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DIFFEOMORPHISMS FROM HIGHER

arXiv:hep-th/9303034v1 5 Mar 1993Preprint-QMW-PH-93-1US-FT/1-91January 1993hep-th/9303034DIFFEOMORPHISMS FROM HIGHERDIMENSIONAL W-ALGEBRASFernando Mart´ınez Mor´as1†, Javier Mas1‡, and Eduardo Ramos2§1Departamento de F´ısica de Part´ıculas Elementales, Facultad de F´ısica, Universidad de San-tiago, Santiago de Compostela 15706, SPAIN2Department of Physics, Queen Mary and Westfield College, Mile End Road, London E14NS, UKABSTRACTClassical W-algebras in higher dimensions have been recently constructed.In this letterwe show that there is a finitely generated subalgebra which is isomorphic to the algebra oflocal diffeomorphisms in D dimensions. Moreover, there is a tower of infinitely many fieldstransforming under this subalgebra as symmetric tensorial one-densities.

We also unravel astructure isomorphic to the Schouten symmetric bracket, providing a natural generalizationof w∞in higher dimensions.† e-mail:fernando@gaes.usc.es‡ e-mail:jamas@gaes.usc.es§ e-mail:ramos@v2.ph.qmw.uk

IntroductionThe purpose of this letter is to give a simple account of D-dimensional classical W-algebras and their intimate connection with the algebras of local diffeomorphisms of a D-dimensional manifold.In general, classical one dimensional W-algebras are defined as nonlinear extensions ofdiff(S1) by tensors of integer weights. These algebras appear naturally in the context of twodimensional conformal field theory.

They are obtained via the centerless c →∞limit of theOPE’s in theories enjoying W symmetry. The canonical example of such a system is providedby the 3-state Potts model and its W3-symmetry.

The classical w3 algebra associated withit is explicitely given by{T(x) , T(y)} = −(T∂+ ∂T)x · δ(x −y){W(x) , T(y)} = −(2W∂+ ∂W)x · δ(x −y){W(x) , W(y)} = (23T∂T)x · δ(x −y)It was shown in [1] that these classical W-algebras also appear as Poisson structures inthe commutative limit of the ring of pseudodifferential operators in one dimension. It wasprecisely this relationship which allowed two of the present authors to generalize this con-struction to higher dimensions.

Nevertheless, one crucial point was missing in [2]. Althoughconjectured, it was not proven that these algebras are extensions of higher dimensional diffeo-morphisms algebras.

We will show in what follows that this is indeed the case, and thereforethat these new algebraic structures fully deserve their name.Before getting into more technical matters, we would like to point out that these classicalW-algebras provide hamiltonian structures for dispersionless KP-type hierarchies [2]. In onedimension these hierarchies play a fundamental role in the planar limit of non-critical stringtheory with c ≤1, as well as in topological models.

It is our hope that these new structureswill come into play in the higher dimensional descriptions of these physical problems. Webelieve that the integrability of the associated hierarchies as well as the relationship todiffeomorphism algebras, support (though weakly) our expectations.In what follows we have tried to avoid, as much as possible, to get into too technical adescription of the subject.

We refer anyone who wishes a detailed analysis of the generalformalism to [2] and references therein.The recipeThe natural arena for the construction of higher dimensional classical W-algebras isprovided by a phase space Y 2D with coordinates (xi, ξi) with i = 1, · · ·, D, and canonicalPoisson bracket given by{f , g} = ∂f∂ξi∂g∂xi −∂f∂xi∂g∂ξi. (1)– 2 –

From now on we will restrict ourselves to homogeneous functions on Y 2D, where thedegree of homogeneity is defined as follows: a function is said to be of degree n if under therescaling ξi →tξi ∀i = 1, · · ·, Df(xi, ξi) →f(xi, tξi) = tnf(xi, ξi).Let us now define the symplectic trace [3][4] as followsTr f =ZdxDdΩξ f(x, θξ)if f is of degree −D and zero otherwise. dΩξ stands for the standard measure of the D −1sphere in the ξ coordinates.

