On the possibility of fractional superstrings

Volume 253, number 3,4

PHYSICS LETTERS B

10 January 1991

On the possibility of fractional superstrings Philip C. Argyres, A n d r 6 L e C l a i r a n d S.-H. H e n r y T y e Newman Laboratory of Nuclear Studies, Cornell University, Ithaca. NY 14853-5001. USA Received 1 September 1990

We explore the possibility of string theories based on a world sheet symmetry that corresponds to an extension of the Virasoro algebra by fractional spin currents. For the special case of world sheet symmetries related to SU (2), the explicit operator algebra satisfied by the symmetry currents is derived. Critical central charges and slope intercepts are proposed.

A string theory is primarily characterized by a world sheet symmetry that is gauged locally [ 1 ]. In this way the original bosonic string in 26 space-time dimensions is characterized by the Virasoro symmetry. To incorporate space-time fermions one introduces world sheet supersymmetry which gives rise to the superstring in 10 space-time dimensions. The superstring is characterized by the super Virasoro algebra, a supersymmetric extension of the Virasoro algebra. A priori, there is no reason to expect that the super Virasoro algebra is the only possible extension of the Virasoro algebra that can be gauged locally to form a string theory. The W-algebras are examples of nonlinearly extended Virasoro algebras which contain additional integer spin currents [ 2 ]. They were studied with the idea of building new string theories in refs. [ 3,4 ]. It was found that one could introduce additional ghosts for the extended currents and construct a nilpotent BRST operator. The models are characterized by a critical Virasoro central extension (c ~r") that is in general greater than 26 (e.g. for the W-algebra associated with SU ( 3 ), c ~" = 100 ). In addition, a classical local world sheet symmetry has been discovered which has the W-algebra as its constraint algebra [ 5 ]. In this letter we explore the possibility of building string theories out o f extensions o f the Virasoro algebra with fractional spin currents. The super Virasoro algebra is just a special case of the algebras we will consider. Just as the supercurrent in the superVirasoro algebra transforms bosonic fields to fermionic fields and vice versa, the fractional spin cur306

rents transform bosonic fields to fractional spin fields and vice versa. We will refer to these algebras as fractional superconformal algebras. If such algebras can be gauged, as in the Virasoro and superVirasoro cases, then, as we will see, c ~ is always a fraction of the bosonic case ( c ~ t = 2 6 ) , and in general is a smaller fraction than the usual supersymmetric case. In some recent work on massive integrable field theories [ 6,7 ], these fractional supersymmetries are realized as symmetries of the particle spectrum. We define the fractional superconformal algebras as follows. Let Gh. denote the W e s s - Z u m i n o - W i t t e n ( W Z W ) model for group G at level K. Consider the current [ 8 ] J¢A)(z) = ~ q,bJ~-, ~ b ( z ) ,

(1)

a,b

where ja_ I are generators of the K a c - M o o d y algebra (J~(z)=~nJ~z-n-I), ~(z) is the chiral primary field in the adjoint representation, and q~b is the Killing form of G. This current Jt/~) generates an extension of the Virasoro algebra. We will refer to this algebra as . ~ / ~ . The Lorentz spin of Jt~) is 1 + A ( ~ ) , where A (q~) is the dimension o f q~a. For instance, for G = S U ( 2 ) at level 2, j(2) is the usual supercurrent; the algebras . ~ ) are the W-algebras. The GA- theory may be bosonized with r = r a n k ( G ) bosons plus some parafermions. By introducing background charges for the bosons, one may thereby form an infinite number of representations of the algebra ,~/~ ~ via a generalized Feigin-Fuchs construc-

0370-2693/91/$ 03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland )

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tion. Unitary representations of this algebra, found by varying the background charge, are the (GK® GL)/GK+L coset models with L varying over positive integers. Indeed, the .~6K) symmetry algebras were first considered in relation to these coset models [8-10]. When G = SU (2), the fractional current J¢~") can be expressed explicitly in terms of ZK parafermion fields and a single boson with background charge [8,9]. In the generalized Feigin-Fuchs construction, the screening operators are V+ = ~,1 exp (ix/} a + ~) and V_ = ~'T exp (ix/2 a _ 0), where q6 and ~,~ are ZK parafermion currents [ 11 ]. ~ is a free scalar field with energy-momentum tensor

To(z)=-½(O¢)2+ix//2ao02¢,

c ~ = 1 - 2 4 a oz ,

To construct j(K) we search for the combination of the above fields that commutes with the screening charges V+ [8,9]. This condition can be translated to a condition on the V±J (~) operator product expansion (OPE):

V+(z)j~K)(w)

1/K.

