc] ~ "' ~

40

_~)3b,;¢~_1+~04 b e x p ( - g\."4.~30L):

Volume 285, number 1,2

PHYSICS LETTERS B

t w e e n the m a c r o s c o p i c and m i c r o s c o p i c loops and could be e x p l a i n e d by i n c l u d i n g the effects o f the h i g h e r m o d e part o f the L i o u v i l l e field. T h i s insight f r o m the topological field t h e o r y app r o a c h and the m a t r i x m o d e l tells us that all the correlation f u n c t i o n s o f the o b s e r v a b l e s are expressed t h r o u g h the integral o v e r the m o d u l i space and satisfy the W a r d i d e n t i t y s u m m a r i z e d in the f o r m o f the V i r a s o r o constraint. In the L i o u v i l l e t h e o r y approach, the ghost fields o f o u r o b s e r v a b l e s m u s t explain these facts, especially they should r e p r o d u c e the V i r a s o r o c o n s t r a i n t u n d e r an interplay with the Liouville field. H o w e v e r , we m u s t leave these i m p o r t a n t p r o b l e m s to future investigations.

Acknowledgement T h e a u t h o r w o u l d like to t h a n k A. T s u c h i y a and T. Eguchi for v a l u a b l e discussions and suggestions on the m a t h e m a t i c a l p r o b l e m s . H e also express his gratitude to T. Eguchi, H. Kawai, K. F u j i k a w a , M. N i n o m i y a , K. O g a w a a n d M. F u k u m a for v a l u a b l e discussions a n d e n c o u r a g e m e n t s . T h i s w o r k is supp o r t e d in part by Soryuusi S h o u g a k u k a i and was supp o r t e d at early stage by the R I M S 9 1 project " I n f i n i t e A n a l y s i s " ( J u n e - A u g u s t 1991 ) held at R e s e a r c h Institute for M a t h e m a t i c a l Sciences, K y o t o U n i v e r s i t y .

References [ 1 ] E. Brdzin and V. Kazakov, Phys. Lett. B 236 (1990) 144; M. Douglas and S. Shenker, Nucl. Phys. B 335 (1990) 635; D.J, Gross and A. Migdal, Phys. Rev. Lett. 64 (1990) 127. [2] M. Douglas, Phys. Lett. B 238 (1990) 176. [31M. Fukuma, H, Kawai and R, Nakayama, Intern. J. Mod. Phys. A 6 (1991) 1385. [4] R. Dijkgraaf, E. Verlinde and H. Verlinde, Nucl. Phys. B 348 (1991) 436. [5] E. Witten, Nucl. Phys. B 340 (1990) 144; R. Dijkgraafand E. Witten, Nucl. Phys. B 342 (1990) 486.

2 July 1992

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