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Area-Preserving Diffeomorphisms, w∞Algebras and w∞Gravity †

arXiv:hep-th/9202086v1 26 Feb 1992CTP-TAMU-13/92Area-Preserving Diffeomorphisms, w∞Algebras and w∞Gravity †E. Sezgin ∗Center for Theoretical PhysicsPhysics DepartmentTexas A&M UniversityCollege Station, Texas 77843February 1991ABSTRACTThe w∞algebra is a particular generalization of the Virasoro algebra with generatorsof higher spin 2, 3, ..., ∞.

It can be viewed as the algebra of a class of functions, relativeto a Poisson bracket, on a suitably chosen surface.Thus, w∞is a special case of area-preserving diffeomorphisms of an arbitrary surface.We review various aspects of area-preserving diffeomorphisms, w∞algebras and w∞gravity. The topics covered include a)the structure of the algebra of area-preserving diffeomorphisms with central extensions andtheir relation to w∞algebras, b) various generalizations of w∞algebras, c) the structure ofw∞gravity and its geometrical aspects, d) nonlinear realizations of w∞symmetry and e)various quantum realizations of w∞symmetry.† Lectures given at the Trieste Summer School in High Energy Physics, July 1991.∗Work supported in part by NSF grant PHY-9106593

1. IntroductionArea-preserving diffeomorphisms of a surface Σ, denoted by SDiff(Σ), arise in diverseareas of theoretical and mathematical physics.

For example, since they are the canonicaltransformations which preserve the Poisson bracket, they naturally arise in theory of dynam-ical systems [1]. In the context of high energy physics, the area-preserving diffeomorphismsof a 2-sphere, SDiff(S2), has been encountered in the theory of relativistic membranes †[2].It arises as a subgroup of the diffeomorphisms of the 3-dimensional world-volume in a light-cone gauge.

It was observed that (in the case of spherical and toroidal membranes at least)the structure constants of the area-preserving diffeomorphisms are those of SU(N) in theN →∞limit (in the case of a torus there are the additional global diffeomorphisms) [2].This fact was used in regularizing the quantum theory of the supermembrane, by consideringthe SU(N) theory and then taking the N →∞limit [4].Area-preserving diffeomorphisms of a 2-torus, SDiff(T 2), has been studied as the analogof the Virasoro symmetry [5]. It turns out that the generators of the Virasoro algebra (withor without central charge) can be constructed as a linear combination of infinitely manySDiff(T 2) generators [6].

Later it was realised that the Virasoro algebra can be embedded inthe area-preserving diffeomorphisms of the 2-plane, SDiff(R2), in a manifest manner [7]. Inthis case, all the other generators of the algebra form a representation of the Virasoro algebraand thus can be viewed as generators with conformal spins 3, 4, ..., ∞.

This observation,opened the way for borrowing many of the techniques used in the study of Virasoro algebraand the 2D conformal field theories.There is a great deal of arbitrariness in choosing basis functions in order to exhibit thestructure constants of the SDiffalgebras explicitly. The problem becomes more acute whenone deals with noncompact surfaces such as R2 or a cylinder, in which case one usually workswith basis functions which diverge at infinity.

Nonetheless, there is a particular choice ofbasis elements appropriate for a surface of cylindrical topology [8], which, though divergentat infinity, it yields a specific set of structure constants. Although, this particular algebra isreferred to as w∞or w1+∞algebra (in the latter case a spin 1 generator is included), SDiffalgebras for other kinds of surfaces, or for the same surface but with a different choice ofbasis elements are also referred to loosely as w∞algebras.

Often it is not so clear whichone of these algebras are isomorphic into each other, especially in the case of noncompactsurfaces, and when infinitely many redefinitions of the generators is required. At any rate,the terminology of w∞will be used to refer to a wide class of higher spin algebras in twodimensions which contain the Virasoro algebra, and are connected to certain area-preservingdiffeomorphisms in a way to be specified carefully case by case, if needed.† The generalization of this group for higher dimensional extended objects, which areknown as the p-branes, is the volume-preserving diffeomorphisms of a p dimensional manifold[3].2

In w∞algebras the commutator of a spin s generator with a spin s′ generator yields aspin (s + s′ −2) generator, and a central extension can arise only in the commutator of twospin 2 generators. Denoting the Betti number of the surface Σ by b1, the algebra SDiff(Σ)admits b1 independent central extensions [9].

An extension of w∞in which the commutator aa spin s generator with a spin s′ generator yields all spins between spin 2 and spin (s+s′−2)arise was constructed explicitly in Refs. [8,10].

This is referred to as W∞algebra and it doesadmit central extensions in all spin sectors.A new field theoretic realization of w∞was found by constructing the so called w∞gravity action [11], which is the analog of the Polyakov’s bosonic string action, incorporatingnow the coupling of all higher spin world-sheet fields to matter fields. Interestingly enough,when quantized, the w∞symmetry of this model deforms into W∞symmetry [12].

In fact,it has been shown that this is a rather general phenomenon which arises even in simplerquantum mechanical systems in one dimension [13].A geometric understanding of w∞gravity is still lacking, but interesting attempts have been made in this direction [14]. Thew∞transformation rules for a scalar field have also been interpreted in the context of anonlinear realization of w∞[11,15], which we shall discuss later.There are a number of other areas in which w∞symmetry arises.

For example, a 2Dsigma model based on an area-preserving diffeomorphisms of a 2-surface turns out to havefield equations that are closely related to the self-dual gravity equations in 4D [16]. This israther interesting, because it suggests a 2D conformal field theory approach to an interesting4D quantum gravity problem.

In fact, self-dual gravity equations have been shown to arise instring theories with local N = 2 supersymmetry on the world-sheet , and a target spacetimeof (2, 2) signature [17]. Furthermore, it has been shown that the loop group of w∞arisesas a symmetry group in the N = 2 superstring theory [17].

More recently, the loop algebraof area-preserving diffeomorphisms of a 2-plane has been discovered as the algebra of aninfinite set of spin 1 primary fields constructed out of a matter field and the Liouville field[18]. A possible connection between the c = 1 bosonic string theory where this symmetryarises and the N = 2 superstring theory has been suggested [17].

Finally, let us mentionthat the area-preserving diffeomorphisms have also been encountered as symmetry groupsin the study of higher spin field theories in 2 + 1 dimensions [18].In this brief review, we shall first describe the algebraic structure of SDiff(Σ) and w∞.After we summarize various generalizations of these algebras, we shall turn to their classicalfield theoretic realizations. In particular, we shall describe the (super) w∞gravity, nonlinearrealizations and various quantum realizations of w1+∞symmetry.2.

