Physics Letters B 274 (1992) 289-297 PHYSIC S LETTERS B North-Holland

The covariant W 3 action

Jan de Boer and Jacob Goeree Institute for Theoretical Physics, University of Utrecht, Princetonplein 5, P.O. Box 80. 006, NL-3508 TA Utrecht, The Netherlands

Received 18 October 1991

Starting with SI (3, R) Chern-Simons theory we derive the covariant action for W 3 gravity.

1. Introduction

Two dimensional gravity has been extensively studied during the last few years. Three different approaches to the subject, namely (i) study of the induced action of 2D gravity in both the conformal (where it reduces to the Liouville action) as well as the light cone gauge, (ii) the discretized approach of the matrix models, and (iii) topological gravity, have all been very powerful (at least for c < 1 ), giving equivalent results.

Higher spin extensions of 2D gravity can also be studied using the above methods. These theories are com- monly denoted as theories of W gravity. Especially W3 gravity in the light cone gauge has been the subject of much recent research [ 1-3]. In this paper we will also concern ourselves with the study o f W 3 gravity, but from a different angle. Believing that the "W3 moduli space" is somehow related to the moduli space of fiat S1 (3, ~) bundles, we will study W3 gravity starting from S1 (3, ~) Chern-Simons theory whose classical phase space is the space of flat S1 (3, ~ ) bundles.

Our analysis resembles the one in ref. [4 ]. In this reference Verlinde showed how the physical state condition in S1 (2, R) Chern-Simons theory can be reduced to the conformal Ward identity, giving as a by-product the fully covariant action of 2D gravity. We will start with SI (3, ~) Chern-Simons theory, and derive the covariant action for W3 gravity. It will turn out that this action describes S1(3, ~) Toda theory coupled to a "W3 back- ground," confirming general beliefs. Although we restrict ourselves to the case of W3 in this paper, we believe that many of our results can be generalized. This will be reported elsewhere [ 5 ].

2. Chern-Simons theory

Chern-Simons theory on a three manifold M is described by the action

S= 4-~l f T r ( A ^ d A + 2 A ^ A ^ A ) , (2.1) M

where the connection A is a one form with values in the Lie algebra g of some Lie group G, and d denotes the exterior derivative on M. In this paper M will be of the form M = E X ~, E being a Riemann surface, for which A and d can be decomposed into space and time components, i.e. A =Ao dt+A, with -~=,4z dz+Az dg, and d = dt O/Ot+~l. Rewriting the action as

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S = ~ i dt Tr[AAO, A+2Ao(dA+AAA)] , Z

(2.2)

we recognize that Ao acts as a Lagrange multiplier which implements the constraint F = dA +A A A = 0. Further- more, we deduce from this action the following non-vanishing Poisson brackets:

2z6 {~/a(z), ~/f(W)} = --k-- r/"~'~(Z-- W), (2.3)

where A_ = 52 ~_"_ T ~, with Tr ( T a T/' ) = r/oh. Upon quantizing the theory we have to replace the above Poisson bracket by a commutator , and we have to

choose a "polarization". This simply means that we have to divide the set of variables (A~, A~) into two subsets. One subset will contain fields X, and the other subset will consist of derivatives 8/8Xi, in accordance with (2.3). The choice of these subsets is called a choice of polarization. Of course we also have to incorporate the Gauss law constraints F (A) = 0. Following refs. [ 4,6,7 ] we will impose these constraints after quantization. So we will first consider a "big" Hilbert space obtained by quantization of (2.3), and then select the physical states ~ b y requiring F(A ) ~'= O.

In ref. [4] it was shown that these physical state conditions for S1 (2, ~) Chern-Simons theory with a certain choice of polarization are equivalent to the conformal Ward identities satisfied by conformal blocks in confor- real field theory (CFT) . More precisely, it was shown that two of the three constraints in F(A)~= 0 could be explicitly solved, leaving one constraint which is equivalent to the conformal Ward identity. In this paper we will generalize these results to the case of S1 (3, ~ ) with a choice of polarization that leads to the Ward identities of the W3 algebra [8] (A different choice of polarization leading to the related W23 algebra was made in ref. [9 ]. ) To explain our strategy, we will in the next section first reconsider the case of S1(2, ~) Chern-Simons theory.