The notation is justified because of the “trace” property1Tr {f , g} = 0.With this machinery it is now simple to construct the analogs of the classical W-algebrasin arbitrary dimension. Let us define the formal generating functionalΛ = ξm +∞Xj=1Uj(xi, ξi),where m > 0, ξ = (PDk=i ξ2k)12, and the Uj’s are homogeneous functions of degree m −j.Therefore Λ can be rewriten asΛ = ξm +∞Xj=1uj(xi, θξ)ξm−j,with θξ denoting the angular coordinates associated with the ξi.The Poisson bracketsamong the u’s are defined via the generalized classical Adler map J (defined below) andlinear functionals on Λ.

The latter are given byFX = Tr XΛ. (2)It is obvious from this and the properties of the trace, that the most general X defining anontrivial functional is of the formX =∞Xj=1Xj(xi, θξ)ξj−m−D.We can now define the “Gel’fand-Dickey” brackets by{FX , FZ}GD = Tr J(X)Z,(3)1We are assuming here that our x-space is compact or, equivalently, that our type offunctions decay fast enough at infinity.– 3 –

withJ(X) = −{Λ , X}⊖Λ + {Λ , (ΛX)−} ,(4)where the subscripts stand for the following projections, if Q = Pk∈Z qk with qk a homoge-neous functions of degree k, thenQ−=Xk≤0qkandQ⊖=Xk≤−D−1qk.We also define Q+ = Q−Q−and Q⊕= Q−Q⊖. Notice by the way that the + (−) projectionis the dual, with respect to the symplectic trace, of the ⊖(⊕) projection.

Moreover, the +and −projections are subalgebras with respect the canonical Poisson bracket defined by (1).This comes out because, as the reader can easily check, if the functions f and g have degreesp and q respectively then {f , g} has degree p + q −1. We would like to remark that our +splitting differs from the usual one because we are excluding from it the components of zerodegree.

This is required in D > 1 if we want to preserve the subalgebra property describedabove.Because of its definition, and the grading properties of the the canonical Poisson bracket,J(X) is bound to have the following form:J(X) =∞Xi,j=1(Jij · Xj)ξm−i,(5)where Jij is a differential operator with coefficients that are at most quadratic in the u’s andtheir derivatives. This together with (2) and (3) implyui(xi, θξ) , uj(yi, θ′ξ)GD = −Jij · δD(x −y)δ(Ω−Ω′),(6)with δ(Ω−Ω′) the delta function associated with the standard measure in SD−1.Although far from obvious, it is a main result of [2] that these brackets define full fledgedPoisson brackets.It is posible to deform the generalized classical Adler map by Λ →Λ + λ, with λ anarbitrary constant, and obtain two different Poisson structures.

ExplicitelyJ(X) →J(2)(X) + λJ(1)(X). (7)whereJ(2)(X) = J(X)J(1)(X) = −{Λ , X}⊖+ {Λ , X−} .

(8)The two Poisson stuctures induced by J(2) and J(1) are said to be coordinated since any linearcombination of them is still a Poisson bracket. For “perverse” historical reasons they arecommonly known as the “second” and “first Gel’fand-Dickey brackets” respectively.

Noticethat, by construction, the first structure induces brackets which are linear in the uj’s.– 4 –

The algebraic structures that we are interested in are only going to appear after imposingcertain constraints in the form of the operator Λ. Explicitely, we are going to set u1 = · · · =uD = 0 for the second structure, and u1 = · · · = um+D = 0, where m is the leading order inΛ, for the first structure.The constraint on J(2) is second class, and its implementation follows Dirac’s prescrip-tion, which in this particular case readsJ(2)ij→Jij −DXm,n=1JinJ−1nmJmj∀ij > Dwhere J−1 is the inverse of the D × D matrix with entries given by the Jnm in (5) and1 ≤n, m ≤D. It is worth pointing out that in spite of potential non-localities, because ofthe term in J−1, the resulting Poisson brackets are local, as a straigthforward computationshows.The constraint on J(1) is much more easily implemented by noticing that the Poissonbrackets of the uj’s with j ≥m + D and the constraints are zero weakly, i.e.

after imposingthe constraints. Therefore in this caseJ(1) →−{Λ , X+}⊖.