The structure of area-preserving diffeomorphisms and w∞algebrasIn two dimensions the area-form is the same as a nondegenerate closed 2-form, i.e. asymplectic 2-form.

Hence, the area-preserving diffeomorphisms are the same as the sym-plectic diffeomorphisms, namely those diffeomorphisms which leave the symplectic form on3

the surface invariant. Unless stated otherwise we shall consider compact orientable surfaces.Denoting the symplectic form by Ω= Ωabdσadσb where σa, a = 1, 2 are the coordinates ofthe surface, it must satisfy the condition LξΩ= 0, where Lξ denotes the Lie derivative alonga vector ξa.

The most general solution to this condition takes the formξa = Ωab(∂bΛ + ωb)(2.1)where Λ(σ) is an arbitrary function and ωa is a harmonic 1-form, i.e. it is curl-free but itcan not be written as a derivative of a scalar globally.

As is well known, on a genus g surfacethere are 2g independent such harmonic 1-forms. Let us expand ωa = crω(r)a , r = 1, ..., 2gwhere cr are arbitrary constants and ω(r)a , are harmonic 1-forms which we normalize asRΣ ω(r) ∧ω(s) = δrs.

We can represent the generators of the area-preserving diffeomorphismsas followsLΛ = Ωab∂bΛ∂a,Pr = Ωabω(r)b ∂a,r = 1, ..., 2g(2.2)These generators obey the algebra[LΛ1, LΛ2] = LΛ12,(2.3a)[Pr, LΛ] = LΛr,(2.3b)[Pr, Ps] = LΛrs,(2.3c)where the parameters appearing on the right-hand sides are given byΛ12 = Ωab∂bΛ1∂aΛ2(2.4a)Λr = Ωabω(r)b ∂aΛ(2.4b)Λrs = Ωabω(r)b ω(s)a(2.4c)From (2.4a) it is clear that SDiff(Σ) algebra is the algebra of functions on Σ relative to aPoisson bracket defined by {A, B} := Ωab∂bA∂aB.Let us now consider the quantum version of the above algebra whose generators we shalldenote by ˆLΛ and ˆPr. The central extension of such an algebra was determined in Ref.

[9].The result takes the form[ˆLΛ1, ˆLΛ2] = ˆLΛ12 +ZΣd2σαΩabωb(Λ1↔∂aΛ2)(2.5a)[ˆPr, ˆLΛ] = ˆLΛr −2ZΣd2σαΩabωbω(r)a Λ(2.5b)[ˆPr, ˆPs] = ˆLΛrs +ZΣd2σβΩabω(r)b ω(s)a ,(2.5c)where α and β are two arbitrary densities.4

In practice, it is very useful to expand the generators of this algebra in a suitable basisand to find the structure constants explicitly. The choice of basis depends on the topologyand geometry of the surface Σ and on the type of functions we wish to expand on it.

Theseissues arise in the very rich and fascinating subject of harmonic analysis.An extensivediscussion of this subject is beyond the scope of this paper. We shall be content by givingfew examples here.

Let us consider first a flat two torus T 2 defined by a square lattice withside 2π. We shall take the symplectic structure to be the Levi-Civita symbol Ωab = ǫabwith ǫ12 = −ǫ21 = 1.

It is natural to choose as basis functions ein·σ where n = (n1, n2)and n1, n2 are integers. Note that this is a complete and orthonormal basis, and furnish aunitary representation of the torus group U(1) × U(1).

In terms of these basis functions wemay expand the parameters Λ(σ) asΛ(σ) =XnΛnein·σ,(2.6)The torus has genus 1, so there are two independent harmonic 1-forms. We can parametrizethe most general harmonic 1-form as ωa = (a1, a2), where the components a1 and a2 arearbitrary constants.

Defining the Fourier components of the SDiff(T 2) generators as followsˆLΛ =XnΛnˆLn,(2.7)we find that (2.5) reduces in this special case to [9][ˆLn, ˆLm] = n × m ˆLn+m + a × n δn+m,0(2.8a)[ˆPr, ˆLn] = nr ˆLn + brδn,0(2.8b)[ˆPr, ˆPs] = cǫrs,(2.8c)where n×m ≡ǫabnamb and br, c are constants given by βr = ǫrscsΛ0Rd2σα and c =Rd2σβ,and r, s = 1, 2 since b1 = 2 for a torus. Note that L0 does not occur in the commutationrelations (2.8).

However, it can be introduced on the right hand side of (2.8c) as ǫrsL0. Itwould be a central charge commuting with all the generators.

Moreover, the central extensionon the right hand side of (2.8c) could be absorbed in a redefinition of L0 [9].An interesting fact about the algebra (2.8a) is that, the Virasoro algebra with a centralextension can be obtained from it as a infinite linear combination of the form LN = P cnNLn.The exact form of the constant coefficients cnN can be found in Ref. [6].A super extension of SDiff(T 2) does exist [9,4].

For completeness, we reproduce it here:[Ln, Lm] = n × mLn+m,[Ln, Gm] = n × mGn+m,{Gn, Gm} = Ln+m,(2.9)5

where Gn are, of course, the superpartners of the bosonic generators Ln. Note that, notmerely is there no central extension possible in the {G, G} anticommutator, but also thecentral term that could be present in the [L, L] commutator in the bosonic case must nowbe absent by Jacobi identities.Another simple example of a compact, orientable surface is a 2-sphere S2.

As basis func-tions it is natural to choose the spherical harmonics Y ℓm(θ, φ). They furnish a representationof the rotation group SO(3), and they form a complete and orthonormal set.

Let us further-more take the symplectic structure to be Ωab = sin−1θǫab, and expand the parameters andthe generators of SDiff(S2) as Λ = Pℓ,m ΛℓmY ℓm and LΛ = Pℓ,m ΛℓmLℓm. Since there existno harmonic 1-forms on S2, there will be no non-trivial central extension and the generatorsLℓm obey the following classical SDiff(S2) algebra[Lℓm, Ljn] = cℓjk (m, n)Lkm+n(2.10)where cℓjk (m, n) are the structure constants, which are essentially the Clebsch-Gordan coef-ficients of SO(3) and can be written ascℓjk (m, n) =Zdθdφǫab∂bY ℓm∂aY jnY k−m−n(2.11)If we take the surface Σ to be a 2-sphere with north and south poles removed, S2\ ±1, which is topologically equivalent to a cylinder, then there will be a nontrivial centralextension.

Its form is somewhat complicated, and it can be found in Ref. [19].We next consider some examples of noncompact surfaces.