(2)

The energy-momentum tensor of the .~¢~t~]2) algebra is then T=T,~+Tp~r,f~m,~o,,, and the total central charge is c = c o + c ( Z ~ ) = 3 K / ( K + 2 ) - 2 4 a ~ . The current J~K) has dimension (K+ 4) / (K+ 2 ). Searching the ZK parafermion spectrum for fields of the appropriate dimension reveals only four potential candidates: the "energy" operators el, e~ of dimension 2 / ( K + 2 ), and the parafermion descendents of these fields gi, ~ of dimension ( K + 4 ) / ( K + 2 ) . However, these fields are not all independent. They are defined as parafermion descendents of the ZK parafermion highest weight operators (spin fields) o2 and a~ --aK_~: e I =A~_I/KO'2,

e~ =A_~/Ka~ ,

~I=A~_I/K_IG2, ¢.~=A_~/~._~o~,

(3)

where A and A * are the modes of the ~,~ and ~,} parafermion currents. A slightly involved exercise in Z~ parafermion current algebra is required to discover the relations between these fields. The main tools are the A and A* "commutation" relations derived in ref. [11]. In particular, one can show that a~= l ~ ~ K A I / K A _ i/KO'2, from which it follows, by (3), that e~=½KA_,/KA]/Ke~ and ~ = I K A _ I _ I / K A ~ / K e l . The AA * "commutation" relation acting on e I then shows that e~=e~ and ~ + g l = ½ K + l ) 0 e l . Thus there are really only three independent fields of the correct dimension to form j¢K): ~1 00, Oe~,and g~.

W(w) +reg., (z-w) 2

(4)

where W is an arbitrary operator. In other words, the residue of the single-pole piece of the OPE must vanish. To compute these OPEs we require the ~'1, ~] OPEs with el, g~. A simple exercise using the Zx parafermion algebra gives

2a2(w) + K+2 (z)el (w )= K z - w -o°'2"w'+ ~ ....

( ¢ ( z ) ¢ ( w ) ) = - l n ( z - w) , a+ = a o - + ~ o +

10 January 1991

qA(z)41(W)=

K 2 + 2 K - 4 ~2(w) K2 (z_w)2

2 K + 4 0~2 (w) - K2 z-w

+

...,

(5)

as well as the daggered versions of these relations. The condition that the single-pole piece in (4) vanishes determines the relative normalizations of the three possible terms in j~K) j~h~)= / K ( K + 4 ) N / 4 c ( K + 2 ) ([V/2e, 0 0 - i a o ( K + 2 ) 0 e , ] + iK(a+ - a _ ) K+4

-C~]) [g~

"

(6)

The terms in square brackets are Virasoro primary combinations. The overall normalization has been chosen for later convenience. Note that this expression for the current differs from the various expressions given in refs. [8,9] but agrees with ref. [ 12 ]. For K= 2, eq. (6) reduces to the usual supersymmetry current, since in this case the parafermion fields are related to the free fermion by 0e~ = ~ =0~u. In the ao--,0 limit, eq. (6) reduces to the bosonized expression for the fractional current ( 1 ) in the SU (2)K model. We now present the explicit form of the .~'~2) algebra. From the SU (2)~ fusion rules and the expression ( 1 ) for j~K) (i.e. when the background charge vanishes), it is easy to show by dimension counting that only the identity operator and j ( K ) can appear with singular coefficients in the jt~)j~K) OPE: 307

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has two fractional currents, denoted ~ a n d ~trt. The identification with our current is j(a)=(t~r/_)_

J(X)(z)J(K)(w) ~ (z-w)-2"(l+(z-w)Z2--~Ac T(w)) + 2t<(c)(z-w) -~ X[J(K)(w)-t-(Z--W)IOJ(t")(W)] (7)

+ reg.,

where A = (K+4)/(K+2). The current j ( K ) ( z ) is Virasoro primary, so it has the standard OPE with T(z). Thus T(z) and j(K) (Z) form a closed chiral algebra. To complete the determination o f the '-q/~UA,)(2) algebra, we must compute the structure constant 2 A-(c). When the background charge vanishes this is a straightforward calculation involving SU (2)/< Ward identities. (A practical method for doing this calculation uses an SU(2)-invariant formalism as described for example in ref. [ 13 ]. ) Using the connection between the Z~< parafermion theory and the SU(2)~< W Z W model [11], one can also use this method to calculate leading terms in the Cle~, e~r/, and r/q OPEs, where r / - i ~ - i ~ is the Virasoro primary field appearing in j(K). Combining these pieces of information with the explicit form for j(h') including background charge, given in eq. (6), we determine the structure constant to be, for K>~ 2, 2K2c2'1 22/<(c) = 3 ( K - ~ ( - / ~ + 2 )

