Commun. Math. Phys. 134, 487-508 (1990) Communications IΠ

MathematicalPhysics

© Springer-Verlag 1990

The Structure of the W^ Algebra

I. Bakas*

Center for Theoretical Physics, Department of Physics and Astronomy, University of Maryland,College Park, MD 20742, USA

Received November 29, 1989

Abstract. We prove rigorously that the structure constants of the leading (highestspin) linear terms in the commutation relations of the conformal chiral operatoralgebra Wm are identical to those of the Diffo 1R2 algebra generated by areapreserving diffeomorphisms of the plane. Moreover, all quadratic terms of the WN

algebra are found to be absent in the limit N -> oo. In particular we show that W^is a central extension of DiίTo 1R2 with non-trivial cocycles appearing only in thecommutation relations of its Virasoro subalgebra. We also propose a representationof W^ in terms of a single scalar field in 2 + 1 dimensions and discuss its significancein the context of quantum field theory.

1. Introduction

The construction of all unitary highest weight representations of the infinitedimensional symmetry algebras that arise in two dimensional conformal fieldtheory has provided a non-perturbative framework for solving a large class ofphysically interesting quantum field theory models (see for instance [1] andreferences therein). One of the most striking results in the classification of rationalconformal field theories was the realization that simple Lie algebras determine thestructure and the operator content of unitary scale invariant 2-dim systems. Inthe chiral operator approach, rational conformal field theories are described asminimal models of extended conformal symmetry algebras W generated by thestress-energy tensor T(z) and other holomorphic fields {ws(z),seJ}, which areassociated with additional conserved currents in the 2-dim world. Typically, thegenerators of ^-algebras are labeled by the vertices of Dynkin diagrams of simpleLie algebras G, which also determine the conformal weight (spin) s of the chiralfields ws(z). The Virasoro algebra

[T(z), T(z')] = (T(z) + T(ϊ))δtZ(z - z') + ~ δ,zzz(z - z') (1)

Supported in part by the NSF grant PHY-87-17155

488 I. Bakas

is the most elementary member in the family of conformal algebras and is associatedwith the simplest Lie algebra Alf

Ί^-algebras are closed (in the sense that they satisfy the Jacobi identity) andhave determining relations which are in general quadratic in nature. Nevertheless,their unitary representations can be constructed in analogy with the Verma modulerepresentations of the Virasoro and Kac-Moody algebras and correspond tounitary conformal field theories with discrete symmetry, depending on the simpleLie algebra G [2]. Using a suitable generalization of the Feigin-Fuksconstruction, it follows that the full spectrum of the anomalous dimensions of the^-invariant primary fields for these theories is expressed in terms of thefundamental weights of G and forms a set of rational numbers. In this fashion,the bootstrap approach to quantum field theory yields a systematic way tounderstand the vacuum structure of string theory as well as construct a periodictable of all types of criticality in 2-dim statistical mechanics, parallel to the Cartanclassification of Lie algebras.

In what follows, we focus our attention on the operator algebra WN which isassociated with the Lie algebra AN_^ and study the details of its structure in thelimit N -* oo. WN is generated by the stress-energy tensor T(z) and a collection ofprimary conformal fields (ws(z); s = 3, 4,..., N} with integer spin s. The interest inthe large N limit behavior of extended conformal symmetries arose from someearlier preliminary considerations which showed that the leading (highest spin)linear terms in the commutation relations of W^ describe an infinite dimensionalsubalgebra, Diffo R2, of all area preserving diffeomorphisms of the plane [3]. Inthis paper we present the details of the proof and show furthermore that thequadratic and the rest of the terms in WN vanish in the limit N->oo, with theexception of central cocycle terms in the Virasoro subalgebra of W^. Our resultsheds some new light in the relations between conformal algebras and areapreserving diffeomorphisms of 2-manifolds that arise as residual symmetries in thelight-cone formulation of membrane theories [4]. It also suggests a conformal fieldtheory approach to the representation theory of Diffo R2 which might be of (some)value in mathematics as well as quantum field theory. The proof we present hererelies on the Hamiltonian description of ^-algebras using the Gelfand-Dickeyalgebraic structure of integrable non-linear differential equations of the KdV type.

There are several motivations and objectives in our program. First notice thatdue to the quadratic nature of the determining relations of WN, there is no naturalgeometric interpretation of higher spin fields with 5 > 2. This problem arises in allhigher spin theories (see for instance [5]) and makes difficult the construction ofconsistent self-interacting gauge theories of massless higher spin fields (in anynumber of dimensions). One possible resolution to the problem is provided by theinclusion of an infinite family of particles with all possible spins. In our case thisprocedure leads to the operator algebra W^ while the connection we find withthe symmetry algebra of area preserving diffeomorphisms assigns a definitemeaning to the role that 2-dim higher spin chiral fields have in geometry.

Second, as far as strings are concerned, we would very much like to have anon-perturbative framework for studying their quantization. Since there is no suchprescription available at the moment, it seems natural to "play it by ear" andadopt some ideas and techniques from the 1/JV expansion of SU(N) Yang-Millsgauge theories (see for instance [6] and references therein). Motivated by the

Structure of the W^ Algebra 489

behavior oϊSU(N) => SΊ/(3) gauge theories in the limit N -> oo and their descriptionin terms of loop dynamics, it is natural for our purposes to embed conventional(W2)-stήngs in a much larger theory (W^-strings) with WN => W2 as the symmetryalgebra on the world sheet. This generalization may accommodate new prospectsin building string theories [7] and in analogy with SU(ao) gauge theories,^-strings could be used for developing a non-perturbative approach to stringquantization itself. Certainly, the association of W^ with the algebra of areapreserving diffeomorphisms we establish in this paper suggests that higherdimensional extended objects (e.g. membranes) could be employed in theformulation of VF^-string theory. This construction is still far from being completedand constitutes a long term goal in our investigation.

Third, we would like to know whether the bootstrap (operator algebra)approach to quantum field theory can be applied successfully to theories in morethan two dimensions. Although we do not have any systematic procedure availablein our disposal for d > 2, we find that a certain class of 3-dim field theory modelscan be approximated by 2-dim conformal field theories that possess an infinitecollection of additional conserved currents with all integer spins. To be moreprecise, we think of the Dynkin diagrams that describe the operator content of^"-algebras (and only for classical non-exceptional simple Lie algebras) as somesuitable discretizations/skeletonizations of a continuous third dimension inspace-time. Passing to the limit N -> oo we effectively obtain a field theory in onedimension higher, provided that the large N limit of the underlying simple Liealgebra (e.g. A^ for W^) is defined as a continual algebra, in the nomenclature ofreference [8]. In this sense, we manage to describe (genuine) higher dimensionaltheories in terms of lower dimensional physics using infinite dimensional structuregroups. We will return to this point later, while discussing the geometricinterpretation of W^ as a subalgebra of the area preserving diffeomorphisms of R2.