In the case of 2-hyperboloid,H2, we can expand the generators of SDiff(H2) in a basis closely related to the sphericalharmonics. In such a basis the structure constants of SDiff(H2) are essentially the Clebsh-Gordan coefficients of SO(2, 1).

Since SO(2, 1) is contained as a subalgebra of SDiff(H2), ifwe identify it with the Lorentz group in 2+1 dimensions, then the possibility of interpretingSDiff(H2) as a higher spin algebra in 2 + 1 dimensions arises.If we take two copies ofSDiff(H2), we can then identify the subgroup SO(2, 1) ⊕SO(2, 1) with the anti de Sittergroup in 2+1 dimensions, and thus obtain its infinite dimensional extension as a higher spinalgebra [20]. This algebra is very similar to the infinite dimensional extension of the AdSgroup SO(3, 2) in four dimensions obtained in [21].H2 is topologically equivalent to a disk and hence its first Betti number is zero.

Con-sequently, SDiff(H2) does not admit a nontrivial central extension. However, if we removethe origin, the resulting surface H2\{0} is topologically equivalent to a cylinder and theassociated area-preserving diffeomorphism algebra will have a nontrivial central extension.In using the general formula (2.5) to compute the central extension, however, care must beexercised in choosing an integration measure such that the relevant integrals are well defined.In what follows we shall mainly concentrate on the classical area-preserving diffeomorphisms.6

The surfaces of interest are those whose area preserving diffeomorphisms can be repre-sented in a basis such that the algebra of area-preserving diffeomorphisms are manifestlyan infinite dimensional extension of the Virasoro algebra, i.e.such that the generatorsof SDiff(Σ) decompose in a simple manner under the Virasoro group. To this end let usfirst consider the area-preserving diffeomorphisms of the upper half plane R2+, with coordi-nates x, y [2].

A choice of basis which would be suitable for expansions of square integrablefunctions on the plane, would be the unitary representation of the Euclidean group in twodimensions, E2. These are essentially the Bessel functions.

However, in such a basis theembedding of the Virasoro algebra would not be manifest. To make the connection withthe Virasoro algebra manifest, a more suitable choice of basis is yℓ+1xℓ+m+1 where ℓ, mare integers [2,22].

To avoid the singularities at the origin, we may remove the origin, andthus consider R2\{0}, which is topologically equivalent to a cylinder. Note that, the basisfunctions diverge as x, y →∞, and they are not orthonormal either.

However, as we will beinterested in functions which admit a Taylor expansion we shall not be concerned about theseproperties of the basis functions. Let us proceed by choosing the symplectic structure to beΩab = ǫab, and expanding the generators and the parameters as Λ = −Pℓ,m Λℓmyℓ+1xℓ+m+1and LΛ = Pℓ,m Λℓmvℓm.

In this basis, the generators vℓm take the formvℓm = yℓxℓ+m−(ℓ+ 1)x ∂∂x + (ℓ+ m + 1)y ∂∂y,m ≥−ℓ−1(2.12)and they obey the following SDiff(R2) algebra[vℓm, vjn] = [(j + 1)(ℓ+ m + 1) −(ℓ+ 1)(j + n + 1)]vℓ+jm+n(2.13)Clearly v0m generate the Virasoro algebra without central extension:[v0m, v0n] = (m −n)v0m+n(2.14)Furthermore, the generators vℓm form a representation of the Virasoro algebra since[v0n, vℓm] = [(ℓ+ 1)n −m]vℓm+n(2.15)This is to be compared with the commutation rule between the Virasoro generators Ln andthe Fourier modes of a spin s conformal field wsm given by[Ln, wsm] = [(s −1)n −m]wsm+n(2.16)Therefore the generators vℓm can be viewed as the Fourier modes (labelled by m) of a con-formal field of conformal spin (ℓ+ 2).Another choice of basis functions for SDiff(R2) considered in Ref. [23] is xs+mys−m.Using the expansions Λ = −P Λsmxs+mys−m and LΛ = P Λsmvsm, one finds that SDiff(R2)algebra in this basis takes the form[vsm, vtn] =(t −n)(s + m) −(s −m)(t + n)vs+t−1m+n(2.17)7

In this basis 12v1n obeys the Virasoro algebra, and by commuting it with vsm we find that vsmcan be viewed as the Fourier modes (labelled by m) of a conformal field of spin (s+1). Notethat in order to avoid negative powers of x and y in the basis functions, we must keep thegenerators vsm with −s ≥m ≤s.

Note also that in order to have integer powers of x and y,s, n must be both integers or both half integers.There are other choices of basis functions which give rise to algebras with a somewhatdifferent interpretation of the generators as Fourier modes of conformal ields. For example,using the polar coordinates r, θ, consider the basis functions rℓ+2eimθ.

Furthermore, choosingthe symplectic structure to be Ωab = r−1ǫab, and using the expansions Λ = −i P Λℓmrℓ+2eimθand LΛ = Pℓ,m Λℓmvℓm, where ℓ, m are integers, we find that SDiff(R2\{0}) in this basis takesthe form[vℓm, vjn] = [(j + 2)m −(ℓ+ 2)n]vℓ+jm+n,(2.18)where ℓ≥−1 and −∞< m < ∞. In this basis, 12v0m obey the Virasoro algebra.

Using thenotation Lm = 12v0m, from (2.18) we obtain[Lm, vℓn] =12(ℓ+ 2) −nvℓn(2.19)Compared with (2.16), this implies that vℓn can be viewed as the Fourier modes (labeled byn) of a conformal field of spin ((ℓ+ 4)/2. We shall call this algebra the twisted w∞algebra.Finally, let us describe a somewhat more convenient choice of basis to describe the algebraof area-preserving diffeomorphisms of a surface with the topology of a cylinder, R × S1.

Letthe coordinates of the cylinder be 0 ≤x ≤2π and ∞< y < ∞. A suitable choice ofbasis functions is yℓ+1eimx [8].

Choosing the symplectic structure to be Ωab = ǫab and usingthe expansions Λ = −i P Λℓmyℓ+1eimx and LΛ = Pℓ,m Λℓmvℓm, we find that SDiff(R × S1)algebra takes the form[vim, vjn] = [(j + 1)m −(i + 1)n]vi+jm+n,(2.20)where ℓ≥−1 and −∞< m < ∞. Evidently v0m obeys the Virasoro algebra and from thecommutator[v0n, vℓm] = [(ℓ+ 1)n −m]vℓm+n(2.21)we see that vℓm are the Fourier modes (labelled by m) of a conformal field of spin (ℓ+ 2).

Inparticular v−1m has spin 1. This algebra is usually referred to as w1+∞algebra.

