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Solutions to Knizhnik-Zamolodchikov

arXiv:hep-th/9306139v1 26 Jun 1993Solutions to Knizhnik-Zamolodchikovequations with coefficients innon-bounded modulesKenji Iohara∗, Feodor Malikov†‡Department of Mathematics, Kyoto University ,Kyoto 606 JapanReceived:AbstractWe explicitly write dowm integral formulas for solutions to Knizhnik-Zamolodchikov equations with coefficients in non-bounded – neither high-est nor lowest weight – sln+1-modules. The formulas are closely relatedto WZNW model at a rational level.1IntroductionLet g be a finite-dimensional simple Lie algebra, ˆg the corresponding non-twistedaffine Lie algebra.

Let λ be a weight of g , M(λ, k) (M(λ, k)c ) be the Verma (contragredient Verma) module over ˆg with the central charge k; for a g−moduleV be denote by V ((z)) the module of formal Laurent series in z with coefficientsin V , regarded as a ˆg−module with the central charge equal to 0.Vertex operator is a ˆg−linear mapΦ(z) : M(λ1, k) →M(λ2, k)c ⊗V ((z)). (1)If highest weights (λ1, k), · · · (λN+1, k) are generic thenM(λi, k) ≈M(λi, k)c, 1 ≤i ≤N + 1 and one may consider a product of vertexoperators ΦN(zN)◦· · ·◦Φ1(z1).

Matrix element ⟨v∗λN+1, ΦN(zN)◦· · ·◦Φ1(z1)vλ1⟩∗e-mail address: iohara@kurims.kyoto-u.ac.jp†Supported by the Japan Society for the Promotion of Science Post -Doctoral Fellowshipfor Foreign Researchers in Japan.‡Address after August, 1, 1993: Department of Mathematics Yale University New HavenCT 06520 USA; e-mail address: malikov@kusm.kyoto-u.ac.jp0

Solutions to KZ equations1related to vacuum vectors is called a correlation function. One of the centralresults of conformal field theory ( see [1]) is that a correlation function satisifiesa remarkable system of Knizhnik-Zamolodchikov equations.

We prepare nota-tions in order to write down the trigonometric form of Knizhnik-Zamolodchikovequations.Letg = h ⊕⊕α∈∆gαbe a root space decomposition. Fix an invariant inner product on g and a basis{hi ∈h, gα ∈gα : 1 ≤i ≤n, α ∈∆} of g so that (hi, hj) = δi,j, (gα, gβ) =δα,−β.

For each µ ∈h∗denote by hµ an element of h satisfying (and uniquelydetermined) by the condition (hµ, h) = µ(h).Setr = 12nXi=1hi ⊗hi +Xα∈∆+gα ⊗g−α.Being an element of U(g) ⊗U(g) r naturally acts on a tensor product of 2g−modules. There are N 2 different ways to make it act on a tensor product ofN g−modules via the following N 2 embeddings of U(g)⊗2 in U(g)⊗N: each ofthem is associated to a pair of numbers 1 ≤i, j ≤N and sendsU(g)⊗2 ∋ω 7→ωij ∈U(g)⊗N,so thatif ω =Xsas ⊗bs then ωij =Xsjz}|{1 ⊗· · · 1 ⊗as|{z}i⊗1 ⊗· · · ⊗1 ⊗bs ⊗1 ⊗· · · ⊗1.For a pair 1 ≤i, j ≤N introduce the following function in 2 complexvariables with values in U(g)⊗N:r(zi, zj) = rijzi + rjizjzi −zj.Theorem 1.1 (Knizhnik, Zamolodchikov) The correlation functionΨ(z) = ⟨v∗λN+1 ΦN(zN) ◦· · · ◦Φ1(z1)vλ1⟩satisfies the following system of differential equations(k + h∨)zi∂Ψ∂zi={Xj̸=irij(zi, zj) −12(λ1 + λN+1 + 2ρ)(i)}Ψ,(2)1 ≤i≤N,where h∨is the dual Coxeter number of g and for each µ ∈h∗µ(i) stands for theoperator acting on V1⊗· · ·⊗VN as hµ applied to the i−th factor of V1⊗· · ·⊗VN.

Solutions to KZ equations2To keep track of the parameters we will be referring to (2) as KZ(λN+1, λ1).The deep theory of KZ equations has been developed by several authors(see e.g. [2, 3, 4, 5] ) in the case when Vi are highest weight modules.