Incidentally, we point out that the idea to use 2-dim models with infinitedimensional structure groups for the description of higher dimensional theories,has already been adopted by Atiyah in his work on the moduli space of Yang-Millsinstantons [9]. In particular, 4-dim self-dual gauge connections with values inG and with topological charge /ceZ, can be thought of as instantons of a 2-dimprincipal chiral model with values in ΩG, the loop space of G. Also, recent workin general relativity has shown that in many aspects, gravity resembles ordinarygauge theories with infinite dimensional structure groups. For example, theself-dual Einstein equations for gravitational instantons are equivalent to Nahm'sequations for BPS monopoles with G^Diff0Z t ( 3 ), the volume preservingdiffeomorphism group of the 3-space J£(3) [10].

We think that all these reasons are very compelling for justifying the study weundertake in this paper. Certainly, the results we obtain in the sequel do notanswer in detail all the questions we have raised; they should be thought of as amodest first step toward the general direction we have outlined. Here is a briefdescription of the way that our material is organized. In Sect. 2 we set up thenotation and review the Hamiltonian formulation of extended conformalsymmetries in terms of the Gelfand-Dickey Lie-Poisson algebra of formalpseudodifferential operators. We find this framework most appropriate andeffective for studying the large N limit behavior of WN. In Sect. 3 we present theresults of somewhat lengthy computations which establish the advertised relation

490 I. Bakas

between Wx and the algebra of area preserving diffeomorphisms of the plane. InSect. 4 we continue this investigation and address the problem of constructingunitary representations of Diff0lR

2 with the aid of 2-dίmZ00-symmetric conformalfield theories. In Sect. 5 we concentrate on the 3-dim description of W^ using acontinual analogue of the Toda field and (Feigin-Fuks) free field representations.Finally, in Sect. 6 we draw our conclusions and indicate some other directions forfuture work.

2. The Chiral Operator Algebra WN

Zamolodchikov's spin N operator algebra WN is generated by the stress-energytensor T(z) and the additional conserved currents (vvs(z)} with spin s = 3,4,..., Nrespectively. Introducing Fourier modes, the commutation relations of WN takethe form [2]

[LM,LJ = (n - m)Ln+m + ^(n3 - n)δn+m^ (2)

[Ln, ws(m)] = [(s - l)n - m]ws(n + m), (3)

[wφ),Mv(jfi)] = £ ZQ''''^^^---.^^)^.^!)..^^). (4){Si £#}{*.}

Here, Ln denote the Fourier modes of T(z) and ws(w) those of ws(z). The identityoperator is included among the generators using the identification / = w0 (i.e.,w0(fe) = δk>0) and the integers {/cj are chosen so that the condition of momentumconservation k± + /c2 + "" + fcp = rc + m is always satisfied. Furthermore, thenormal ordering prescription is implicitly introduced on the right-hand side ofEq. (4). The central charge c is the only free parameter in the operator algebraWN, whose defining relations (2)-(4) are non-linear (quadratic or higher polynomial)and satisfy the Jacobi identity.

WN is clearly a conformal algebra because it contains the Virasoro as subalgebra(see Eq. (2)). Determining the structure constants C for arbitrary N is not an easytask and requires extremely lengthy computations. For any given pair of spins(s,s'X the structure constants C are (in general) different from zero only ifs1+s2 + — + S p £ s + s'-2. We also assume that C(0,0;0,...,0;c) = 0 for all s, s'.The only terms in (4) which are linear in the generating fields {wj occur when all{sj but one (call it s") are zero. Obviously, we have to demand that s" ̂ s + sf - 2,as well as s" ̂ N. Then, for sufficiently large N, the leading (highest spin) linearterm in (4) has s" = s -f s' - 2. Later we will show (among other things) that whenN-> oo, Cs

sfs'~2(n,m; ft -f m;c) is independent of c and in fact

C*sts'~2(H,m;m + m) = n(s' - 1) - m(s - 1). (5)

We mention for completeness that central charge terms occur in Eq. (4) whenSi = s2 = ••• = sp = 0 and C£r° Φ0.

There are several ways to describe the detailed structure of the conformaloperator algebra WN. The most direct method (though not very practical) isprovided by the constraint of Jacobi identities in the commutation relations (2)-(4).This leads to an algebraic system of equations for the structure constants C^},

Structure of the Wm Algebra 491

which turns out to be very complicated to solve when N is sufficiently large.Equivalently, the explicit calculation of the operator product expansion ws(z)ws,(zf)for any two primary conformal fields vvs and ws, requires knowledge of the singularas well as (some) of the non-singular terms in the operator product expansion offields with spin less than 5 and s'. Although the closure of the resulting operatoralgebra puts severe constraints on the values of all possible free parameters(coefficients) that appear, this procedure does not seem to be the most efficient forour purposes.

Alternatively, one may use the free field representation of W-algebras andintroduce a multi-component free massless bose field Φ(z) [1,2]. For WN, thecomponents of Φ, {φa\ a = 1, 2, . . . , N — 1}, have two-point correlation functions ofthe form

(6)

The generators T(z), ws(z) (s = 3, 4, . . . , N) of WN are composite fields of Φ, i.e.,

(7)

ws(z) = £ Mαι -as:dφaι(z) -dφas(z): + derivative terms, (8){ l ^ α ί ^ J V - l }

with appropriately chosen numerical constants Mfll fls.The structure of the operator algebra WN is determined entirely by the Lie

algebra characteristics of A N _ 1 . In particular,

(9)A = ι

is the Weyl vector of AN_ 1 and the N vector hk (k = 1, 2, . . . , N) which are defined by

f>k = 0, Mm = 3*.*-^ (10)k = ι ιy

form an overcomplete system in (N — l)-dim Euclidean space. In this notation,C i

[hk — hk+l\k = 1,2,..., Λ Γ — 1} are the N —I simple roots of AN_± and < £ hk\U=ι

ί - 1, 2, . . . , N - 1 i the fundamental weights. Therefore, p2 = ̂ (N* - N). Also, α0

is a loop counting parameter of the order — 1/αJ ~ ft, i.e. Planck's constant. Theasymptotic behavior of the (N - l)-component free boson Φ(z) is Φ(z) ~ 2iα0p log z,which insures the fall off rate

Γ(z)~z~4; w s(z)~z-2 s(s = 3,. . . , iV) (11)

as z -> oo. The latter condition is necessary for conformal symmetry to be unbroken.Another method for deriving the commutation relations of WN is provided by

the Toda field theory for the Lie algebra AN,1 [11]. In this case, one introducesa collection of scalar fields φa(z9 z) (α = 1, 2, . . . , N — 1), which satisfy the coupled

492 I. Bakas

system of differential equations

ddφa(z, z) = exp (Y Kabφb(z, z)\ (12)\b=l /

Here Kab is the Cartan martix of AN_l. There is always a se_t of Characteristic(N - 1) chiral fields w<+)(dφ f l, d

2φa, . . . , dsφa) and similarly w<->(^«, 32φfl, . . . , 3sφα)

(s = 2, 3, . . . , N) associated with the Toda equations (12). The simplest one,

T(z) := w<2

+) = - 1 X 3ψβ(z, z)Kabdφb(z, z) + Σ 520«(*> 4 (13)

coincides with the holomorphic component of the improved stress-energy tensorof Toda field theory, while T(z):= w(

2~}(z) is defined similarly and yields the

antiholomorphic component. The explicit expression of w(

s

±] with s — 3, . . . , N is alittle bit more complicated and we refer the reader to reference [11] for furtherdetails. Nevertheless, it is important to point out that all the characteristic fieldsw^±} are conserved, i.e. dw4+) = 0 and dw(

s~] = Q, by virtue of the equations of

motion (12). Moreover their conformal weights are s = 2, 3, . . . , N respectively andgenerate two (commuting) copies of the operator algebra WN, one holomorphicand one antiholomorphic, depending on the choice of the + or — signs. Sincewe are considering chiral operator algebras, it is sufficient to concentrate only onthe holomorphic sector generated by (w* + )(z)}.