( 3 ( K + 4 ) 2 1 1) kK(K+2) c "

(8)

Here CII 1 is the SU(2)~< structure constant for the OPE of two spin-1 primary fields to give a spin- 1 field. The SU (2) structure constants have been calculated in ref. [13]. For K = 2 , c t t t = 0 by the SU(2)K fusion rules, so 22 vanishes identically, giving the superVirasoro algebra. For K = 4 , Clll = 1, and 24 reduces to a simple expression, 9c242= 2 (8 - c), also derived from associativity arguments for the spin-~ algebra introduced in ref. [14] #l. Note that the d~)(2) algebra is actually smaller than the spin- 4 algebra which

Note that we disagree by a factor of x/2 with the value of c~~j given in ref. [ 13] for K=4. 308

10 January 1991

The algebras ,~/(G*<) for G ¢ SU (2) are highly nonlinear algebras due to the presence of composite operators in the algebra (see e.g. the S U ( 3 ) W-algebra). This fact has been a serious obstacle in the construction o f string models based on these algebras. When G = SU (2) the situation is improved because there are no composite operators. Nevertheless, the algebra is non-linear due to the manner in which the generator c (the central extension) enters the algebra through the function 2/<(c), eq. (7). This implies that given two representations of the algebra, J l m and J(2~') , in general IaIt toKt a) ----1/ (IK~ ) , t"~ ~ " + l ® J 2(K) does not form a representation of the same algebra. We have not ruled out a more complicated comultiplication A(j(^'))e ~'#(ASH(2)) (/,) ® '?l(A(s(J)2)),where 41(A~)2)) is the universal enveloping algebra ,~" that v . < ,~ S U(K) (2), would allow one to tensor together representations to form new representations. We now turn to the computation of the likely values of the critical central charge and the slope intercepts for a potential string model based on gauging the algebra .~/~A-). Recall the case of the usual bosonic string [ I ] . (The Virasoro algebra corresponds to -,eSU(2) ()) in our classification.) One begins with a rep._,.s resentation of the Virasoro algebra at some c. A primary field satisfies the gauge conditions L~>o l ~ ) = 0. It is a physical state if it also satisfies the on-shell condition Lo I ~Uphy,) = a i Vphy, ). The parameter a is called the slope intercept. Descendent fields are given by L~_'~LU2... l q)). A descendent IX) is a null state if it is also primary, which implies that it is orthogonal to all primary fields, including itself. A spurious state is a physical null state. In a string theory a spurious state corresponds to a gauge degree of freedom that decouples from scattering amplitudes; such spurious states are required for unitary. At c = c cm, the number of spurious states is greatly enhanced, and in fact there are enough of them to allow a no-ghost theorem. Generalizing the Bilal-Gervais analysis for the Wstring [ 4 ] and Thorn's no-ghost proof [ 15 ], one may argue directly from the Kac formula in deriving c crit. The generalized Kac formula gives the allowed eigenvalues of Lo for null states in a module of given highest weight. Let us derive the Kac formula for the

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algebras ,~¢~) from the generalized Feigin-Fuchs (FF) construction. The idea is that the generalized FF construction provides us with degenerate conformal families for the algebras ~¢~m, and such representations are characterized by null vectors. For concreteness consider the SU (2) case. An ansatz for the ~ ¢ ~ 2 ) primary fields is qbmU),= exp (ix/~am, 0 ) O'2j ,

ot,,,=½(1-m)a+ +l(1-n)a_

,

(9)

where tr2; is a spin field of the ZK parafermion theory of dimension A ° ) = j ( K - 2 j ) / ( K 2 + 2 K ) , and n, m are integers. The parameter j is an SU (2) spin that labels the primary fields of the SU(2)~: theory, i.e. j = 0 , ½, ..., ½K. If we parametrize the background charge ao in terms of c, the central charge of the ,~/~K]2 ) algebra, then we can write the dimensions of the ~ )mn a~ u) ---i ~t m n + A
~J,.. =~4(c-c0) - ~6[ ( m + n) cxFL~-co +(m-n)

cx/c~-cl] z.