It hasalso been realized that this theory is relevant to physics applications in the casewhen (λi, k) is either integral or generic.Indeed, if conflicting with the aboveassumptions, some of M(λi, k) are reducible then the product ΦN(zN) ◦· · · ◦Φ1(z1) does not exist unless each of the operators Φi(zi) can be pushed downto a mapΦi(zi) : L(λi, k) →L(λi+1, k) ⊗Vi((zi)),(3)where L(λ, k) stands for an irreducible highest weight module with the highestweight (λ, k).In the case when (λ, k) is an admissible weight (for example,dominant integral weight) [6] the last condition reduces to the singular vectordecoupling condition: matrix elements of Φi(zi) related to singular vectors ofM(λi) vanish. It is known[7] that if each (λi, k) is dominant integral theneverything goes through nicely, in particular, the Schechtman-Varchenko inte-gral solutions to (2) come from products of vertex operators (3).

However if thecentral charge is not integral it has been realized ( [10], see also[8, 9]) thatthe singular vector decoupling condition implies that Vi is neither highest norlowest weight module. Though some results for such models were obtained in[8, 9], where in particular the connection to quantum hamiltonian reduction wasrevealed, not much is known about KZ equations in this case.In [11] a new method of constructing solutions to (2) was proposed whichseems to be relevant to the problem.

Let G be a complex Lie group related tog, F = G/B be a flag manifold and F 0 ⊂F be the big cell. There is a familyof embeddings of g into the algebra of order 1 differential operators on F 0πµ : g →Diff 1(F 0), µ ∈h∗.This makes the space of analytic functions on F 0 into a huge g−module.

Dif-ferent g−closed subspaces give realization of different g−modules. For example,contragredient Verma modules are realized in the space of polynomials on F 0, µbeing the highest weight and a constant function being a highest weight vector;this observation has been extensively used recently with regards to Wakimotomodules [12, 13, 14].

The spaces of multi-valued functions give modules withquite different properties, the simplest example being that of sl2: in this casethe big cell is C, contragredient Verma modules are realized in C[x]; the spacexνC[x, x−1], ν ∈C is also closed under the action of sl2 and the embeddingπµ, µ ∈C makes it into generically irreducible sl2−module. This module istransparently neither highest nor lowest weight one.Regarding V in (1) as a g−module realized in functions on the big cell oneidentifies elements of V ((z)) with functions of 2 groups of variables: x and z,where x stands for a ( vector ) coordinate on the big cell and z is a coordinateon C. Likewise, the correlation functionΨ(z) = ⟨v∗λN+1, ΦN(zN) ◦· · · ◦Φ1(z1)vλ1⟩

Solutions to KZ equations3is identified with a function of x(1), . .

. , x(N); z1, .

. .

, zN where x(i) is a coordinateon the i−th copy of the big cell, zi ∈C, 1 ≤i ≤N. One of the advantages ofthis functional realization is that the emebedding πλ : g →Diff 1(F 0) lifts tothe mapping of the group G: for g ∈g the exponent exp(−tg) is a well-definedoperator.Let W be the Weyl group of bg, w = rm1rm2 · · · rml ∈W be a decomposi-tion(not necessarily reduced),where rm denotes the reflection at the correspond-ing simple root.

Set,βj = 2(rml+2−j · · · rml · λ1, αml+1−j)(αml+1−j, αml+1−j)+ 1, 1 ≤j ≤l.GivenΨold(z) = ⟨v∗λN+1, ΦN(zN) ◦· · · ◦Φ1(z1)vλ1⟩,setΨnew =lYj=1Γ(−βj)−1Z{exp(−tlFml) · · · exp(−t1Fm1)Ψold}lYj=1t−βj−1jdt1 · · · dtl,(4)where the integration is carried out over an arbitrary cycle of the highesthomology group related to the multi-valued integrand.In (4) it is set thatEi, Fi, Hi, 0 ≤i ≤rk g are canonical Cartan generators of ˆg and Ei, Fi, Hi, 1 ≤i ≤rk g are the ones coming from the inclusion g ⊂ˆg.Theorem 1.2[11] Ψnew is a solution to KZ(λN+1, w · λ1).Theorem 1.2 works as follows: given a solution to KZ it generates new oneslabelled by elements of the affine Weyl group. In our notations the simplestsolution to KZ(λN+1, λ1) is given by◦Ψold=Qi

In particular,◦Ψold is independentof x′s. The purpose of this paper is to explicitly write down the integralΨnew =◦Ψold ×lYj=1Γ(−βj)−1Z{exp(−tlFml) · · · exp(−t1Fm1)1}lYj=1t−βj−1jdt1 · · · dtl,(5)for g = sln+1, generalizing the calculation carried out in [11] for sl2.