If we excludethe spin 1 generator, we still have a closed algebra, known as the w∞algebra.Experience with Virasoro algebra suggests that in order to have a nontrivial unitaryrepresentation of w1+∞it should admit a central extension. Starting directly from (2.15)and searching for central extensions by means of checking the Jacobi identities, one finds8

that it is allowed only in the spin 2 sector. Denoting the central extension that arises in thecommutator of vim with vjn by cij(m, n), it takes the familiar formcij(m, n) = c12(m3 −m)δi,0δj,0δm+n,0,(2.22)where c is an arbitrary constant.An extension of w1+∞which does contain a centralextension in all spin sectors exists, and it is referred to as W1+∞.

It turns out that w1+∞algebra can be viewed as a contraction of the latter. Alternatively, w1+∞can be interpretedas the classical limit of the quantum algebra W1+∞.w1+∞algebra admits two natural subalgebras.One of them, which we shall denoteby w+1+∞has the generators vℓm with the restriction ℓ≥−ℓ−1, ∞≤m ≤∞, and theother one, w−1+∞generated by vℓm with the restriction ℓ≤−ℓ−1, ∞≤m ≤∞.

Anotheruseful subalgebras is the Cartan subalgebra. There are a number of ways of choosing it.

Forexample, vℓ0 are infinitely many mutually commuting generators, thus forming the Cartansubalgebra †. Another set of mutually commuting generators are vℓ−ℓ−1 and vℓℓ+1.

Thesealgebras will play a role when we discuss the nonlinear realizations of w1+∞.3. Generalizations of the w∞AlgebrasThere are a number of extensions of the area-preserving diffeomorphisms.

Among themare the N = 1 [24] and N = 2 [22] supersymmetric extensions of w∞. The N = 2 super w∞algebra takes the form [22,25][vim, vjn] =(j + 1)m −(i + 1)nvi+j + c8(m3 −m)δi,0δj,0δm+n,0[vim, Jj−1n] =jm −(i + 1)nJi+j−1m+n{¯Gαr , Gβs } = 2vα+βr+s −2(β + 12)r −(α + 12)sJα+β−1r+s+ c2(r2 −14)δα,0δβ,0δr+s,0[vim, Gαr ] =(α + 12)m −(i + 1)rGα+im+r[vim, ¯Gαr ] =(α + 12)m −(i + 1)r¯Gα+im+r[Ji−1m , Gαr ] = Gi+αm+r[Ji−1m , ¯Gαr ] = −¯Gα+im+r.

[Ji−1m , Jj−1n] = c2mδi,0δj,0δm+n,0,(3.1)where the notation is self explanatory. In fact, this is the algebra of symplectic diffeomor-phisms on a (2, 2) superplane, i.e.

a plane of two bosonic and two fermionic dimensions [24].† It is interesting to note that, since in particular [v00, vn0 ] = 0, and v00 is the usual Hamil-tonian H = L0 = v00, we can interpret vn0 , n ≥1 as infinitely many conserved quantitiesthat commute with the Hamiltonian.9

The N = 1 super w∞algebra can be obtained from the above algebra by truncation, ordirectly as an algebra of the symplectic diffeomorphisms of a (2, 1) superplane [24,22]. Fori = j = α = β = 0, the algebra (3.1) reduces to the well known N = 2 superconformalalgebra.Yet another extension is called the topological w∞algebra denoted by wtop∞[26].

It isobtained from the N = 2 super w∞algebra by a twisting procedure introduced by Witten[27]. The idea is to identify one of the fermionic generators as the nilpotent BRST chargeQ, and to define bosonic generators which can be written in the form ˆvim = {Q, something}.This is the higher-spin generalization of the property that holds for the energy-momentumtensor of a topological field theory.

A suitable candidate for the BRST charge is Q = −¯G0−12.We then define the generators of wtop∞to be Gim+ 12 and define ˆvim as ˆvim = −{Q, Gim+ 12}. Itcan be easily shown that these generators obey the algebra[ˆvim, ˆvjn] =h(j + 1)m −(i + 1)niˆvi+j−ℓm+n[ˆvim, Gjn+12] =h(j + 1)m −(i + 1)niGi+j−ℓm+n+12{Gim+12, Gjn+12} = 0(3.2)Note that the structure constants for [ˆvim, Gjn+12] are the same as those for [ˆvim, ˆvjn], and thatthe algebra is centerless.

A field theoretic realization of W top∞is given in Ref. [26].Another interesting extension of w∞involves a Kac-Moody sector.

It can be obtaineddirectly by a suitable contraction of the W1+∞algebra with SUN) symmetry found inRef. [28].

The generators of the algebra are vim and Ji,am where i is an integer such thati ≥−1 and a = 1, ..., N2 −1 labels the adjoint representation of SU(N) and they have thecommutation relations[vim, vjn] = [(j + 1)m −(i + 1)n]vi+jm+n + c12(m3 −m)δi,0δj,0δm+n,0[vim, Jj,an ] = [(j + 1)m −(i + 1)n]Ji+j,am+n[Ji,am , Jj,bn ] = 12f abcJi+j+1,cm+n+ 116kmδi+1,0δj+1,0δm+n,0(3.3)where f abc are the structure constants of SU(N), and the central extension, c, of the Virasoroalgebra, and the level, k, of the Kac-Moody algebra are related to each other by the Jacobiidentity: c = Nk. The generators of the above algebra (without central extension) can berepresented as followsvℓm = −iyℓeimxh(ℓ+ 1) ∂∂x −imy ∂∂yiJℓ,am = −itayℓ+1eimx,(3.4)10

where ta are the generators of SU(N).It would be interesting to find a field theoreticrealization of this algebra.There also exists an infinite dimensional generalization of w∞algebra which is relatedto the symplectic diffeomorphisms in four dimensions. The generators are labelled as vℓ,⃗kmwhere ⃗k = (k1, k2) and they obey the algebra [29][V ℓ,⃗km , V j,⃗ℓn ] = [(j + 1)m −(ℓ+ 1)n]V ℓ+j,⃗k+⃗ℓm+n+ ⃗k ×⃗ℓV ℓ+j+1,⃗k+⃗ℓm+n,(3.5)Another infinite dimensional extension of the w∞algebra is the loop algebra of w∞.