(10)

10 January 1991

the minimum of j + 1 and K - j - 1. We define the spin j of a state according to the SU (2) description of its quantum numbers; we also define the spin-j sector as all states of spin j. For example the spin of the state corresponding to the primary field q~)~ is ofcoursej. Thus the states in a module do not have fixed spin and obviously the states in a fixed spin sector span different modules. Since Lo commutes with spin, a priori the different j sectors will have different slope intercepts. We will need the dimensions of the null states. The result is that in the module [ ~ ) ~ ] with m, n > 0 and with n - m = 2j mod K, there exists a null state IXU') ) with spin j' satisfying n + m = 2 j ' m o d K and dimension mn

dz~ ~=zl,~, + ~ - +A u') .

(12)

The argument relies on standard ideas in the FeiginFuchs construction [ 16 ], and we only sketch it. Consider the operation of the screening charges V+ on the primary field ~)~, U) Q(p)cb --mn

Above, Co= 3 K / ( K + 2 ) , and ct =Co +24/K. The module [ q:' ~),, ] with highest weight A~), is generated by the operation of negative modes o f J ~t~>(z) and T ( z ) on the primary field ~)~. Recall that the current in the form ( 1 ) is associated with the adjoint representation of SU(2). Thus, from the SU(2) WZW fusion rules, j(t¢)(z) can change the s p i n j of states it acts on by one unit. In particular, acting on the primary state q~.... u) j(K) (z) has the allowed modings

=

~ (z2j/(K+2)+p--IJ--2(j+I)/(K+Z)--p p~Z

..].. Z - - 2 / ( K + 2 ) + p - I j _ p "l- z - - Z ( J - - I ) / ( K + 2 ) + P J 2 j / ( K + 2 ) _ p _ I )

[ ~ m(J) .>

,

(11) where the first, second and third terms in the sum correspond to states with spins j + 1, j and j - 1, respectively. Actually, the SU(2) fusion rules restrict the spin of the final state to lie between IJ - 11 and

= ? dZl dz2 ... dzpV+ (zl) V+ (z2) ... V+ (Zp)~,~),,, (13) where the zj contour integration is inside the z~ contour for j > i, and all contours start and end at zt. For the integration over the closed Zl contour to be well defined, we n e e d p = m and 2j= ( n - m ) mod K. (To derive this restriction on n, m one uses the repeated action of ~fl on the spin fields given in ref. [ 11 ]. ) Since the screening operators commute with J (~) (z) and T ( z ) in the sense ofeq. (4), Q(") maps a statez in ~rq~u)],,,, to some state in r,:pu') ..... ] with the same weight and spin. In particular, this means that a state Z u') in [ ~ ) , ] with weight A ~ , , is mapped by the BRST charge Q(m) to the primary field ~ , , . Thus Z u') is the null state satisfying eq. ( 12 ). As tbr the bosonic and superstrmg, we constrain c crit by requiring an enhancement of spurious states. Let us consider null states in the modules [ q ~ j + ~]. From the above analysis, we are guaranteed at least one null state Z °') with conformal weight A~'l!2j+l, with 2j' = 2 j + 2 mod K. For simplicity, consider the 309

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null states in the j = 0 module. The lowest null state is a t j ' = 1, with weight A~_~?,I= 2 / ( K + 2 ) , irrespective ofc. This null state can be written as [SI ' ) ) = J - 2 / ( K + 2) wj,1 "~ (0) ) ,

(14)

where the J moding follows from eq. ( 11 ). (For K = 2, j ' is actually zero. ) We now require an additional null state in the same j = 1 sector of the same weight 2 / ( K + 2 ). This conformal weight 2 / ( K + 2 ) defines the slope intercept of the j = 1 sector. The next spurious state in the j = l sector lives in the tr rn(o) "~I,K+ 1 ] module, and can be written, for a some coefficient, 1S(21) ) = ( J - 2 / ( K + 2 ) - I +oLL_IJ_2/(K+2))

ch(O) ~-.~-+ ) .