Solutions to KZ equations4Remark. The integral representation of Ψnew in Theorem 1.2 is nothingbut the conventional definition of F βlml · · · F β1m1 · Ψold.The latter comes fromlooking at the “matrix element”⟨v∗λN+1 ΦN(zN) ◦· · · ◦Φ1(z1)F β1m1 · · · F βlmlvλ1⟩.Though the expression F β1m1 · · · F βlmlvλ1 is not understood as an element of M(λ1, k),the powers are chosen in such a way that it formally satisfies the singular vectorconditions [11, 15], which makes the statement of Theorem 1.2 almost obvious.One can similarly consider an expression⟨Eβ′1m1 · · · Eβ′lml · v∗λN+1 ΦN(zN) ◦· · · ◦Φ1(z1)vλ1⟩,for appropriate β′1, .

. .

β′l and write down another solution in the form close to(4) but with F ′s replaced with E′s or combine both methods or, finally, applythem to other solutions obtained in [4, 8, 9].As to relation of our solution (4) to correlation functions, we have been ableto verify in simplest cases that (4) indeed gives a matrix element of a productof vertex operators and hope that (4) will prove useful for investigation of otherrational level models.Acknowledgements. Our thanks are due to M.Jimbo for his interest inthe work.

Results of the work were announced when F.M. visited the NationalLaboratory for High Energy Physics in Tsukuba.

F.M. is obliged to H.Awataand Y.Yamada for their hearty hospitality and interesting discussions.2Integral formulas for soilutions of KZ(λN+1, λ1)2.1Main resultHere we are going to write down the integral (5) in the case of g = sln+1, ˆg =dsln+1.

In this case there are 3(n + 1) Cartan generators Ei, Fi, Hi, 0 ≤i ≤n,where Ei, Fi, Hi, 1 ≤i ≤n are the ones coming from the inclusion g ⊂ˆg.Explicitly the generators are described as follows. If eij = (ast) is an (n + 1) ×(n + 1) matrix then Ei = eii+1, Fi = ei+1i, Hi = eii −ei+1i+1, 1 ≤i ≤n andE0 = en+11 ⊗z, F0 = e1n+1 ⊗z−1 ( see [16] for details).

The g−weight µ isconsidered as a vector (µ1, . .

. , µn), µi = µ(Hi).

The embeddingπµ : sln+1 →Diff 1(F 0), µ = (µ1, . .

. , µn)is calculated in [12] (see also [13]).

To recall this result we choose coordinatesof the big cell F 0 to be {xij : 1 ≤i < j ≤n} identifying as usual the big cell

Solutions to KZ equations5with the subgroup of matrices1x11· · ·· · ·x1n1..................xnn01.For 1 ≤i ≤n set∂xij :=∂∂xij.Then πµ acts on Cartan generators byEi 7→−∂xii −nXj=i+1xi+1j∂xij,Fi 7→xiiiXj=1xji∂xji −i−1Xj=1xji−1∂xji−1−nXj=i+1xij∂xi+1,j+i−1Xj=1xji∂xji−1+µixii.Here xij = 0 unless 1 ≤i ≤j ≤n.The matrix e1n+1 may be written ase1n+1 := [· · · [E1, E2], · · ·], En].Using the above formulas one proves the followingLemma 2.1πµ(e1n+1) = −∂x1nFrom now on till the end of this section we omit writing πµ identifying Liealgebra elements with their images under πµ.The action of the Lie algebra ˆg on a function on F 0 × C∗is determined bythe evaluation map g ⊗zk 7→zkg. In particularF0 = e1n+1 ⊗z−1 7→−z−1∂x1n.The result of exponentiation of these formulas is given byLemma 2.2 .