We intend to return to the free field and Toda field theory representations ofWN when we discuss Wx from a 2 + 1 viewpoint using the continual analogue ofthe algebra A^. Meanwhile, for the actual computation of the commutationrelations of W^ we adopt an alternative (though equivalent) formalism originatedfrom the Hamiltonian description of integrable non-linear differential equationsof the KdV type. More specifically, we are going to use the Gelfand-Dickey algebraof formal pseudodifferential operators, GΌ(AN_1), with the simple Lie algebraAN_l as its label [12], and study the structure of WN in the large N limit, JV-> oo.We point out that Gelfand-Dickey algebras have already been employed quitesuccessfully in the classical description of extended conformal symmetries [13, 14].In this framework, the calculation of the structure constants C^1? becomes just acombinatorics problem. Of course, to make exact contact with the corresponding(quantum) chiral operator algebras of 2-dim conformal field theory, one has tonormal order all quadratic (and higher) terms that appear in the commutationrelations of fields with spin s ̂ 3. Although the structure of the algebra remainsunchanged and consistent with Jacobi identity, the normal ordering prescriptiondeforms (in general) the classical values of the structure constants C. How-ever, this deformation is insignificant when JV-»oo [15] and the classical(Gelfand-Dickey algebra) calculation of W^ yields the exact quantum mechanicalanswer. We will return to this point later.

Next, we review (briefly) the basic theory of Gelfand-Dickey algebras,GΌ(AN_1), associated with the ^-series of simple Lie algebras (see [12-14] formore details). Let

LN = dN + u2(z)dN-2 + +uN(z) (14)

be an Nth order differential (Lax) operator with N — 1 coordinate (potential)

Structure of the Wx Algebra 493

functions {^(z); i = 2,3,..., JV}. We also consider (formal) pseudodifferentialoperators A(z) = /4_(z) + A + (z) with

/4_(z)= £ dkαfc(z); ^ + (z)= Σ ak(z)dk (15)

and introduce the notation res/4 =<z_1(z). Then, to any local functional/[w2,...,wN], we assign the (formal) operator sum

"-1 , δ/Λ r

/= X (? *- + d NxN(f). (16)

The variable xN(f) is chosen so that the following condition is satisfied:res [LN, Xf~] = 0. With this in mind, the Gelfand-Dickey bracket between any twofunctionals /[M] and g[u\ is defined to be

(17)where

VXf(LN) = LN(XfLN)+ - (LNXf) + LN. (18)

In all formulas, we use Leibniz's rule for the multiplication of operators (bothdifferential and formal). In particular, for all k> 0, the following identities are true:

* k\z)dk-m, (19a)

= o\m

oo /If i m __ 1 \

d~ka(z)= £ (-H }a(m\z)d-k-m. (19b)m=o \ m J

It was shown in references [13, 14] (see also [2]) that under the GD-bracket(17), the coordinate functionals u2, «3, . . . , UN form a closed conformal algebra withquadratic determining relations. In this case, the Virasoro subalgebra is generatedby u2(z),

{u2(z)9 u2(z')}N = (u2(z) + u2(z'))δtz(z - z') + ^(N3 - N)δ,zzz(z - z'\ (20)

and the central charge is c = I 2 p 2 . However, the rest of the coordinate fields us(z)(s = 3, . . . , N) are not primary. Primary conformal fields ws(z) with spin s = 3,...,Nare obtained using appropriate (polynomial) combinations of all (tφ); i ̂ s} andtheir derivatives. They are of the (general) form

HΦ)= Σ A^.^W u^z), (21){i},{k}

with kl + ••• + kp + /! -f •-• + ip = s ̂ N. Then, T(z):= u2(z) and (ws(z); s = 3,...,N}generate the extended conformal symmetry algebra WN, as desired. All thecalculations are fairly straightforward but quite lengthy. In fact, the numericalconstants A{^.{k} in Eq. (21) are not easy to compute for arbitrary N. Explicit resultsare only available for small values of N, e.g. 2, 3, 4. Nevertheless, we find thatwhen N -> oo the structure of the operator algebra WN simplifies considerably andexact expressions for all structure constants become available. This is the subjectof the next section.

494 I. Bakas

3. The Algebraic Structure of W^

According to Eq. (16), the (formal) operators

*MZ') = SS~N- lδ(z - z') + d'NxN(us(z')) (22)

are assigned to the coordinate functions us with s = 2,3,..., N. The variables XN(US)are not arbitrary. Recall that the definition of the GΌ(AN_1) algebra imposes theconstraint res [LN, XUs] = 0, for all values of s. This condition is equivalent to thefollowing differential equation:

x>s(z')) = (~Ϋ ~ - Z>P(Z))(%P+* (23)Λ V p = θ f c = l \ K /

Here MO(Z) =l,uί(z) = Q and the derivatives are taken with respect to z. Then, theGelfand-Dickey bracket (17) between any two (coordinate) fields us and us, is givenby

K(z), us(z')}N = $dz(A + B + C + D), (24)

where

m + 1 -(δ(z - z)up(z}}*\δ(z' - z)ιv(z)f+m)

m j " '(25a)

with p, k subject to the restriction p + k ̂ s — 1,

N oo s- l - fc-p

B- Σ Σ Σ (-r

's-l-p-i

m(25b)

with p, fc subject to the same restriction p + k 5Ξ s — 1,

c=- Σ s 1 Ys~ί~1)w,+p +*+ιM^(z-3«p (s))Wδ(z'-2) (25c)p,p' = 0 fe = 0 \ k /

with p subject to the restriction p ̂ s — 1, and

J>= - Σ S I 1(S"^~1V..P + P'+*M^(Z-^^^ (25d)

with p subject to the same restriction p ^ s — 1. We have w0(z)= I,w1(z) = 0 asbefore and all derivatives in (25a-d) are taken with respect to the integrationvariable z.