(15)

This state Is2~) ) corresponds to the null state with dimension a ~_1) + ~. Setting d ~_~~.A+, = 2 / ( K + 2 ) fixes the critical central charge to be ccm=co+cl-

6K 24 K+~ + K"

(16)

Even though we have only used two spurious states to determine c c~", the idea is that, in analogy to the no-ghost theorem [ 15 ], they are sufficient to ensure the removal of all negative norm states in the theory. This is because, by the structure of the .~/~) symmetry algebra (7), all spurious states can be written in the form of either (14) or (15). In fact, it is easy to see that there is an infinite tower of spurious states for the values o f c c~" and slope intercept that we have derived. Note that K= 1, 2 correspond to the known values of ccr"=26, 15 for the usual bosonic and superstrings. We remark that c cm decreases as K increases, and accumulates at ¢crit=6. An interesting case occurs at K = 4, which is the spin-~ algebra mentioned above, which has ccr"= 10. AS above, we define the slope intercept ao) o f the spin-j sector to be the conformal weight of the physical states in that sector. The above analysis fixed a~ l ) = 2 / ( K + 2). The same reasoning applied to the other sectors gives ao, ) =A~'~!2j+~. As described above for the j' = 1 sector, when c = c cm there is the requisite enhancement of spurious states. When c = c cr", the slope intercepts take the simple form au)= 310

~.1 + AO.j+AO. )

(17)

10 January 1991

where 2j' = 2 j + 2 mod K. F o r j ' S 0 , ½, we can choose

j=j'-I in (17) giving a u , ) = 2 j ' ( K - 2 j ' + 2 ) / K ( K + 2 ) . F o r j ' = ~, the canonical choice o f j is ½ ( K - 1 ), which gives at~/2)=2K+ 1/K(K+2); for j ' = 0, we chose j = ½( K - 2 ), giving a~ 3). However, a naive BrinkNielsen-type argument [ 17 ] relating this number to the slope intercept cannot be formulated. The same line of reasoning can be applied to the .~/&A-~ algebra. The generalized FF construction involves r = r a n k ( G ) bosons, and the generalized parafermion theory associated with G [ 18,10 ]. One finds the analogous equation to (10) with Co=C~4h* d / K = Kd / ( K + h* ), and thus

2Kd

4h*d

c crit- - + K + h* K

'

(18)

where h* is the dual Coxeter number of G, and d is the dimension of G. Note that for G = SU ( 3 ), (d = 8, h * = 3 ) and K = 1, the known formula ccr"= 100 for the SU (3) W-string is recovered [3]. Note also that c crit decreases as K increases. It is interesting to consider the K = o o limit of the above results. In this limit, it is not difficult to realize that the algebra we are gauging is the usual K a c Moody algebra. This is because as K--+oo, the adjoint primary field q~ has dimension zero and should be identified with the identity operator; thus there is no adjoint operator with which to form a singlet, as in ( 1 ). Furthermore, the dimension of the current J ~ ) is equal to 1 in this limit. Since the algebra consists of the ordinary bosonic currents J~(z), a ghost con-

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struction is known in this case. Namely, one introduces fermionic ghosts and antighosts for each current, c a and b a ( a = 1, 2 ..... d) of d i m e n s i o n 0 and 1 respectively. The standard analysis yields a ghost central charge of - 2 for each ghost c o m p o n e n t [ 19 ]. Therefore cgh°st= - 2 d . As usual c crit--[-cg h ° s t = 0 for a string construction, yielding cCm= 2d, in agreement with (18). Note that this interpretation does not involve introducing additional ghosts for the Virasoro algebra; this is due to the fact that the Virasoro generators are composite operators by the Sugawara construction, and should not be independently gauged. In fact, the BRST operator corresponding to gauging the K a c - M o o d y currents has been shown to be nilpotent precisely when the level L = - 2 h * [20]. This value of L can be alternatively understood as follows. As m e n t i o n e d above, the ,~¢~K) algebras are related to the (G,~®GL)/GK+L coset models, where the level L is a function of the background charge. By comparing central charges, we can determine the effective value of L corresponding to a critical string model. The central charge of the coset model is CK,L =

Kd/ (K+h*) + L d / (L + h * ) - (K+ L )d/ (K+ L + h* ). Equating with c cm (16), we find Lcrit= - (K+ 2h*)2+_K 2 2(K+2h*) '

(19)

which is negative and non-integer in general (of course, K is still a positive integer). As K ~ m, we obtain LC"it= -2h*, in agreement with the above result. A string model consists of a representation of d & K) with c = c mr. We have not yet found a non-trivial example of such a representation that would qualify as a string theory in the usual sense. However, a representation with c= c c'i' is obviously obtained by tuning the background charge in the generalized FF construction, or equivalently by choosing L = L crit in the coset description. For example for G = SU (2) with K = 1, the coset theories with L = - ~ or - ~7~ and are non-unitary. However, as is well known, the usual string is n o n - u n i t a r y because of the Minkowski metric; unitarity is restored after the world sheet symmetry is gauged so that negative norm states are removed; the same can happen here. Some insight into the m e a n i n g of these fractional level cosets can be arrived at as follows. In analogy with the case of the bosonic string such a model is best thought of as a non-critical string [ 21 ] with no matter contribution.