Forexample, the loop algebra of SDiff(R2) in the basis (2.17) is[vsm(σ), vtn(σ′)] =(t −n)(s + m) −(s −m)(t + n)vs+t−1m+n (σ) ∂∂σδ(σ −σ′)(3.6)This concludes the brief survey of some of the salient features of the area-preserving dif-feomorphisms, w1+∞algebras and their generalizations. We now turn to their field theoreticrealizations.4.

w∞Gravity and SupergravityBefore we describe field theoretic realizations of w∞symmetry, it is useful to review asimple field theoretic realization of the Virasoro symmetry, and to formulate it in a languagethat lends itself readily to a w∞generalization. To this end consider the LagrangianL = −14√−hhij∂iφ∂jφ,(4.1)where hij is the inverse of the worldsheet metric hij, (i, j = 0, 1), h = dethij and φ is a realscalar.

This Lagrangian clearly possesses the 2D diffeomorphism and Weyl symmetries. Itis convenient to parametrize the metric as follows [25,14]hij = Ω2h++1 + h++h−−1 + h++h−−2h−−,(4.2)where Ωis an arbitrary function which drops out in the action, and the light-cone coordinatesare defined by x± =1√2(x0 ± x1).

In terms of these variables the Lagrangian (4.1) becomesL = 12(1 −h++h−−)−1(∂+φ −h++∂−φ)(∂−φ −h−−∂+φ). (4.3)It turns out to be very useful to rewrite this Lagrangian in the following first order formL = −12∂+φ∂−φ −J+J−+ J+∂−φ + J−∂+φ −12h−−J2+ −12h++J2−,(4.4)11

where J± are auxiliary fields which obey the field equationsJ+ = ∂+φ −h++J−,J−= ∂−φ −h−−J+,(4.5)These equations define a set of nested covariant derivatives [30]. Solving for J± and sub-stituting into (4.4) indeed yields (4.3).

Thus the two Lagrangians are classically equivalent,though in principal they may be quantum inequivalent. The action of the Lagrangian (4.4)is invariant under 2D diffeomorphism transformations, with a general parameter k+(x+, x−),given by [11]δφ = k+J−δh++ = ∂+k+ −h++∂−k+ + k+∂−h++δh−−= 0δJ−= ∂−(k+J−)δJ+ = 0,(4.6)and 2D diffeomorphisms with parameters k−(x+, x−), which can be obtained from the abovetransformations by changing + ↔−everywhere.The W symmetric generalization of (4.4) is now remarkably simple.

With the furthergeneralization to the case in which the fields φ and J± take their values in the Lie algebraof SU(N) the answer can be written as follows [11]L =tr−12∂+φ∂−φ −J+J−+ J+∂−φ + J−∂+φ−Xℓ≥01ℓ+ 2A+ℓtr Jℓ+2−−Xℓ≥01ℓ+ 2A−ℓtr Jℓ+2+. (4.7)Note that A0+ = h++ and A0−= h−−.

The equations of motion for the auxiliary fields nowreadsJ+ = ∂+φ −Xℓ≥0A+ℓJℓ+1−,J−= ∂−φ −Xℓ≥0A−ℓJℓ+1+. (4.8)The Lagrangian (4.7) possesses the w∞symmetry with parameters k+ℓ(x+, x−) that gener-12

alize (4.6) as follows [11]δφ =Xℓ≥−1k+ℓJℓ+1−δA+ℓ= ∂+k+ℓ−ℓXj=0[(j + 1)A+j∂−k+(ℓ−j) −(ℓ−j + 1)k+(ℓ−j)∂−A+j]δA−ℓ= 0δJ−=Xℓ≥−1∂−k+ℓ(J−)ℓ+1δJ+ = 0,(4.9)and W transformations with parameters k+ℓ(x+, x−) which can be obtained from above bythe replacement + ↔−everywhere. It is important to note that we must setk−1± = −1NXℓ≥1k±ℓtr Jℓ+1∓,(4.10)to ensure the tracelessness δφ and δJ± in the transformation rules above.

The Lagrangian(4.8) has also Stueckelberg type shift symmetries which arise due to the fact that for SU(N),only (N −1) Casimirs of the form tr(J±)ℓ+2 are really independent, while the rest can befactorize into products of these Casimirs. For a further discussion of these symmetries, seeRef.

[11].There exists an interesting chiral truncation of the w gravity theory discussed above.It is achieved by setting A+ℓ= 0. In that case from (4.10) we have J+ = ∂+φ and J−=∂−φ −Pℓ≥0 A−ℓtr(∂+φ)ℓ+1.

In this case, it is more convenient to work in second orderformalism. Thus, substituting for J± into the Lagrangian (4.7), we obtain [11]L = 12tr∂+φ∂−φ −Xℓ≥01ℓ+ 2Aℓtr(∂+φ)ℓ+2,(4.11)where we have used the notation A−ℓ= Aℓ.

This Lagrangian has the following symmetry[11]δφ =Xℓ≥−1kℓ(∂+φ)ℓ+1(4.12)δAℓ= ∂−kℓ−ℓ+1Xj=0[(j + 1)Aj∂+kℓ−j −(ℓ−j + 1)kℓ−j∂+Aj](4.13)The Lagrangian (4.11) has also the appropriate Stueckelberg symmetry. Using this symmetryone can obtain [11] the chiral W3 gravity of Ref.

[31]. Note that the interaction term in this13

Lagrangian has the form of a gauge field × conserved current1ℓ+2tr(∂+φ)ℓ+2. It is importantto note that the OPE of these currents do not form a closed algebra, while they do closewith respect to Poisson bracket.

Hence, the w∞symmetry described above is a classicalsymmetry, as expected.The w∞gravity with N = 2 super w∞symmetry has also been constructed [32]. Forreaders convenience we summarize the result of Ref.

[32] for chiral N = 2 super w∞here.Consider two real scalar superfields φ and ¯φ. Let the superspace coordinates be Z = (z, θ),and define the covariant derivatives D = ∂θ −θ∂and ¯D = ∂¯θ −¯θ¯∂.