With this result in our disposal we study now the leading behavior of thecommutation relations (24). Notice that for sufficiently large N (i.e., N ^ s + s' — 2)the spin of all leading linear terms is s + s' — 2. The term (25a) will provide a

Structure of the Wm Algebra 495

contribution ~ us+s, _ 2 only if (p = 0; p' = s + sf - 2) or (p = s + s' — 2; p' = 0). Sincep -f fe ̂ 5 — 1, the second possibility is immediately ruled out. On the other hand,the constraint k + k' + m = s + sf — p — ρf — 1 = 1 implies that either (k = 1;k' = m = 0) or (kf = 1; k = m = 0) or (m = 1; k = kr = 0). Therefore, the total highestspin linear contribution from A is

- (N - s + l)δ f(z - z)δ(z' - z)us+s,_2(z)

-(N-s' + l)δ(z-z)(δ(z'-z)us+s,_2(z)\2 (26)

+ (5 - l)<5(z - z)(δ(zf - z)us+s,_2(z)\;.

It is easy to see that the second and fourth terms (25b, d) do not contribute at allto this order, while the only relevant term in (25c) has p = 0, p' = s + 5' - 2, k = 1and equals to

- (s - \}(δ(z - z)us+s,_2(z)\;δ(z' - z). (27)

Putting the expressions (26), (27) together and performing the necessaryintegrations, we obtain the following result (for N ̂ 5 + 5' — 2):

{us(z\II,,(Z')}N = [(s - lK+s,_2(z) + (sf - lK+,.2(z')]5 ,(z - z')

+ (lower spin terms). (28)

The lower spin terms are local functionals of {ut(z)} with i<s + s' — 2. Theirstructure is determined by all other terms in Eqs. (25a-d) and turns out to bequite complicated indeed.

It is clear that when N-> oo, most of the numerical coefficients that appear inthe commutation relations (24) diverge rapidly. For example, the central charge cof the Virasoro algebra (20) blows up to infinity like oo3. However, the infinitieswe encounter here are not characteristic of the chiral operator algebra W^. Theyoriginate from the normalization of the fields (ws(z); s = 2,3,..., N} used in thedefinition of the Gelfand-Dickey algebra GΌ(AN_ί). At this point we realize thatthe standard choice (14), (18) of the operators LN and VX(LN) is rather special,because the value of the central charge c = Λ/"3 — N is fixed. On the other hand weknow that the commutation relations of WN allow for arbitrary values of c.Therefore, to obtain a (classical) Hamiltonian description of WN for all values ofN and c, it is necessary to modify our definitions by introducing appropriaterescalings in the (basic) variables of the theory.

First we consider the Virasoro subalgebra (20) and define the quantities

Lv = — { , }N (29)C

for all c Φ 0. It follows that

[u2(z\ u2(z'KN = (U2(z) + u2(zf))δft(z - z') 4- ̂ 3^* «,(* - *') (30)ΛP 12

with arbitrary central charge for all N = 2,3,..., oo. For consistency, we also haveto rescale the rest of the generating fields {us(z)}. Recall that in the free fieldrepresentation of WN9 conformal fields of weight s are ~ (3Φ(z))s. Then, for s = 2,

Eq. (29) implies that dΦ(z) has been modified by a factor of ^/c/ΛΓ3. Therefore, it

496 I. Bakas

is natural to define

/ c Y/2

us(z) = ( — } us(z\ for s = 2,3, . . . ,Λf . (31)\N3J

This is the only consistent prescription applicable to all values of N9 including oo.Moreover, it does not affect the coefficients of the (leading) highest spin linearterms in Eq. (28). From now on, we adopt the rescaled variables (ύs(z)} in ourdescription of WN and its large N limit. For convenience (and to simplify thecalculations) we may choose c = 1; arbitrary values of c will be considered in thenext section.

Our next task is the computation of all terms in the commutation relations[us(z\ύs,(z'}~]^ w*tri arbitrary s and s'. Using Eqs. (24), (25) and with the definitions(29), (31) in mind we obtain:

08._2(z')δSt2)δ"'(z - z') + &δs,2δs,t2δ'"(z - z')

1 {s^p + 2}

+ ̂ Σ (s'-l-p)5^ P,P'^2

+ ύp(z')ύp,(z'))δ'(z-zf)

- J{S=Σ 2}(^- 1 -P)^2 P,p'^2

1 { s Z p ' + 2}

-~ Σ (5/- ! -p)δ

+z Σ (s'-i-p^^.-a.^^^ ωβ^ω-M2)";^))^-^).^ p,p'Z2

(32)

Few clarifying remarks are in order. The final result (32) has been organizedso that the subscript (spin) of all contributing fields is strictly ^ 2. For examplethe second term on the right-hand side (~ ύs_2) is present only if s — 2 ̂ 2; similarrestrictions apply to all other terms. The derivatives are always taken with respectto the variable z and not z'. The calculation is straightforward but quite lengthy.We find that many individual terms in (25a-d) diverge when N -» oo (even afterintroducing the rescaling (29) and (31)); however, their net contribution to thebracket [us(z\ϋs,(z')']^ is zero. We also point out that some of the quadratic termsin Eq. (32) are not manifestly antisymmetric; nevertheless one can easily verify thatthe total expression we have obtained is in fact antisymmetric, as required. Alsothe Jacobi identity is automatically satisfied, because Gelfand-Dickey algebrashave been designed this way.

Structure of the WΛ Algebra 497

The commutation relations (32) provide a Hamiltonian description of theextended conformal algebra Wm. Setting s = s' = 2, we obtain the Virasoro(sub)algebra with c = 1. Unfortunately, the generating fields {ίϊs(z); s^ 3} are notprimary (in general). We have

[S2(z), ύs(zf)^ = lu,(z) + (s - l)u,(z>W(z - z') + ^us-2(z')δ'"(z - z'} (33)

for all s^4. Only ύ3(z) is a primary field with conformal weight 3. Also, thedetermining relations (32) are quadratic in nature. The simplest example wherequadratic terms occur is provided by [w3(z), us(z')']σo with s ̂ 3. In this case we have

- (fi2(z)fi,_ ,(Z) + U2(Z')US. tf))δ'(z - Z'}

z)ύs^(z] - U2(z)ύ's_ ,(z))δ(z - z'\ (34)

Notice that the right-hand side of Eq. (34) involves linear terms with spin higherthan s as well as quadratic terms. This is a characteristic behavior of higher spinalgebras, in any number of space-time dimensions.

The standard attitude in higher spin theories is that consistent self-interactinggauge theories of massless particles with s > 2 require the inclusion of an infinitefamily of fields with all integer spins s = 2,3,...,ΛΓ->oo. In terms of thecorresponding infinite dimensional algebra of gauge transformations this meansthat all quadratic terms should disappear when N-+OO. On these grounds weexpect that only the leading (highest spin) linear terms will actually contribute inthe large JV limit of WN. Certainly, this behavior is not manifest in Eq. (32).Nevertheless, we are going to show that all quadratic terms are trivial, in the sensethat there exist fields

w,(z) = δ,(z)+ £ ^^fiίΐ^ δ^ίz) (35)ttu*}

with linear commutation relations. A{$} are numerical constants which will becalculated shortly. The integers {i} and {k} are not arbitrary but satisfy theconstraint kί + — h kp + i l + — h ip = s, for all 5 = 4, 5, . . . . We also assume that4}8} = 0 i f j > = l.