10 January 1991

This is because the non-ghost c o n t r i b u t i o n to c in this case comes solely from the generalized F F fields, which is equivalent to Liouville theory for SU(2 ) and K = 1. In making this analogy, the Liouville theory is replaced by a boson plus a Z/¢ parafermion coupled through a term in the action that is the screening operator (for S U ( 2 ) ) ; this formally can be taken as a definition of a pure "fractional supergravity". It is a pleasure to thank D. Bernard and Z. Hlousek for discussions. This work was supported in p a n by the National Science Foundation.

References [ 1] See e.g.M.B. Green,J.H. Schwarzand E. Witten, Superstring theory, Vol. 1 (Cambridge U.P., Cambridge, 1987). [2] A.B. Zamolodchikov,Theor. Math. Phys. 65 (1985) 1205; V.A. Fateev and S.L. Lykyanov,Intern. J. Mod. Phys. A 3 (1988) 507; A. Bilal and J.-L. Gervais, Phys. Len. B 206 (1988) 412; Nucl. Phys. B 318 (1989) 579. [ 3 ] J. Thierry-Mieg,Phys. Lett. B 197 ( 1987 ) 368; see also K. Schoutens,A. Sevrinand P. van Nieuwenhuizen, Commun. Math. Phys. 124 (1989) 87. [4] A. Bilal and J.-L. Gervais, Nucl. Phys. B 326 (1989) 222. [5] C.M. Hull, Phys. Lett. B 240 (1990) 110. [6 ] A.B. Zamolodchikov,Landau Institute preprint (September, 1989). [7] D. Bernard and A. LeClair. Cornell preprint CLNS 90/974, Residual quantum symmetriesof the restricted sine-Gordon theories, Saclay preprint SPhT-90-009 (January 1990); C. Ahn, D. Bernard and A. LeClair, Nucl. Phys. B 346 ( 1990) 409. [8 ] D. Kastor, E. Martinecand Z. Qiu, Phys. Lett. B 200 ( 1988) 434. [9 ] J. Bagger, D. Nemeschanskyand S. Yankielowicz,Phys. Rev. Lett. 60 (1988) 389; F. Ravanini, Mod. Phys. Left. A 3 (1988) 397. [ 10] P. Christe and F. Ravanini,Intern. J. Mod. Phys. A 4 (1989) 897. [ 11 ] A.B. Zamolodchikovand V.A. Fateev, Sov. Phys. JETP 62 (1985) 215. [12] D. Bernard and A. LeClair, Phys. Len. B 247 (1990) 309. [ 13] A.B. Zamolodchikov and V.A. Fateev, Sov. J. Nucl. Phys. 43 (1986) 657. [14] F.A. Fateev and A.B. Zamolodchikov,Theor. Math. Phys. 71 (1988)451. [ 15 ] C. Thorn, Nucl. Phys. B 248 (1984) 551; B 286 ( 1987 ) 61. [ 16] B.L. Feigin and D.B. Fuchs, Funct. Anal. Appl. 16 (1982) 114; V. Dotsenko and V. Fateev, Nucl. Phys. B 240 (1984) 312; G. Felder, Nucl. Phys. B 317 (1989) 215. 311

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[ 17 ] L. Brink and H.B. Nielsen, Phys. Lett. B 45 (1973) 332. [ 18] D. Gepner, Nucl. Phys. B 290 (1987) 10. [ 19] D. Friedan, E. Martinec and S. Shenker, Nucl. Phys. B 271 (1986) 93. [20] T. Kawai, Phys. Lett. B 168 (1986) 355; Z. Hlousek and K. Yamagishi, Phys. Lett. B 173 ( 1986 ) 65; A. Bilal and J.L. Gervais, Phys. Lett. B 177 (1986 ) 313.

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[21 ] V.G. Knizhnik, A.M. Polyakov and A.B. Zamolodchikov, Mod. Phys. Lett. A 3 (1988) 819; F. David, Mod. Phys. Lett. A 3 (1988) 1651; J. Distler and H. Kawai, Nucl. Phys. B 321 (1989) 509.