3. Sl(2, R)

In order to understand how one obtains the Virasoro Ward identity from S1 (2, ~ ) [ 10 ] Chern-Simons theory, let us first recall how one can in general obtain Ward identities from zero-curvature constraints. Given operator valued connections A and X ~, the zero curvature condition reads

F~=:Od-OA+[A,X]: ~ = 0 . (3.1)

Here the dots denote normal ordering, which simply amounts to putting all 8/8X to the right. (Here X is some arbitrary field. ) I f we take ~2

A 0 = ( 6 / 8 # 10)' (3.2)

and put X~ 2 =/z, we can solve for the remaining components of A, if we require that the curvature operator must have the form

F = ( 0 , ~ ) . (3.3)

Eq. (3.1) reduces to one equation which is precisely the Virasoro Ward identity for c = 6k

nl Note that from now on ,4 will be denoted as A. n2 In the following 8/'6X should in fact read (2re/k) (6ff6X), for all fields X.

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8 +½03#) ~ = 0 ( [ ~ - # o-2(oit)] ~ (3 .4)

If we start with S1 (2, ~ ) Chern-S imons theory and pick a polarization, then A and A consist of three fields and their variational derivatives. On the other hand, the Virasoro Ward identity contains just one field. So in order to obtain the Virasoro Ward identity as zero curvature constraint of Chern-Simons theory, we must introduce two extra degrees of freedom without changing the contents of the zero curvature constraints. We can in this case s imply introduce extra degrees of freedom by performing a gauge transformation. Under such a gauge transformation the curvature transforms homogeneously: F-- ,g - IFg , and F = 0 will still give the Virasoro Ward identity. If we require F to remain of type (3.3), g must be of the form

g=(g,I O ) \g21 gi-11 " (3.5)

We can parametrize g via a Gauss decomposit ion:

(1 0~(e* 0 ) g= (3.6)

Z 1] \0 e -~ '

and we find that F-+ F e 2~. The gauge transformed A, A are

Ag=( Z+0O e -2° "] e 2~ (02'-Z2+8/Sit) - Z - 0 O J ' (3.7)

r ig=( ½0it+Zit + 00 e -2~ ) e 2~ (It 8/Sit-½02it-itX2-ZOit+SZ) -½0it-Zi t -0# " (3.8)

If we pick the following polarization:

A=( A,, A12 ) \ 5/(~-"~12 --All ' (3.9)

.gf--½8/Sa,l ~z~12 =lk - 8/8A,2 ½8/8Ali J ' (3.10)

and let F(A, A) act on a wavefunction e s ~v[it ], where S still has to be determined, we find that F(A, A) e%V= 0 e s F ( A ' , A' ) ~v=0, with ~3

=( A,, A,2 ) A' ~kSS/81~12..at_~)/~).<~l 2 --Aix ' (3.11)

~,= ( - ½8S/8A~, - ½8/8A,~ .4,2 ) \ -8S/SA,z-8/8A,2 ½8S/8A,l+½8/SA~l " (3.12)

To proceed, suppose that we want to match (3.11 ) and (3 .12) with (3 .7) and (3 .8) . This gives

A I , = Z + 0 ~ , m l 2 = e -20 , . e~ lz= i t e -2~ • (3.13)

Using these expressions, we can express the variational derivatives with respect to A and .g in terms of those with respect to Z, 0 and It. One finds

~3 Here we ignored terms of the type 82S/8X8X ', that involve delta-function type singularities that have to be regularized somehow. Therefore the validity of our discussion will be limited to the semi-classical level. In the full quantum theory we expect corrections to the expressions given in this paper.

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5 5 5 ( 8 5 5 ) 5 = e 20_.~ ( 3 . 1 4 ) 8A~ - 8Z' 8A~ - ½e2° O ~ + ~ - 2 / z g - ~ # , 8-~,2 8/z"

Quite remarkably, the terms containing 8/8/z in (3.11 ) and (3.12 ) agree precisely with those in (3.7) and (3.8). As Tdepends only on/z, we can omit the 8/8~ and 8/8Z terms in (3.12). In conclusion, we see that (3.11 ) and (3.12 ) are exactly identical to ( 3.7 ) and ( 3.8 ) if the following relations hold:

8S 8S 8S - - = e 2° (½02/z"]-/zZ2"~-Z 0 /Z-- ~ ) , - - e 2~ (OZ-Z 2) (3.15) 8All -- -- 0/Z--2Z/Z--20O, ~AI2 ~-~12 '

Before continuing, we will first rewrite these expressions in a form that is more suitable for generalizations to other cases. Let G be the subgroup of S1(2, ~) consisting of all g of type (3.5), and let A = o~ (oo), which is equal to (3.2) up to terms containing 8/8/z. Furthermore, let (Ao)~2=/z and let Ao be such that F(A, Ao) is of type (3.3). For this Ao one finds F(A, Ao) o o = (-03~,/2o). To rewrite the polarizations, define projections Hk and Hi on the Lie algebra sl (2, ~ ) via

The polarization (3.9) and (3.10) is such that the fields are in/-/'~A and HkA, and that the derivatives are in HitA and HtkA, where Hi t = 1 --Hk and Htk = 1 - H i . Now (3.7) and (3.8) read, up to terms containing 8/8/z, A = g - ~Ag + g - ~ Og and A = g - ~Aog + g - ~ ~g. Therefore, ( 3.13 ) can be compactly formulated as

H i A = H i ( g - l A g + g - ' 0g) , (3.16)

Hkd=Hk(g-~Aog+g -~ ~g), (3.17)

and eqs. ( 3.15 ) become

8S =H*i (g-~..4og+g -~ Og) , (3.18)

8//iA

8S 8~-k.~=Htk(g- Ag+g -~ Og). (3.19)

Surprisingly, these equations can be integrated ~4 to give

S = ~ d 2 z T r ( n { A H k A ) - ~ d 2 z T r ( A O g g - ' ) - F w z w [ g ] , (3.20)

where Fwzw is the Wess -Zumino-Wi t t en action

Fwzw[g] = ~ d 2 z T r ( g - ' Ogg-' ~ g ) - ~ T r ( g -~ dg) 3 . (3.21) B

For S1 (2, N) we find that

s= ~ d~z (/zaz-/zx~-2~0z-a0;0), (322)

which indeed solves (3.15). To make contact with the results of ref. [4 ] we have to redefine O - ' - ½~o, and X--' ½~o+ ½0~0. Then the action becomes S=So(co, ~o,/z) + S~(~o,/z), where

#4 The proof of this fact will be given elsewhere [5].

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kf So=- ~ dZz [ ½ # o 9 2 - o J ( ~ - 0 # - / z 0 t P ) ] , (3.23)

and SL is a chiral version of the Liouville action:

(3.24)

The actions (3.23), (3.24) are precisely the same as the ones found in ref. [ 4 ]. Altogether we have now shown how the Virasoro Ward identity follows from S1 (2, E ) Chern-Simons theory. In general, the procedure consists of three steps: (i) pick a polarization and parametrization of the components of A and A, (ii) move A and .4 through a term of the form e s, and (iii) perform a gauge transformation. In the next section we will apply these steps to the case of SI (3, E) Chern-Simons theory.

4. Sl(3, R)

The Ward identities of the W3-algebra can be obtained in the same way as the Virasoro Ward identity was obtained in the previous section. We start with

(o Z) A= 0 0 , (4.1) 8 /8v 8/8#

and put.413 = v and.,{23 =/1. The remaining components of A are fixed by requiring F to be of the form

(°o °i) F = 0 . (4.2)

F3 t F32

As was shown in e.g. ref. [ 1 ], F3~ and F32 are directly related to the W3-Ward identities. The subgroup G of S1 (3, ) that preserves this form of F consists of all g~ SI (3, E) satisfying g~ 3 =g23 = 0. TO parametrize these g, we will

again use a Gauss decomposition: (1 00)(e 0 0 0 e p-'~ g = 01 1 0 0 1

03 02 1 0 0 e -p 0 0

(4.3)

Under a gauge transformation F--,g- ~Fg we find

F~l =e'~+P F31 +0t e'~+#F32, (4.4)

F32 =Z e~+PF31 + (Z01 ea+#+e2#-a)F32, (4.5)

which clearly shows that F~[p , v] = 0 .~ (g-IFg)~[#, v] =0. The polarization we choose is such that it is invariant under the subgroup G of SI ( 3, ~):