At this point it is instructive to compare Eq. (35) with (21). Recall that thecoordinate (Lax) fields {us(z)} of the Gelfand-Dickey algebra GΌ(AN_1) are notprimary in general. However, for appropriate choices of the structure constantsA$}9 the polynomial combinations (21) yield primary conformal fields ws(z) ofweight s. The classical (Hamiltonian) description of WN (for all N = 2,3,...) isformulated entirely in terms of the field variables {ws} and not {us} (see [1 1, 13, 14]).For this reason we need to know which are the appropriate combinations toconsider for the generators of W^. It is quite remarkable that the choice (35) notonly eliminates all non-linear terms, but also provides the primary conformal fields(generators) of W^. The calculations are once more a little bit complicated; weonly indicate how to proceed and then state the result for the general form of thenumerical coefficients A(^}.

The key formula is provided by the commutation relations (34), which playthe role of a "generating function" for all ws(z) with s ̂ 4. To be more specific, westart with w3(z):= ύ3(z) which is a primary conformal field when N-* oo. We find

498 I. Bakas

that

[w3(z), w3(z')]00 = 2(w4(z) + w4(z'))<5'(z - z'), (36)where

w4(z) = δ4(z)-iδl(z). (37)

Next we calculate the bracket between w3(z) and w4(z'). We find that

[w3(z), w4(z');L = (2w5(z) + 3w5(z'))5'(z - z'), (38)where

w5(z) = w5(z)-α2(z)M3(z). (39)

Iterating this procedure we obtain a whole tower of fields of the form (35). Thenext few members are given by the following expressions:

w6 = M6 - tϊ2w4 - \ύ\ + \ύ\, (40)

w7 = UΊ - u3iί4 - U2u5 + M 2w 3, (41)

W8 = W

8 - U

2U6 - U

3U

5 - \U\ + M

2M3 + W

2W

4 - i«2» (

42)

w9 = ίϊ9 — u2ίϊ7 — ίϊ3M6 — u4M5 H- M|MS 4- 2w2M3M4 4 ^w3 — w|u3, (43)

W i o = W l O ~ Ms ~ M? ~ Mβ ~ i"5 + "2^6 + "2«4

4- 2M2w3w5 4 ulu4 - ύ\ύ± - \u\u\ 4 \u\ (44)

and so on. For simplicity we have suppressed the z-dependence of the variables ύ.All these fields are designed so that for all s ̂ 3 we have

[w3(z), w^z')^ = [2ws+ x(z) + (s - l)ws + 1(z')]5'(z - z'). (45)

It is interesting to notice that no derivative terms appear in the expressions for{ws}. Therefore, the structure constants A$} in Eq. (35) are independent of {fe}. Itis very straightforward now to guess the general formula for ws with arbitrarys ̂ 4. It is given by

- * » - " ? ' ' w w " ' W A (46)

with riiii 4- n2i2 4 — h npip = s. We emphasize that here, the integers n1 , n2, . . . , np

denote powers and not derivatives of the fields uil9ui2,...,uip. Also the subscriptsiι,i 2» » ϊ p (spins) are not ordered in any particular way and their values are all^ 2. For example, to obtain w7, we partition 7 as <7 1 >, < 3-1, 4-1 >, <2 1, 5-1 > and<2 2, 3 1>. Then, we write down the corresponding polynomial combinationsM7,£3M4,M4M3,M2M5,M5M2, M2M3, M2u3w2, M3M2 and weight them with + 1, - 1, -£, -|,— j, 4j, +j, 43- respectively. Summing up all these terms we recover Eq. (41).Similarly we construct any other field ws that satisfies the relation (45).

Using the expression (46) and the commutation relations (32) it is possible toshow that

[wβ(z), *,.(/)]„ = [(5- l)ws+s,_2(z)4(s'- l)uw_2(z')]<5'(z-z')

<5S',2<Πz-z') (47)

is true for all values of 5 and s' = 2, 3, 4, ____ Therefore, the structure of the extendedconformal algebra W^ is linear and the assertion we made earlier is proven. From

Structure of the W^ Algebra 499

now on we concentrate on Eq. (47) and discard the intermediate variables ύ usedin the course of this investigation.

The final piece of information comes from a recent paper by Bilal [15]. It wasshown that in the JV-»oo limit, the quantum (commutator) algebra of (w5(z);s = 2,3,..., N} reduces to the classical (Poisson bracket) algebra W^. In particular,the Gelfand-Dickey calculation provides the dominant contribution to thestructure constants of WN when N is large. To put it differently, the normal orderingprescription deforms the numerical coefficients in (25a-d) only by a negligibleamount which has no effect on the final result for [w^zXw^z')]^. It is then safeto conclude that the chiral operator algebra W^ of two dimensional (quantum)conformal field theory is described precisely by the commutation relations (47).

24. Unitary Representations of Diffg R

Introducing Fourier components ws(n) (neZ) for the fields w5(z) (s = 2, 3,...), weobtain

K(n), nv(m)] = [(s' - l)π - (s - l)m]ws+s,_2(n + m) + ̂ δn + m^2δs^2) (48)

with c = 1. Arbitrary values of the central charge will arise if we make use of Eqs.(29) and (31) with c Φ 1 in general. For this reason we may consider Eq. (48) asthe (quantum mechanical) commutation relations of the chiral algebra W^ witharbitrary central charge c. For convenience we also drop the subscript (oo) of theGD-bracket (47). The observation that the structure constants of W^ are given byEq. (5) was made first in reference [3], where the leading order (highest spin)behavior of W^ was studied. The present result completes this investigation andshows that no other terms appear in the commutation relations of W^, apart froma central (cocycle) term in the Virasoro subalgebra.

It is quite natural now to look for a geometric interpretation of the infinitedimensional symmetry algebra (48). This is certainly possible because thedetermining relations of W^ are linear. We will see that the right framework forour purposes is provided by the algebra of area preserving diffeomorphisms of the2-plane, Diff0R

2. To be more precise let us consider the Lie algebra of allHamiltonian (i.e., divergenceless) vector fields on R2 with commutation relations

Here, { /, g] denotes the Poisson bracket between any two functions /, g on R2, i.e.

™-ψτ-ττ (50)dx dy dx dy

and

is the Hamiltonian vector field associated with /. The coordinate functions (x, y)on R2 have been chosen so that {x9y} = 1. The symmetry algebra (49) arises in1-dim classical mechanics and generates all canonical transformations on the phase

500 I. Bakas

space R2. In this setting, the variables x and y represent the position and conjugatemomentum (respectively) of a particle with one degree of freedom.