The currents whichclassically generate the N = 2 super w∞algebra are [32]wℓ= Dφ(∂φ)ℓ+1D¯φ,forℓ= −1, 0, 1, 2, ...= (∂φ)ℓ+ 32D¯φ + 12DhDφ(∂φ)ℓ+ 32D¯φiforℓ= 12, 32, ...(4.14)In terms of these currents the chiral N + 2 w∞supergravity action takes the form [32]S =Zd2ZhDφ ¯D¯φ +∞Xℓ=−1Aℓwℓi(4.15)The transformation rules for the matter fields are [32]δφ =∞Xℓ=−1,0,...kℓDφ(∂φ)ℓ+1 +Xℓ=−12, 12 ,...hkℓ(∂φ)ℓ+ 32 −12DkℓDφ(∂φ)ℓ+ 12iδ¯φ =∞Xℓ=−1,0,...n−kℓ(∂φ)ℓ+1D¯φ −(ℓ+ 1)DkℓDφ(∂φ)ℓD¯φo+∞Xℓ=−12, 12,...n12Dkℓ(∂φ)ℓ+ 12 −(ℓ+ 32)Dkℓ(∂φ)ℓ+ 12D¯φ+ 12(ℓ+ 12)DDkℓDφ(∂φ)ℓ−12D¯φo(4.16)The gauge fields transform as follows [32]δAℓ= ¯Dkℓ+Xj=12,1,...h(12 −j)Aj∂kℓ−j+52+ 12(−1)2jDAjDkℓ−j+52+ (ℓ−j + 3)∂Ajkℓ−j+52i(4.17¬a)for ℓ= −12, 12, ..., andδAℓ= ¯Dkℓ−2ℓ+32Xj=1,2,...Ajkℓ−j+52+Xj=12,32,...h(12 −j)Aj∂kℓ−j+52+ 12(−1)2jDAjDkℓ−j+52+ (ℓ−j + 3)∂Ajkℓ−j+52i(4.17¬b)14

for ℓ= −1, 0, ... The spin 12 transformations can be included as in Ref.

[32]. The nonchiralversion of this theory has also been constructed [32].Going back to the bosonic version of nonchiral w∞gravity, an interesting geometricformulation of it has been given by Hull [14].

Consider the case of a single real scalar φ.According to Ref. [14], the symmetry algebra of nonchiral w∞gravity is a subalgebra of themore general set of transformationsδφ =∞Xn=2λi1i2···in−1(x)∂i1φ∂i2φ · · ·∂in−1φ ≡Λ(xi, yi),(4.18)where yi = ∂iφ and λi1i2···in−1(x)(n = 2, ...) are the infinitesimal parameters which aresymmetric tensors on the worldsheet.

It is easy to show that these transformations satisfythe Poisson bracket algebra on the phase space with coordinates xi, yi. The action proposedin Ref.

[14] isS =Zd2x˜F(x, y)(4.19)where ˜F(x, y) has the expansion˜F(x, y) =∞Xn=21n˜hi1i2···in(x)yi1yi2 · · · yin,(4.20)where ˜hi1i2···in(x) are gauge fields which are tensor densities on the world-sheet. Invari-ance of this action under the transformations (4.18) requires the imposition of the followingconstraint on the parameter Λ(x, y) [14]ǫikǫjl∂2Λ∂yi∂yj∂2 ˜F∂yk∂yl= 0(4.21)The expansion of this equation gives an infinite number of algebraic constraints on theparameters.The first such constraint is λij˜hij = 0.Other constraints relate the traceof a rank ≥3 parameter to the products of lower rank parameters and gauge fields.

Theinvariance of the action under the transformations (4.18), in addition to the constraint (4.21),also requires that the gauge fields transform as [14]δ˜hi1···ip =pXn=2−(p −n + 1)(n −1)(p −1)∂j˜hj(i1···in−1λin···ip) −λj(i1···ip−n˜hip−n+1···ip)−(n −1)˜hj(i1···in−1∂jλin···ip) + (p −n + 1)λi1···ip−n+1∂j˜hip−n+2···ip)j(4.22)It turns out that a gauge invariant condition can be imposed on the gauge fields whichreduces at the linearized level to the constraint present in w∞gravity. This constraint is [14]det∂2 ˜F(x, y)∂yi∂yj= −1(4.23)15

The expansion of this equation gives an infinite number of algebraic constraints on the gaugefields. The first two constraints are det˜hij= −1 and ˜hijk˜hij = 0.

Other constraints relatethe trace of rank≥4 gauge fields to the products of lower rank gauge fields. The procedure forsolving the constraints (4.21) and (4.22) in terms of unconstrained gauge fields and gaugeparameters, as well as the attendant generalized Weyl symmetries have been outlined inRef.

[14].The equation (4.23) has a nice geometrical interpretation. With an appropriate identi-fication of ˜F(x, y) with a Kahler potential, (4.23) can be interpreted as the Monge-Ampereequation for a Kahler metric of a self-dual four-manifold [14].

Other relations between w∞and self-dual geometry have been discussed in [7,16]5. Nonlinear Realizations of w∞The field theoretic realization of w1+∞in terms of a single real scalar as given in (4.12)can be understood within the framework of nonlinear realizations [11].

Denoting the coordi-nates of a cylinder by (x+, y), the scalar field φ(x+, x−) can be considered as parametrizingthe coset space w1+∞/w∞. The generators of this coset are v−1m = −imeimθ∂y, and thereforewe can choose the coset representative to be e∂+φ∂y.

If we denote the coset generators byK, and the subalgebra generators by H, then one can see from (2.19) that the structure ofthe algebra is [H, H] ⊂H, [H, K] ⊂H + K, [K, K] = 0. Thus the coset w1+∞/w∞is not asymmetric space, and the K generators do not even form a linear representation of H (exceptwhen the subalgebra element lies in the Virasoro subalgebra).

Although we can apply thegeneral theory of non-linear realizations to this situation, we should not be surprised to findthat some of the features are non-standard. In particular, the action of H on the coset willin general be non-linear.Acting on the coset representative with a general G transformation g, one hasge∂+φ∂y = e∂+φ∂yh,(5.1)where h is an element of the subalgebra H. For an infinitesimal transformation g = 1 + δg,one therefore finds∂+δφ =e−∂+φ∂y δg e∂+ϕ∂yG/H(5.2)Taking δg = kℓ(x+)yℓ+1 we find that ∂+δφ = ∂+kℓ(∂+φ)ℓ+1.

Hence we haveδφ = kℓ(x+)(∂+φ)ℓ+1(5.3)which is indeed the global w1+∞transformation discussed in the previous section. Thisresult can also be derived in the Poisson bracket language.

Corresponding to (5.2) we haveδφ =e−Adφ δg|y=0=δg + {φ, δg} + 12! {φ, {φ, δg}} + · · ·|y=0(5.4)16

Setting y = 0 amounts to restriction to the coset direction. To generalize this construc-tion to the case of the non-chiral bosonic w∞model in the light cone gauge, we consider twocopies of the cylinder with coordinates (x+, y) and x−, ˜y), so that we have a total of fourcoordinates (x+, x−, y, ˜y).