{A++A_ Al2 AI3 "~ A = / A21 - 2 A + A2 3 ~ , (4.6)

k 8/8d~3 8/8d,3 A+-A_ /

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- -~8/8A+ - ½8/8A - 8/8A2, d , , \

A= -8 /8A ,2 ~8/8A+ -~23 ) , (4.7) -8 /8A,3 -8/8A23 -~8 /SA++½8/SA_

and the projections Hi and Hk are given by

,)(ooi) H k | e --2a f = 0 0 , (4.8) \ g h a - b 0 0

. ( 4 . 9 ) H,~ e - 2a = - 2a f b \ g h a - b 0 a

The matrices A and Ao are also found completely analogously to the SI (2, Y~ ) case, one simply requires F(A, Ao) to be of type (4.2), to find ( IZ) A= 0 , (4.10)

0

/ ' o~+~o2v ~+0v v '~ Ao = [ - 02/t- ~03u -½02v /t ) . (4.1 l)

\ 031~+ 204p --02]/ - -103/ / -- 0]./-- ~02/./

It is straightforward to read off the explicit expressions for H~A and Hk,4 from (3.16) and (3.17). They are

A+ = / [01 --02 + (0 2 --03)• e2a-f l - - •00, e2" -~+O(a- f l ) ] ,

A_ =~[0 , + 0 2 + ( 0 ~ - 0 3 ) z e 2 " - a - Z 0 0 1 e 2 " - ~ + O ( a + f l ) ] ,

-412----(20~--02)zWe/~-2e~+Z2(02--03) e2a - /3 -Z2 001 e 2a / % Z 0 ( 2 a - f l ) + 0Z,

A13 = - Z e 2~-# ,

A 2 1 = ( 0 3 - - 0 1 + 0 0 1 ) e 2 a - # ,

23 --~

-~23 = ( P - v0~ ) e "-2/~ . (4.12)

Again, we want to construct an action S, such that A' and A', defined through F(A, A) e s T = 0 ~ eSF(A ', A' ) ~ = 0 , are equal to the connections A g and Ag (the generalizations of (3.7) and (3.8)) that are the gauge transforms with g as in (4.3) of the connections A and J mentioned in and below (4.1). If A g=A' and Ag =A' are satisfied, F(A' , .4' ) ~ = 0 is equivalent to the statement that ~satisfies the W3-Ward identities. The connec- tions ,4' and A' can be obtained from (3.9) and (3.10) by first replacing 6/6X~ by 6/6X~+fS/ fX, everywhere, followed by putting all terms in 8/8X, that do not contain 8/8/z or 8/8v equal to zero, as ~depends only on Iz and v. Comparing these A' and A' with A g and Ag yields a set of equations for fiS/SX~, analogous to (3.15). These equations are necessary, but not sufficient, because the 8/8/z and 8/8v dependence ofA' and A' must also be equal to the 8/6/~ and 8/8v dependence ofA g and Ag. Whether this is the case has been verified by explicitly computing the 8/8)(, in (4.6) and (4.7) in terms of 8/8/~, 8/8v, and the functional derivatives with respect to the fields in the Gauss decomposition (4.3). It turns out that this dependence is indeed precisely the same. The computations involved here are rather cumbersome, whether there is a more direct way to see this, is under

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current investigation [ 5 ]. Due to this remarkable fact, we know that if we now solve (3.18 ) and (3.19 ) for this case, F e s ~u= 0 will be satisfied (with this choice of parametrization and polarization) if and only if ~u satisfies the W3-Ward identities. Again, S is given by (3.20), and reads

S= ~ dEz { - ½A ° Ooti Oas-q)iA° Oas-0Z (00, + ~ - 0 3 ) e eat-a2

+P[ ( 0 - 02)~2 +0201 -03] + v ( 0 - 02) (03 -0201 )}, (4.13)

where a I = ~, O~2 ~-- ] ~ and A o is the Cartan matrix of S1 (3, Bq), A o= ( 2 - -~ 2). This action is important if one wants to compute inner products of wave functions ~v in Sl (3, E) Chern-Simons theory. This will be the topic of the next section.