Next, we introduce the basis elements

/m.» = * l"+V+1; weZ (52)

for the generating functions of Diff0R2. It is straightforward to verify that in this

basis the commutation relations (50) become

{fm,n,fm;n } = [(m + D(n' + 1) - (m' + !)(« + l)]/m+m.,n+n, (53)

This is a double graded algebra which contains the Virasoro as subalgebra (withcentral charge c = 0). Indeed, the basis elements {/m0;me2£} satisfy the relations

{/m,o>/m',o} = (m - m')/m+m<,0 (54)

There are many other interesting subalgebras of (53). Some of them have alreadybeen considered in the mathematics literature, in connection with certain problemsin (co)homology [16,17]. Explicit calculations have shown that the homologiesof arbitrary fixed dimension of the Lie algebra of polynomial Hamiltonian vectorfields on R2 are finite dimensional. Also, the subalgebra of polynomial vector fieldsgenerated by all even functions in x and j; has been used extensively in thecohomology theory of differential operator algebras. For our purposes it is naturalto consider the subalgebra of Hamiltonian vector fields

^O} (55)

which are polynomial only in the y variable. We will prove shortly that W^ is acentral extension of Diffo (R2).

Notice that the infinite dimensional space (55) is a module for the Virasoroalgebra whose action is defined as follows:

Λ,o /m,n:={Λ,o,/m.J = ~[(m+ l)-(n+ !)(*+ l)]/m+M. (56)

It can be readily checked that the central charge c is zero. Equation (56) has animportant meaning in conformal field theory. It states that (for fixed n) /m π arethe basis elements of a primary conformal field of weight Δ = — (n + 1). Indeed,

in the holomorphic representation Lk= — zk+i — , primary fields transformdz

according to the rule

Lk(zm+1dzΔ)=-[(m+l) + Δ(k+ l)]zm+k + 1dz4. (57)

Therefore, Diffo ^2 can be viewed as an infinite dimensional higher spin con-formal symmetry algebra. The Fourier components of the generating conformalfields {ws} are /m π and the allowed values for the weight (spin) s are1 - Δ = n + 2 = 2, 3, 4, . . . . (Recall that Δ and 1 - Δ are dual to each other.) Tomake exact contact with the commutation relations (48), we introduce the variables

^2. (58)

Then Eq. (53) yields

{ws(n), Hv(m)} = [(s' - l)n - (s - l)m]ws+s,_2(n + m), (59)

Structure of the W^ Algebra 501

which is identical to Eq. (48), up to central charge terms. We conclude immediatelythat W^ is a central extension of Diffo R2, with non-trivial cocycle appearing onlyin the commutation relations of its Virasoro subalgebra.

This result is quite important for the geometric interpretation of higher spinsymmetries in two dimensions. Area preserving diffeomorphisms are certainly easyto comprehend geometrically. However, we have no good explanation for theirorigin at the moment, other than the algebraic identification of W^ with Diffo R2.Why area preserving diffeomorphisms and not some other infinite dimensionalLie algebra? We think that 2-dim conformal field theories with an infinite numberof additional conserved currents can be interpreted as (2 + l)-dim quantum fieldtheories. The higher dimensional viewpoint we propose in Sect. 5 is very suggestiveand could provide a physical answer to this question.

Diff^ R2 is also interesting from a mathematical point of view. We point outthat the full algebra of area preserving diffeomorphisms of R2 admits the followingdecomposition:

DifΓ0 R2 = Diff + R2 Θ H θ Diffo R2. (60)

Here, DiffJ R2 is generated by {/m>n; weZ, n ̂ 0}, H by {/m>_ x = xm+*; weZ} andDiffo R2 by {/m>ll;meZ,n< —1}. It is clear that H is an abelian subalgebraof Diffo R2, i.e. {/m,-ι,/m',-ι} =0. Also, notice that the Belavin-Polyakov-Zamolodchikov duality between conformal fields of weight A and 1 — A [1] impliesthat

(Diffo R2)* £ Diffί R2 Θ H. (61)

However, "self-dual" fields have weight Δ* = Δ, i.e. Δ = 1/2, which is not integer.Since the basis element {/ I Π f_1;meZ} have Δ = 0, H is not a Cartan subalgebraof Diff0R

2. Also, one may'verify directly that {^Difr^R2}^ Diff^R2 and so(60) is not a Cartan decomposition of Diff0R

2. In fact we have no such candidatefor Diffo R2. On the other hand, area preserving diffeomorphisms of compact2-surfaces, e.g. the torus S1 x S1, share many common features with simple Liealgebras. As we will see later, Diff^S1 x S1) ̂ A^.

Unitary representations of area preserving diffeomorphisms pose a difficultproblem in mathematics. Of course we already know a unitary representation ofDiff0 R

2 from elementary quantum mechanics. It is defined by assigning theoperators

F = ihξf + f-y^ (62)dy

to all classical functions / on R2. ξf is the Hamiltonian vector field (51) and theHubert space where the operators (62) act consists of all square integrable functionson R2, L2(R2). This representation is known as prequantization in the theory ofgeometric quantization (see for instance [18]). Unfortunately, it is not relevant forquantum field theory because L2(R2) cannot accommodate an infinite number ofdegrees of freedom. Nevertheless, it is possible to gain some information aboutthe (field theoretic) representations of Diff^ R2 using the connection we foundwith the symmetry algebra Wn.

It is known that unitary (highest weight) representations of the chiral operatoralgebras WN (N = 2,3,...) correspond to 2-dim conformally invariant models with

502 I. Bakas

(global) TLN symmetry [1,2]. These theories are minimal models of the WN algebras(in the sense that they have a finite number of conformal building blocks) and fallinto classes (series) depending on N. For any (fixed) N, the only allowed valuesfor the central charge c are given by the rational numbers

p = N + l,N + 2,.... (63)P(P + i

Moreover, the spectrum of anomalous dimensions of W^-invariant fields is

Π(Kί9.. 9KN-l9K\9 9KN-'L) — ~ J (64)24p(p + 1)

where {kj and {fcj} are all positive integers subject to the constraints

N-l J V - 1

and {ωj are the fundamental weights of AN_l satisfying

o}i'0}j = 5 for i^j- (66)

It is clear that when N -> oo, the only values of c allowed by unitarity are

r _ Oλr. h _ 1 0 ^ 4 (£Ί\Lk — L*"> Λ — *» A J? M 5 V° ' J

and the number of conformal (building) blocks of all W^-minimal models is infinite.This procedure suggests that highest weight (Verma) module representations ofthe centrally extended Diffo R2 algebra (48) can be constructed in analogy withthe Virasoro algebra. In particular, we introduce a normalized state | f t> (so that</ι |/ι> = 1), with the property

ws(n) |/ι>=0, for n>0, (68a)

ws(0)|Λ> = A J Λ > (68b)

for all s ̂ 2. Then, the Verma module of W^ is obtained by successive applica-tion of the operators {ws(n);s^2, rc<0} on |h>. For central charge ck = 2k(k = 1,2,3,...) the roots of the corresponding Kac determinant (hs=2) will be givenby Eq. (64) in the limit N -> oo. Notice that many of these numbers are zero. Sinceω t - f—hω^_ 1 = p, we have that f ι( l , l , . . . ; l , l , . . .) = 0. Also for certain otherchoices of {kt} and {k'J, the numerator in (64) diverges (typically) like N and thedenominator like N2. However, it is also possible for some of the h's to be infinite.