The coordinates y and ˜y play the rˆole of “momenta” conjugateto x+ and x−, and so one can think of the four-dimensional space as being the cotangentbundle T ∗(Σ) of the 2-torus Σ. The area-preserving transformations on the two cylindersmay be described by introducing the symplectic formΩ= dx+ ∧dy −dx−∧d˜y,(5.5)and using this to define a Poisson bracket {f, g} = Ωij∂if ∂jg in the four-dimensional space.The symplectic diffeomorphisms which leave this form invariant are δxi = {xi, δg}, andthe global non-chiral w∞transformations correspond to restricting δg to have the formδg = k−ℓ(x+)yℓ+1 + k+ℓ(x−)˜yℓ+1.

Their action on φ is given by (5.4), where Adφδg is nowtaken to be Ωij∂iϕ ∂jδg [11].It is instructive to consider some other nonlinear realizations of w1+∞involving morethan a single scalar field. We have considered elsewhere [15] the nonlinear realization basedon the coset w↑1+∞/Vir+, where w↑1+∞is generated by vℓm with m ≥−ℓ−1, and Vir+is generated by v0m with m ≥0, i.e the positive modes of the Virasoro algebra.

In thisconstruction we view the coordinate x+ itself as a coset parameter associated with theVirasoro generator L−1 = v0−1. We choose the coset representativek = e−x+v0−1 Yℓ̸=0e−φ{ℓ},(5.6)where we have used the notation φ{ℓ} ≡P∞m=−ℓ−1 φℓmvℓm.

Note that there are infinitelymany initial Goldstone fields φℓm corresponding to the coset generators vℓm, ℓ̸= 0. Excludingthe generator v−10which does not correspond to a Goldstone field but to the coordinate x+,all the remaining generators of the coset form a linear representation of the subalgebra.

Wecan construct the Cartan-Maurer form as followsP = k−1dk = E0−1v0−1 +Xℓ̸=0Eℓmvℓm +Xm≥0ω0mv0m,(5.7)where Eℓm and ω0m are all 1-forms, i.e. Eℓm = dx+Eℓm.

Next we look for a maximum set ofcovariant constraints with which we can eliminate the inessential Goldstone fields. Such aconstraint isEℓm = 0,for ℓ̸= 0(5.8)As a consequence of we find thatE0−1 = −dx+,ω0m = 0φℓm = −1ℓ+ m + 1∂+φℓm−1,for m ≥−ℓ(5.9)17

The last relation shows that only the Goldstone fields φℓ−ℓ−1 (the “edge fields”) survive theconstraints, and all the other Goldstone fields can be expressed in terms of their derivatives.The transformation rules for the surviving Goldstone fields can be derived as follows.The action of the group w↑1+∞on a coset representative, which we shall generically denotee−φ(x), is given byge−φ(x) = e−φ′(x′)h,(5.10)where h is an element of the divisor subgroup Vir↑. For infinitesimal transformations, wehaveeφ(x)+¯δφ(x)(1 + δg)e−φ(x) = 1 + δh,(5.11)where¯δφ(x) ≡φ′(x′) −φ(x)= φ′(x) + δx+∂+φ(x) −φ(x)≡δφ(x) + δx+∂+φ(x).

(5.12)Projecting (5.11) into the coset direction yields the formulaeφ¯δe−φG/H=eφδge−φG/H. (5.13)Upon the use of (5.9), the variation (5.12) simplifies to ¯δφ{ℓ} = δφ{ℓ} −[φ{ℓ}, v0−1]δx+.Substituting this result in (5.13) we find that all the δx+ terms cancel pairwise except the−δx+ in the v0−1 direction.

Studying (5.13) level by level in spin, the transformation rulesfor all the Goldstone fields can now be derived. To this end it is useful to consider the firsttwo levels of dressing of δg which we parametrize as followsδg =Xℓ,mαℓmvℓm,(5.14)where the αℓm are constant parameters.

Consider a spin-ℓtransformation with parameterα{ℓ} ≡Pm αℓmvℓm. Definingex+v0−1α{ℓ}e−x+v0−1 =Xmβℓm(x+)vℓm,(5.15)we find thatβℓm+1(x+) =−1ℓ+ m + 2∂+βℓm(x+).

(5.16)Note that this is the same relation as that satisfied by the fields φℓm. Proceeding on to thenext level of dressing, we defineXℓ,meφ{−1}βℓm(x+)vℓme−φ{−1} =Xℓ,mγℓm(φ{−1})vℓm(5.17)18

The field dependent parameters γℓm can be straightforwardly computed in terms of βℓm+1(x+),and as a consequence one finds thatγℓ−ℓ−1 =1(ℓ+ 1)!δℓ+1γ−10δyℓ+1 ,(5.18)γ−10=∞Xℓ=−1βℓ−ℓ−1(x+)yℓ+1(5.19)where we have introduced the notation y ≡−∂+φ−10 . In terms of these quantities, Eq.

(5.13)yields the resultsδx+ = −γ0−1(5.20a)δφ−10= −γ−10 . (5.20b)δφ1−2 = −γ1−2 −2φ1−2∂+γ0−1 + ∂+φ1−2γ0−1(5.20c)δφ2−3 = −γ2−3 + γ0−1∂+φ2−3 −3∂+γ0−1φ2−3 + γ1−2∂+φ1−2 −∂+γ1−2φ1−2(5.20d)...Note that only fields and parameters corresponding to the left edge, i.e.

φℓ−ℓ−1 occur in theseresults. Moreover, Eq.

(5.20b) is precisely the w1+∞transformation rule of Eq. (4.12), afteridentifying φ and kℓwith (−φ−10 ) and βℓ−ℓ−1, respectively.It may be verified that the action L0 = 12∂+φ−10 ∂−φ−10transforms by a total derivativeunder the full set of w↑1+∞transformations δφ−10= −P∞ℓ=−1 kℓyℓ+1.

Free second-orderscalar actions involving the higher left-edge Goldstone fields φℓ−ℓ−1 cannot be made becausethey do not have Lorentz weight zero, since the Lorentz weight of φℓ−ℓ−1 is −(ℓ+ 1). Onemay, however, couple the higher Goldstone fields to currents built from φ−10 .

One way to thiswould be to construct a composite w1+∞connection Aℓ−ℓ−1 built out of the “edge scalars”,and to use it in the w∞gravity action of Ref. [11], thus obtaining the global w1+∞symmetricLagrangianL = 12∂+φ−10 ∂−φ−10−∞Xℓ=01ℓ+ 2Aℓ(−) −ℓ−1(φ)(−∂+φ−10 )ℓ+2.

(5.21)The construction of the composite connection requires further work. The idea is essentiallyas outlined above, though, the construction may have to be based on a different coset spacethan the one considered here [15].6.