Solutions to KZ equations6(i) If µ = 0 then1) exp(−tF0) : xkl 7−→ z−1t + x1n(k,l)=(1,n)xklotherwise2) exp(−tFi) : xkl 7−→xki1+xiitfor l = i−(xki −xki−1xii) t + xki−1for l = i −1xilt + xi+1lfor k = i + 1xklotherwise(ii) Genericallyexp(−tFi).ψ(x) = (1 + xiit)µiψ(x′), 1 ≤i ≤nwhere x′ is given by the substitution of the item (i) while action of F0 is inde-pendent of µ.ProofIf µ = 0 then all F’s are vector fields. The problem of evaluating an exponentof a vector field is, actually, a problem of the theory of ordinary differentialequations: the exponent of a vector field is an element of a 1-parametric familyof diffeomorphisms generated by the vector field and, therefore, is given by ageneral solution to the corresponding system of o.d.e.’s.

In our case the systemand the solution are (resp. ):1):˙xkl = δk1δlnz−1 =⇒x1n = z−1t + x1n;2):˙xii = x2ii=⇒xii =xii(0)1−xii(0)t˙xji = xiixji=⇒xji =xji(0)1−xii(0)t˙xji−1 = xji −xji−1xii=⇒xji−1 = (xji(0) −xji−1(0)xii(0)) t + xji−1(0)˙xi+1j = −xij˙xij = 0 for j ≥i + 1=⇒xij = xij(0)=⇒xi+1j = −xij(0)t + xi+1j(0),which completes proof of the item (i).

As to the item (ii), one shows that anyorder 1 diffrential operator is conjugated to a vector field by the multiplicationby a function it annihilates. This implies (ii) since Fi · (x−µiii) = 0.

Q.E.D.Now by using all this one can calculate the integrand of (5).But to formulatethe result it is convenient to give some more notations.SetT = (tij) :=x11· · ·· · ·x1n1...............01xnn;

Solutions to KZ equations7for 1 ≤ik ≤ik−1 ≤· · · ≤i1 ≤j , j + k ≤n + 1,Iji1,i2,···,ik := {j + 1 −i1, j + 2 −i2, · · · , j + k −ik}Jjk := {j, j + 1, · · · , j + k −1}T ji1,i2,···ik := (tij)i∈Iji1,i2,···,ik,j∈JjkQji1,i2,···ik :=det(T ji1,i2,···ik)for k > 01for k = 0Introduce a collection of functions on the big cell along with an ordering onit.Definition We writeQji1,i2,···,ikFl−→Q′def⇐⇒exp(−tFl)Qji1,i2,···,ik =(11+xjjtnQ′t + Qji1,i2,···,ikofor l = jQ′t + Qji1,i2,···,ikfor l ̸= jIn the definition it is assumed that µ = 0.Lemma 2.31)Qji1,i2,···,ikFj+r−1−ir−→Qji1,i2,···,ir+1,···,ikfor 1 ≤r ≤k2)Qji1,i2,···,ikFj+k−→Qji1,i2,···,ik,1for k ≥03)Qjj,i2,···,in+1−jF0−→(−1)n−jQji2−1,···,ik′ −1z−1wherek′ −1 := ♯{r : r > 2, ir > 1}Otherwise,QFl−→0The proof of this lemma is a standard calculation of linear algebra usingLemma 2.2; in particular we use Laplace expansion of a certain determinant toprove 2).The above definition suggests to introduce the following n+1−colored graphΓ. The set of vertices of Γ is the set of all Q ̸= 0 such that1Fj1−→Q1Fj2−→Q2 −→· · · −→Qr−1Fjr−→, Qfor some j1, .

. .

, jr. It follows from Lemma 2.3 that each vertex is of the form(−1)(n−j)rQji1,i2,···ikz−r.

Define a function on the set of vertices byl((−1)(n−j)rQji1,i2,···ikz−r) = (n + 1)r +kXp=1ip.

Solutions to KZ equations8Two vertices P, Q are connected by an edge of the color i if and only ifPFi−→Q.With any vertex Q ∈Γ associate a set P(Q) of all oriented paths connecting 1and Q.Lemma 2.4 All γ ∈P((−1)(n−j)rQji1,i2,···ikz−r) are of the same lengthl((−1)(n−j)rQji1,i2,···ikz−r).ProofLemma 2.3 shows that if there is an edge going from P to Q then l(Q) =l(P) + 1. The lemma now follows from the obvious remark that l(1) = 0.Q.E.DWe are in a position to write down the integral (5).Recall that W isthe Weyl group of bg and w = rm1rm2 · · · rml ∈W is a decomposition (notnecessarily reduced),where rm denotes the reflection at the corresponding sim-ple root (αm).m can be viewed as a map from I1(= {1, 2, · · ·, l}) to I2(={0, 1, · · ·, n}).Therefore,m−1(j), j ∈I2, is a subset of I1.