All structural details in the representations of WN are determined by the simpleLie algebra AN-ίt When Λf-»oo, the number of vertices in the correspondingDynkin diagram becomes infinite and so does the dimensionality of the Cartanmatrix. Then it is natural to trade the infinite number of Lie algebra labels withone continuous parameter. But this is equivalent to introducing an extra dimensionin space-time and reformulating the underlying field theory in 2 + 1 dimensions.Of course, we have to make additional regularizations in quantum theory to avoid

Structure of the W^ Algebra 503

divergencies from the A ^-degrees of freedom. For instance, in the free fieldrepresentation for the stress-energy tensor T(z), the summation over a has to beregulated appropriately when JV->oo.

It is (more or less) obvious now that the systematic study of W^ and its unitaryfield theoretic representations require a higher dimensional viewpoint. In the nextsection we present a general framework for this study and derive some (partial)results. However, the complete picture is still lacking.

5. W^ From a 2 + 1 Viewpoint

The key idea for the rest of our investigation comes from the recent work ofSaveliev and collaborators [8]. These authors proposed a new class of infinitedimensional Lie algebras-the continual generalizations of Z-graded Lie algebras,which have an infinite dimensional Cartan subalgebra and a continuous set ofroots. Their approach incorporates many physically interesting symmetries,including arbitrary current algebras, the algebra of Poisson brackets and thealgebra of vector fields on a manifold (diffeomorphisms). However, for our needs,we only have to consider the large N limit of AN and its continual realization.Somewhat different descriptions of A^ have also been discussed in refs. [19-21].For notational purposes we review first the main constructions of ref. [8] andthen apply them to W^. This way we set the basis for a new representation ofW^ in terms of a single scalar field in 2 -I-1 dimensions.

Let Xi9 Hh Yi (ί = 1,2,..., N - 1) be the system of Chevalley-Weyl generatorsfor A N _! with commutation relations (see for instance [22])

Of course, there is no summation over repeated indices in the equations above— 1) x (N — 1) Cartan matrix of AN_ί9and K is the (non-degenerate)

2__ j

K =

-1

2

-1

-1

2

0 2 -1

-1 2

(70)

It is convenient to introduce a parameter At (which depends on N) and rescalethe generators [Xi9Hi9 Yt} as follows:

j _~ (71)

Then the structure of the defining relations (69) remains the same, with the onlydifference that δy and Ktj are replaced by

-Kίj~ (72)

504 I. Bakas

When N -> oo, it is possible to choose At appropriately so that δ and K becomedistributions of some continuous variable ί. In this case we interpret Eqs. (72) asdiscretized versions of the generalized functions δ(t — t') and 3C (ί, ί') = — δ "(t — tf)respectively. With this prescription in mind, it is natural to introduce aone-parameter family of generators {3Γ(t)9Jt?(t)9<3f(t)} which satisfy the com-mutation relations

'}~] = 0; {X(t\ ®(t')-} = δ(t - t)#(t\ (73a)

(73b)The set of generators (71) is a discretization of {X(t\J^(t\9(t)} and Eqs. (73) canbe thought as the defining (Weyl) relations of A ao. In analogy with finite dimensionalsimple Lie algebras, the rest of the A^ generators are obtained by taking successivecommutators of the $Γ's and '̂s and imposing Serre's condition.

The limiting procedure we adopt here is well motivated by (certain) physicalconsiderations in elementary quantum mechanics. To be more precise, let S[q] =^\dtq(t)2 be the action of a 1-dim free particle propagating in time from ί0 to tf.The quantum mechanical propagator of this system, (q0,t0\qf9tfy, is computedin a standard way by introducing a time skeletonization (ί0, tl = ί0 + Aty . . . , tf =t0 + (N — 2) At). In other words, we think of the Dynkin diagram of AN_^

1 2 3 N-l

0 - 0 - 0 ----- 0 (74)

Co) (tf)

as representing the propagation of a free particle in TV — 2 discrete time steps Atfrom the original to the final configuration. Introducing small fluctuations ξ(t)around the classical path of the system, q(t) = qcl(t) + ξ(t) one finds that

o I <?/ȣ/> = Cexp< -SciM f > where C is a constant written in terms of the

δ2Sdeterminant of Jf(ί,ί')'= - =-δ"(t-t'). When N is finite, detJf is

δq(t)δq(t)approximated by (A t) 3 det X, the determinant of the Cartan matrix (70). To obtainthe complete quantum mechanical answer for the propagator, we pass to thecontinuum limit At-+Q by letting N->oo. Since JΓ(ί,ί') is precisely the definingdistribution of the commutation relations (73), the analogy with techniques usedin the path integral quantization of a free particle shows that Saveliev's descriptionof A oo as a continual Lie algebra is very natural physically.

It is advantageous to adopt the notion of continual algebras and try toreformulate conformal field theories with an infinite collection of additional (higherspin) conserved currents in terms of 3-dim physics. In this case, as we have alreadyindicated, the extra (third) dimension of space-time is associated with thecontinuous set of roots of A^. The picture we envision here is best described bythe field theoretic representations (6)-(13) of WN in the limit N-> oo. Notice thatin the free field as well as the Toda field representations of WN9 the componentsof the scalar field Φ, {φa\ a = 1, 2, . . . , N — 1}, are labeled by the roots of the simpleLie algebra AN-V. Therefore, in the large N limit, it is appropriate to replace theinfinite collection of fields {φa} by a single scalar field Φ(t) which depends on a

Structure of the WΛ Algebra 505

continuous variable t. Of course the z and/or z dependence of Φ(t) has beensuppressed to simplify the notation.

It is necessary now to construct the continual analogue of the free and Todafield theory representations of W^ This way we hope to obtain an intrinsic 3-dimdescription of Diffo R2 and its unitary representations. Using a single scalar fieldΦ(z,z, ί), the Toda field equations (12) for A^ assume the following form:

Φ"M,ί)) (75)

Then it is straightforward to verify that the current

w2(z) := J AΛ'[ - $dΦ(z9 z9 ί) Jf (ί, t')dΦ(zy z9 1') + δ(t - t')d2 Φ(z9 z, f)] (76)

is conserved, i.e. <5w2(z) = 0, as a consequence of Eq. (75). Comparison with Eq.(13) shows that w2(z) can be regarded as the (improved) stress-energy tensor T(z)of the continual A^ Toda field theory with action

S[Φ] = ldzdzdt[^dΦ(z, z, t)jΓ δΦ(z, z, ί) + e* *<*•* •'>]. (77)

Clearly, S[Φ] describes a 3-dim field theory whose equations of motion are givenby (75). Here we have used implicitly the (short-hand) notation

Jf Φ(z, z, ί) = \άϊ JΓ(t9 f')Φ(z, z, tf). (78)

In analogy with the results obtained in ref. [11], it is possible to construct aninfinite list of higherjcharacteristic) fields (ws(δΦ(ί), d2Φ(t\ . . . , 5sΦ(ί)); s = 2, 3, . . .}with spin s, so that <9ws(z) = 0 for all values of s. These fields will certainly providethe representation of W^ we are looking for. Since their existence is guaranteedby the general theory of Toda models, it is reasonable to expect that (ws(z)} ares-fold integrals of the form

-I- higher derivative terms. (79)

t^. . . , ίs) and the coefficients of all other terms are generalized functions writtenin terms of δ(t) and its derivatives. They are the continual analogue of the formulasderived by Bilal and Gervais in [11].