Quantum Realizations of w∞The realization of w1+∞in terms of the currentsvℓ(z) =1ℓ+ 2(∂+φ)ℓ+2(6.1)19

is necessarily a classical one, since in commuting two of these currents which involves anoperator product expansion, there will be terms coming from multiple Wick contractions ofthe basic building blocks ∂+φ, giving rise to terms that violate the closure of the algebra.The single contraction, on the other hand, corresponds to evaluating the Poisson bracketof the two currents, evidently gives a closed algebra. If one modifies the currents (6.1) asVℓ(z) = P cℓmnp(√¯h)m(∂nφ)p where cℓmnp are a set of constant coefficients, then the algebrawill of course close on these currents.

(On dimensional grounds (n + 1)p + m = 2ℓ+ 2).Although this closure may in some sense be considered as trivial, in fact there is a way ofchoosing the coefficients (which amounts to using a certain basis for the algebra) in sucha way that the currents V ℓ(z) have a definite transformation property under an SL(2, R)and moreover they are quasi-primary fields with respect to a natural Virasoro subalgebra.The resulting algebra is the W1+∞algebra.At the level of quantum field theories thisphenomenon amounts to the deformation of the classical w1+∞symmetry to a quantumW1+∞symmetry [12]. The former can be obtained in the ¯h →0 limit of the latter one.

Thisphenomenon has been found in the study of quantum w∞gravity as well as the study of aquantum mechanical system on a circle [13].Turning our attention to the issue of constructing a quantum realization of w1+∞, onepossibility is to construct the w1+∞currents in terms of b-c ghost systems. As a first step,one constructs the BRST charge [33]Qgh =If ij(∂, ∂)−12cicjbi+j,(6.2)where bi(z) and ci(z) are anticommuting ghosts satisfyingbi(z)cj(w) ∼δijz −w,(6.3)or equivalently, the anticommutator {bi, cj} = δij and f ij(m, n) = (j + 1)m −(i + 1)n arethe structure constants of w1+∞.

The notation f ij(∂, ∂) indicates that the Fourier-modeindex m is replaced by a partial derivative that acts on ci only, and the index n is replacedby a partial derivative that acts on cj only. From Qgh, we can derive the ghostly quantumrealization vigh of w1+∞vigh = {Qgh, bi},(6.4)This formula yields [33]vigh(z) =Xj≥0(i + j + 2)∂cj bi+j + (j + 1)cj ∂bi+j,(6.5)which indeed generate the w∞algebra:vigh(z)vjgh(w) ∼i + j + 2(z −w)2 vi+jgh (w) + i + 1z −w∂vi+jgh (w) + δi0δj0c/2(z −w)4.

(6.6)20

The operator terms on the right-hand side come from single contractions. The central termin the spin-2 sector (the only one that occurs in the w∞algebra) has a central charge thatis formally divergent.

After zeta-function regularization, one finds c = 2 [34,33].Using the structure constants of the topological algebra wtop∞given in (3.2), by means ofthe method outlined above, one can also construct its ghostly quantum realization as follows[26]ˆvigh(z) = (i + j + 2)∂cjbi+j + (j + 1)cj∂bi+j −(i + j + 2)∂γjβi+j −(j + 1)γj∂βi+j,Gigh(w) = (i + j + 2)∂cjβi+j + (j + 1)cj∂βi+j,(6.7)where β, γ are the commuting ghost fields, and summation over j is understood. A detaileddescription of topological w1+∞is given in Ref.

[26].Finally, let us mention a quantum realization of the loop algebra of SDiff(R2) due toWitten [23], which has emerged in the study of string theory with two dimensional targetspacetime. The theory is characterized by the stress tensorTzz = −12(∂X)2 −12(∂φ)2 +√2φ −2b∂c + c∂b(6.8)where X is the matter field, φ is the Liouville field and b, c are the ghost fields, all of whichobey the usual OPE rules.

A quantum realization of of the loop algebra of SDiff(R2) is givenby [23]vsn(z) =e−i√2Xs−neis√2Xe√2(1−s)φ(6.9)Acting on a polynomials in [23]x =cb +i√2(∂X −i∂φ)ei√2(X+iφ)y =cb −i√2(∂X + i∂φ)e−i√2(X−iφ),(6.10)the generators defined in (6.9) obey the algebra (3.6). In particular v1/21/2 acts like∂∂y andv1/2−1/2 acts like∂∂x, and hence commute, on the plane.

However, as operators they do notcommute, yielding a central extension term [23].7. CommentsThe subject of w∞algebras is a rapidly growing one.

Here we have attempted to givesome of the salient features of this subject, and necessarily have omitted a number of topics,some of which are briefly mentioned in the introduction. Many of these topics are relevantin one way or another to the question of how to construct new string theories with higherworld-sheet, and possibly higher spacetime symmetries.21

The w∞algebra is a certain N →∞limit [35] of the WN algebra [36]. Some workhas already been done on string theories based on WN symmetry [37,38,39].A typicalfeature which has arisen is that although one expects a higher slope and therefore higherspin massless fields in the spectrum [40], it turns out that due to the necessary presence of abackground charge at least for one of the scalars in the theory, the slope is pushed back to avalue in such a way that there are no massless higher spin fields after all [38,39].

At the end,one finds the spectrum of the usual string and new massive trajectories. In this author’sopinion this state of affairs is somewhat disappointing, because the symmetry enlargementone intuitively expects is buried in the complexities of the massive trajectories, if at all there.Of course, there may be new field theoretic realizations of W-algebras still to be discoveredwhere the situation may be dramatically different.A string theory based on w∞, or its quantum deformation W∞has not been consideredso far.

It would be very useful to accumulate field theoretic realizations of these symmetrieswhich might play a role in constructing a sensible w∞string theory. Such a theory mayas well look like a topological field theory, since there would be infinitely many physicalstate conditions to satisfy.

We would like to speculate, however, that there may exist a w∞string theory where only a tower of higher spin massless fields in target spacetime would arisein the spectrum, corresponding to the infinitely many higher spin world-sheet symmetries,and that the usual string theory with its massive states may arise as a result of some sortof “spontaneous” symmetry breaking. Clearly a lot remains to be done, and there couldbe some surprises ahead in the search for a string theory (or possibly a theory of higherextended objects, such as supermembranes) with higher symmetries than those which havebeen realized so far.ACKNOWLEDGEMENTSI would like to thank Eric Bergshoeff, Steve Fulling, Chris Pope and Kelly Stelle fordiscussions.

I also would like to thank Professor Abdus Salam, the International AtomicEnergy Agency and UNESCO for hospitality at the International Center for TheoreticalPhysics where lectures on the topics of this review article were delivered.This work ispartially supported by NSF grant PHY-9106593.22

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