Set,βj = 2(rml+2−j · · · rml · λ1, αml+1−j)(αml+1−j, αml+1−j)+ 1, 1 ≤j ≤landKw(t1, t2, · · · , tl) =lYj=1Γ(−βj)−1 × {exp(−tlFml) · · · exp(−t1Fm1)lYj=1t−βj−1j1},where 1 is viewed as an element of V1 ⊗· · · ⊗VN equal to the unit function onthe product of N copies of the flag manifold. With any pathγ : 1Fj1−→Q1Fj2−→Q2 −→· · · −→Qr−1Fjr−→Q, r = l(Q)associate a polynomial in t′s:fγ(t) =Xp1<...

( This is the only point where the decomposition w = rm1rm2 · · · rml enters thecalculation.) Denote by Γj the subgraph of Γ consisting of all vertices connectedwith Qj1 by an oriented path.

It is equivalently defined as a subgraph generatedby all vertices (−1)(n−j)rQji1,i2,···ikz−r with the fixed superscript j. SetP jw(x, z; t1, t2, · · · , tl) =lXl′=0XQ∈Γj:l(Q)=l′QXγ∈P(Q)fγ(t). (6)

Solutions to KZ equations9Theorem 2.5Kw(t1, t2, · · · tl) =lYj=1Γ(−βj)−1NYp=1nYj=1{P jw(x(p), zp; t1, t2, · · · , tl)}µ(p)jlYj=1t−βj−1j,(7)where µ(p) = (µ(p)1 , . .

. , µ(p)n ), 1 ≤p ≤N, is a highest weight of Vp and x(p), 1 ≤p ≤N, is a coordinate in the p−th copy of the flag manifold.This theorem can be proved by induction on l using Lemma2.3 and Lemma2.4.Let M be the local system of continuous branches of Kw(t1, · · · , tl) over theDomain of Kw(t1, · · · , tl)(say D ).Then finally we obtainTheorem 2.6 For any σ ∈Hl(M, D),the integral◦ΨZσKw(t1, t2, · · · , tl)dt1dt2 · · · dtlsatisfies the system KZ(λN+1, w · λ1)Remark.

Theorem 2.6 gives solutions as an integral over a certain cycledepending on parameters (x, z). These cycles belong to a homology group ofa complement to a collection of hypersurfaces Kw(t1, t2, · · · , tl) = 0 with coef-ficients in a local system defined over this complement.

Note that generically(l > 2), and much unlike the case of Schechtman-Varchenko integral formulas,Kw(t1, t2, · · · , tl) = 0 is a union of hypersurfaces not isomorphic to hyperplanesand, therefore, investigation of the integral cannot be carried out by usual meth-ods. We have already encountered with the same phenomenon in a different butrelated framework.

As we argued in the Introduction, our integral formulasare intimately related to ˆg or g−modules extended by complex powers of a Liealgebra generators. Rigorous treatment of such modules requires considerationof a Lie algebra action on sections of a local system defined over a complementto – highly non-linear – set of “shifted” Schubert cells on a flag manifold; fordetails see [11].Note also that if l = 1, 2 then Kw(t1, t2, · · · , tl) = 0 is isomorphic to a unionof affine hyperplanes and the number of cycles can be calculated using resultsof [5].2.2Some examplesTheorem 2.5 produces rather an algorithm to write down the kernel of theintegral ( 5) than a completely explicit formula for it: ( 7) relies on ( 6), whilethe latter is a linear combination of explicitly given polynomials Qji1...ikz−r withcoefficients in the form Pp tp1 · · · tpr determined by the combinatorial data.

Wehave been able to “resolve” the combinatorial part of the formula in the cases

Solutions to KZ equations10g = sl2, sl3. Although the sl2−case was treated in [15], we discuss here both ina unified way for completeness.The sl2−case.

In this case the flag manifold is CP1, the big cell is C ⊂CP1. Fix a coordinate x on C. Then the matrix T ( via which the polynomialsQji1,...,ik are defined) is given by T = (x).

The set of all Qji1,...,ik consists of 2elements: Q1 = 1, Q11 = x. The vertices of the graph Γ are all of the form:Aǫi(x, z) = z−ixǫ, ǫ = 0, 1, i = 0, 1, 2, 3, .