It is unfortunate that the explicit expressions for the generating fields {ws(z);5 ̂ TV} of WN are complicated and quite difficult to derive. Nevertheless we thinkthat the Toda field representation of WN simplifies considerably in the limit N -> oo.In fact we only need to know the continual analogue of the chiral field w3(z),because the rest of the generators (ws(z); 5^4} can be obtained by successiveapplication of the commutation relations (45). Recall from Sect. 3 that Eq. (45)acts as a recursive relation for all generating higher spin fields of the infinitedimensional Lie algebra Wx. This procedure provides a practical and efficient wayto construct any field theoretic representation of W^ once the explicit (continual)expressions for w2(z) and w3(z) are known. However, the results we have obtainedso far are incomplete and we postpone their presentation for future publication[23].

It is clear now that similar considerations apply to the free field representationof W^. For this reason we do not repeat the arguments we gave earlier. However^there are a few additional remarks we would like to make next. It is known that

506 I. Bakas

the free field representation of WN (for JV = 2,3,...) is described in terms of theHeisenberg (oscillator) algebra

ίAa(n),Ab(m)l = nδn+mt0δa^ n,meZ; a,b= 1,2,...,N- 1. (80)

The operators Aa(n) arc the Fourier components of dφα(z) and the system ofconformal fields (ws(z); 5 = 2,3,..., N} belongs in the universal enveloping algebraof (80) [1,2]. Therefore, when JV-> oo, there is an infinite number of Lie algebra(A^) degrees of freedom and additional regularization are required in field theory.This point has to be taken into account when we construct the continual analogueof the Feigin-Fuks (free field) representation, or any other representation of W^.

Finally, it would be interesting to develop a BRST quantization of field theorieswith W^ as the underlying symmetry algebra. It is known that the BRST operator<2 of the chiral operator algebra WN (N = 2,3,...) is nilpotent, provided that thecentral charge of the theory is [24]

c = 2 £ (6s2-65+1). (81)s = 2

Notice that when Λf-> oo, c diverges like oo3 which is not very satisfactory. In thiscase we have an infinite collection of ghost and conjugate-ghost fields (bs(z),cs(z))and each one contributes 2(6s2 — 6s + 1) to the total value of the central charge(81). Here, we also expect that appropriate regularization of the A^ -algebra degreesof freedom will renormalize the unwanted divergencies in the ghost sector of thetheory and produce a finite answer for the "critical" value of c. This problem willbe addressed elsewhere.

6. Conclusions

We have shown that the infinite dimensional symmetry algebra Wx of 2-dimconformal field theory is a central extension of DϋΓ^ IR2. We also proposedrepresentations of W^ in terms of 3-dim scalar field theories. This description,although incomplete for the moment, could help us understand why area preservingdiffeomorphisms arise in the large N limit of extended conformal symmetries. Itis crucial to realize that in this case, the symmetry algebra of area preservingdiffeomorphisms does not refer to the 2-dim world M on which chiral W^~conformal field theories are defined. If that were the case, the transformations(z, z) -> (σ^z, z), σ2(z, z)) with Jacobian J(σγ, σ2) = 1 would mix the (local) light-conecoordinates z, z and violate chirality. Two dimensional theories in the fixed areagauge break scale invariance and therefore fail to be conformal. In order to givea geometric interpretation to the algebra of area preserving diffeomorphismsassociated with W^ one has to introduce an auxiliary surface (membrane) andreformulate the theory in 2 + 1 dimensions. Then, according to Eq. (54), pointcanonical transformations on the membrane induce conformal transformationson M.

It is quite remarkable that both infinite dimensional Lie algebras W^ and A^are described in terms of area preserving diffeomorphisms of 2-surfaces. Thelatter is identified with Diff^S1 x S1), the algebra of area preserving diffeo-morphisms of the torus [8,20,21]. Although we have no good explanation

Structure of the WΛ Algebra 507

for this occurrence, it is tempting to speculate that membrane dynamics couldprovide a non-perturbative framework for string quantization. In particular, itwould be very interesting to construct a consistent theory of strings with W^ asthe conformal symmetry algebra on the 2-dim world-sheet and study its relationwith ordinary H^-string theory. Work toward this direction is in progress.

Finally, we point out that there are several integrable systems in 2 + 1dimensions that may be viewed as 2-dim models with infϊni' iimensional structurealgebras. Some examples (including the Λ^-Toda field thury (75)) have alreadybeen considered in ref. [25]. However, an intrinsic 3-dim description of theirquantization is still lacking. We think that a systematic study of these problemswill improve our present understanding of the relations between 2 and 3dimensional physics. It could also lead to new classes of exactly solvable quantumfield theories, which are quite different from those considered by Witten [26].

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representations of W^. Mod. Phys. Lett. A (to appear)

Communicated by N. Yu. Reshetikhin

Note added in proof. After this work was completed I realized that the tower of higher spin fields(46) can be obtained from the generating function

w(t) = log(l + fl(ί)),

where

w(ί)= Σ ws(z)ίs, u(t)= Σus(z)ts

s=2 5=2

and log(l + x) = x - x2/2 + x3β - x4/4 + .It also became clear that the large N limit of WN algebras is not uniquely defined. It turns

out that the commutation relations of W^ depend on the background charge α0 used in theFeigin-Fuks free field realization of the W-generators. However, different limiting proceduresfor the quantum commutation relations of WN at large N (depending on the choice of α0) donot affect the leading structure (59), but only the subleading terms associated with derivativesof the fields ws(z) and central terms. This is so, because the structure constants of the purelypolynomial w-terms are independent of the background charge and do not deform uponquantization. Since the rescaling (29), (31) does not change the structure constants of purelypolynomial terms, the results of our analysis are valid for them in all cases.

If the background charge is not independent of N but is chosen so that α0 ~ 1/ΛT, in the limitN -> oo the resulting W^ algebra will be well-defined and no rescaling of the generators isnecessary. In this case, the quantum commutation relations will be given by Eq. (59), up tosubleading terms associated with derivatives of the fields ws(z) as well as derivatives of ό-functions.However, it is quite difficult to compute the form of these deformation terms directly using theFeigin-Fuks representation. Recently, Pope, Romans and Shen made an ansatz for the completequantum structure of W^ which admits central terms in the commutation relations of all higherspin fields, by requiring linearity and compatibility with the Jacobi identities [27]. Subsequently,it was shown that their algebra provides an adequate description of W^ (with α0 ~ 1/JV, asN -> oo) using the large N limit of WN minimal models (Z^ parafermions) [28].