5. The inner product and W3 gravity

The wave functions 5v[/z, v] that solve the W3-Ward identities, can be obtained from a constrained Wess- Zumino-Witten model [ 1,11 ]. This means that at this stage we know the complete wavefunction e s ~u. These wavefunctions solve the holomorphic Wa-Ward identities, and can therefore be seen as effective actions of chiral W3-gravity. However, in ordinary gravity there is a nontrivial coupling between the holomorphic and the anti- holomorphic sectors, and this is where part of the geometry of two-dimensional quantum gravity comes in. In this section we will consider the coupling between the holomorphic and anti-holomorphic sectors of W3-gravity, hoping that it will lead to an understanding of the geometry underlying the W3-algebra. This nontrivial coupling appears when computing inner products of wavefunctions in SL (3, E) Chern-Simons theory. The expression for such an inner product is

( ~ul I ~2 ) = ~ D(/-/iA) D (//kA) D(HtkB) D(/-/~, B) e v+s+s 7'1 [~t, v] ~P2 [/2, ~] • (5.1)

The nontrivial coupling is due to the K~ihler potential V, which is associated to the symplectic form defined by (2.3). To find an expression for this K/ihler potential, we first give the definitions of B and/~, the variables on which the anti-holomorphic wave function ~P2 depends. Let H be the subgroup of S1(3, ~) consisting of all elements he S1 (3, R) satisfying h3t = h32 = 0; H can be conveniently parametrized by a Gauss decomposition. Define the connection Bo by requiring that (Bo)31 = ~ and (Bo)32 = f / , and that m F(Bo, A t ) = 0 , where A t is the transpose of A. Then

B=hBoh -1 - Oh h-1 ' (5.2)

.B=hAth- l -Oh h - l , (5.3)

and the anti-holomorphic action ~q is given by

S= ~ d2zTr(II~BIlkB)+ ~ dZzTr(Ath - 1 0 h ) - F w z w [ h ] . (5.4)

In terms of A and B, the I~hler potential is given by

k V= ~ f dEz Tr(//~A I-I~,B-FIk.,~II~B). (5.5)

The total exponent K= V+ S+ g occurring in the inner product (5.1) is now a function of g, h, and {/~, v,/~, ~}. This "action" K is part of the covariant action of W3-gravity. The complete covariant action is given by K+ Sw [/l, v] + gw [/~, ~'], where Sw [p, v ] and Sw [/~, ~ ] are the chiral actions for W3-gravity that were constructed in ref.

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[ 1 ]. In the case o f S1(2, N), Kis equal to the Liouville action in a certain background metric, plus an extra term depending on ¢t,/?t only. K represents the W3-analogon of the Quil len-Belavin-Knizhnik anomaly. Clearly, it will be interesting to have an explicit expression for it. One can work out such an explicit expression, by simply substituting all the expressions given above. The result of all this is a large, non-transparant expression. How- ever, upon further inspection, it turns out that K is invariant under local SI (2, ~ ) × N symmetry transformations, which can be used to gauge away four degrees of freedom. Actually, one can prove (see ref. [ 5 ] ) that K depends only on the product gh. Using this the action can be greatly simplified by introducing a new Gauss decomposi- tion for gh, which is now an arbitrary element of SI ( 3, N). To be more specific, we take

(1 oo)(e l o o)( i i3) gh= Pl 1 0 0 e ~2-*'1 0 1 - 2 -

P3 P2 1 0 0 e ~ 0

(5.6)

Substituting this, one sees that the action depends simply quadratically on P3, P3. Therefore, ignoring subtleties arising from the measure when changing variables from A and B to gh and/~, v,/~, u, we can perform the fDp3 Dp3 integration. The resulting action can be written in the following form:

k f ( O~ogO~oj+ ~e-"'J~',-AU(p,+O~o~)(~+O~oj) K = ~ d2z ½A 0

- e ~' - - '~ ( ~ - 1 0 v - vp~ ) (~+ ½~+ @, ) - e - ~ ' - ~ vo

(~+ ½0V+ Pp2) (fi-- ½0V-- tYff2) +/.zT+ p W+1i7~+ t7I~7~ , (5.7) __ e (,o2- 2~[ /

where we defined T, W, T, Wthrough the following Fateev-Lukyanov [ 12] construction:

(O-p2) (O-p, +P2) ( 0 + p , ) -- 03-4 - T O - W+ ½0T,

(0-~2) (O-P, +~2) (0+~,) = 03+ T~+ w+ ½~T, (5.8)

and we sh i f t ed /1 - , / t - ½0 v and/~ ~/Tt + ½ 0p. The first part o f K is precisely a chiral SI (3) Toda action, confirming the suspected relation between W3-gravity and Toda theory. Actually, one would expect that in a "conformal gauge", the covariant W3-action will reduce to a Toda action. Indeed, if we put v = P = 0 in K, then we find that K is also purely quadratic in p~, P2, P~, f12. Performing the integrations over these variables as well, we find that

) K[ (,Ol, (,o2, ~, fi] = ~ d22 ½~'~'O,(,o,O,,~iAV+4~e-~'"e*+R¢.~ +K[cz, fi] , (5.9) i

where K[It, 2] is the same expression as was derived in refi [4], namely

k f d a z K[U, 12]= ~ (1- /~f l ) '[O~Ofl--½1~(Ofl)2--1fl(Olt) 21 (5.10)

and ~ is the metric given by ds 2 = ] dz +/z dYl 2. In the case of S1 ( 3, N), ~. (0, with ~ being one half times the sum of the positive roots, is just given by q~ + (o2. The action ( 5.9 ) is the same Toda action that was originally present in K in a chiral form, and the integration over p~, p~, P2, P2 has the effect of coupling it to a background metric

L Of course, the most interesting part of the action is the part containing v, ~. Unfortunately, if we do not put

u= ~=0, we can integrate over eitherpj, p~ or overp2, P2, but not over both at the same time, due to the presence of third order terms in K. Another clue regarding the contents of the action (5.8) can be obtained by treating

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the second and third line in (5.8) as perturbations of the first line of (5.8). This means that we try to make an

expansion in terms of/.t,/2, v, ~. The saddlepoint of the p-terms is at p~= - 0¢i and ~i = - 0~. From (5.8) we can now see that T, W, T, W are, when evaluated in this saddle point, the (ant i - )holomorphic ene rgy-momentum tensor and W3-field that are present in a chiral Toda theory,

T = - ½A ij O~oi Orpj - ~," O2(p ,

W = -- 0~01 [ (0~02 )2.4. 102~02 __ 02~01 ] .4. ½03~01 __ ( 1'--'2 ) , (5.11 )

and similar expressions for T, W.

This suggests that the full action K contains the generating functional for the correlators of the energy-mo- men tum tensor and the W3-field of a Toda theory, "covariantly" coupled to W3-gravity. The presence of the third order terms in W, if" in (5.8) prevents us from computing the action of this covariantly coupled Toda theory.

Detailed proofs, that were omitted here, as well as generalizations to other W-algebras, will be the subjects of a future publication [ 5 ].

This work was financially supported by the Stichting voor Fundamenteel Onderzoek der Materie (FOM).

References

[ 1 ] H. Ooguri, K. Schoutens, A. Sevrin and P. van Nieuwenhuizen, The induced action of W3 gravity, preprint ITP-SB-91 / 16, RIMS- 764 (June 1991).

[ 2 ] K. Schoutens, A. Sevrin and P. van Nieuwenhuizen, Nucl. Phys. B 349 ( 1991 ) 791. [3] C.M. Hull, Nucl. Phys. B 353 (1991 ) 707. [4] H. Verlinde, Nucl. Phys. B 337 (1990) 652. [ 5 ] J. de Boer and J. Goeree, in preparation. [6] S. Elitzur, G. Moore, A. Schwimmer and N. Seiberg, Nucl. Phys. B 326 (1989) 108. [7] E. Witten, Commun. Math. Phys. 137 (1991) 29. [ 8 ] A.B. Zamolodchikov, Theor. Math. Phys. 65 ( 1985 ) 1205. [ 9 ] A. Bilal, W-algebras from Chern-Simons Theory, CERN-preprint CERN-TH 6145/91, LPTENS 91 / 17.

[ 10] A. Bilal, V.V. Fock and I.I. Kogan, On the origin of W-algebras, CERN preprint CERN-TH 5965/90 (December 1990). [ 11 ] M. Bershadsky and H. Ooguri, Commun. Math. Phys. 126 (1989) 49. [ 12] V. Fateev and S. Lukyanov, Intern. J. Mod. Phys. A 3 (1988) 507.

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