. .. Further, Γ coincides with Γ1 and isgiven by1F1−→xF0−→z−1F1−→z−1xF0−→z−2F1−→z−2x · · · .Observe that the Weyl group of bsl2 is a free group generated by 2 reflectionsr0, r1 and, therefore, each element is uniquely expanded as eitherr0r1 · · ·orr1r0 · · ·the second one being relevant to our calculation.

Settingw = r1r0 · · · rǫ|{z}l,one obtainsPw(x, z; t1, . .

. , tl) (= P 1w(x, z; t1, .

. .

, tl)) =1Xǫ=0l−ǫXi=0z−ixǫσi+ǫ(t1, . .

. , tl),σj(t1, .

. .

, tl) =X0≤i1

The matrixT is given by T =x11x121x22. The set of all Qji1,···,ik consists of 5 elements:Q1 = 1, Qj1 = xjj(j = 1, 2) Q111 = x11x22 −x12, Q22 = x12.The graph Γ and its subgraphs Γ1, Γ2 are given byΓ1 :Q11F2−→Q111F0−→−z−1F1−→−z−1Q11F2−→−z−1Q111· · ·F1ր1F2ցΓ2 :Q21F1−→Q22F0−→z−1F2−→z−1Q21F1−→z−1Q22· · ·

Solutions to KZ equations11The Weyl group W of csl3 is realized as a group generated by reflections at acertain collection of affine lines on the plane ( see [16] ). These lines producea covering of the plane by triangles, called alcoves, which W acts on effectively.Looking at this action one obtains a collection of elements of W so that a reduceddecomposition of any element of W is contained in it:Put c := r0r1r2(c is called a Coxeter element), then any w ∈W can bewritten as w = scktc−lu where s = e, r2, r1r2 , u = e, r2, r2r1 ,t = r0, r0r1, r0r1r0if kl ̸= 0 and if kl = 0, then t can also be equal to e.Further one obtainsP jw(x, z; t1, t2, · · · , tl) =lXl′=0Ajl′f jl′(t), j = 1, 2,wheref jl′(t) =Xp1

. .

l} →{0, 1, 2}is a function determining a reduced decomposition of w,A1l′ =(−z)−qifl′ = 3q(−z)−qx11ifl′ = 3q + 1(−z)−q(x11x22 −x12)ifl′ = 3q + 2,A2l′ =z−qifl′ = 3qz−qx22ifl′ = 3q + 1z−qx12ifl′ = 3q + 2.References[1] Knizhnik V., Zamolodchikov A., Nucl.Phys. B 247 (1984) 83 - 103[2] A.Tsuchiya, Y.Kanie, Adv.

Stud. Pure Math.

16 297-372[3] V.Schechtman, International Mathematics Research Notices 3 (1992), 39-49[4] V.Schechtman V., A.Varchenko, preprint, Max-Planck-Institut fur Mathe-matik MPI/89-51, 1989, Letters in Math. Phys.

20 1990, 279-283[5] V.Schechtman , A.Varchenko, Invent.Math. 106, 139-194[6] V.G.Kac, M. Wakimoto ,Proc.

Nat’l Acad. Sci.

USA 85 (1988) 4956-4960[7] B.Feigin, V.Schechtman, A.Varchenko, Letters in Math.Phys. 20 (1990),291-297

Solutions to KZ equations12[8] Furlan P., Ganchev A.Ch., Paunov R., Petkova V.B., Phys.Letters 267(1991) 63-70[9] Furlan P., Ganchev A.Ch., Paunov R., Petkova V.B., preprint CERN-TH.6289/91, accepted for publication in Nucl.Phys.B[10] Awata H.,Yamada Y., KEK-TH-316 KEK Preprint 91-209, January 1992[11] B.Feigin, F.Malikov, preprint RIMS-894 September 1992, to appear in Ad-vances in Sov.Math. [12] B.Feigin , E.Frenkel , Usp.Math.Nauk (=Russ.Math.Surv.) 43 (1988) 227 -228 ( in Russian )[13] P.Bowknegt, J.McCarthy, K.Pilch, Progress of Theoretical Physics, Supple-ment No.102 70 (1988) 67-135[14] H.Awata, A.Tsuchiya, Y.Yamada, Nuclear Phys.

B365 (1991) 680-696[15] Malikov F.G., Feigin B.L., Fuchs D.B., Funkc.Anal.i ego Pril.(=Funct.Anal. Appl.) 20(1988) 2, 25-37 (in Russian)[16] Kac V.G.

Infinite-dimensional Lie algebras,Cambridge University Press1990

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