(9)Notice that after the reduction the linear part of J(2) is given byJ(2)linear = −{Λ , X}⊖ξm +ξm , (ΛXξ−1)⊖ξ,(10)which can be seen to be isomorphic to (9) under the map Λ →Λξ−m. Another important factis that, upon the imposition of the constraints in the second structure, the Poisson bracketsinvolving the field uD+1 with any other of the uj’s are directly linear, therefore isomorphic tothe ones obtained using the first structure after the relabeling that maps uj →uj−m.

This isa crucial feature which drastically simplifies the explicit construction of the diffeomorphismsubalgebra from the second Gel’fand-Dickey brackets.The D = 1 and D = 2 Poisson brackets defined by (6) were explicitely computed in [2]and we now briefly describe their main features.For D = 1 and m = 1 the second structure2 coincides with a limit n →∞of the classicalwn algebras after setting the constraint u1 = 0. We would like to stress that this limit isintrisically non-linear and therefore non-isomorphic to the standard w∞.

However, the firststructure, after imposing the constraint u1 = u2 = 0 turns out to be exactly w∞, as expected.For D = 2 it was shown that there is a subalgebra isomorphic to the algebra of diffeo-morphisms in two dimensions. This subalgebra plays an analogue role to the one of Virasoroin the one dimensional case.

Moreover, it was shown that the first structure has a subalgebragenerated by symmetric tensor-one-densities that offers a natural generalization of w∞intwo dimensions [5], and is related to the Schouten bracket. In what follows we will showthat this lower dimensional properties extend for arbitrary dimension D.2 It can be shown using the techniques developed in [1] that for any D the second structureis isomorphic for all values of m ̸= 0, while the first can be shown to be independent of mby explicit computation.– 5 –

The algebra of diffeomorphismsLet us begin by recalling some simple facts about diffeomorphisms. In local coordinatesinfinitesimal diffeomorphims are defined through the map xµ →xµ + ǫfµ(x).

The algebragenerated by these transformations is isomorphic to the algebra of vector fields ⃗f ≡fµ(x)∂µ,i.e.h⃗f , ⃗gi= (fµ(∂µgν) −gµ(∂µfν))(x)∂ν. (11)In a field theory invariant under diffeomorphisms the above algebra will be implementedvia Poisson brackets3 , i.e.

there must be a Lie algebra homomorphism given by⃗f = fµ(x)∂µ →Q⃗f =ZdDxfµ(x)Pµ(x). (12)where the Pµ are the generators of infinitesimal diffeomorphisms and such thatnQ⃗f , Q⃗goPB = Q[⃗f ,⃗g].

(13)For the left hand side of (13) we getnQ⃗f , Q⃗goPB =ZdDxZdDy fµ(x)gν(y) {Pµ(x) , Pν(y)}PB(14)whereas for the right hand side, because of (11), we should getQ[⃗f ,⃗g] =ZdDx(fµ(∂µgν) −gµ(∂µfν))(x)Pν(x). (15)Equating both sides we obtain 4{Pµ(x) , Pν(y)}PB = (Pν∂µ + ∂νPµ)x · δ(x −y).

(16)Notice that for dimension one (16) is, up to an irrelevant global sign, nothing but thecenterless Virasoro algebra.3 Strictly speaking, in classical field theory only spatial diffeomorphisms will be representedvia Poisson brackets in this way. The splitting between spatial and time coordinates requiredin the canonical formalism is obviously not invariant under arbitrary diffeomorphisms and,moreover, requires the introduction of a metric.

Consequently the Poisson brackets amongspatial and timelike diffeomorphisms have, in general, an explicit dependence on the metric.In this sense D should be considered the dimension of the spacelike coordinates. Nevertheless,this should not concern us for the abstract manipulations that follow.4We are only considering, as is usually the case, diffeomorphisms with compact support,so boundary terms can be consistently neglected.– 6 –

Let us first show how to obtain (16) from the first Gel’fand -Dickey bracket. After thereduction of setting the first m + D fields equal to zero, the brackets are obtained from (3)with J(1) given in (9).

To the vector field ⃗f we associate the linear functional Q⃗f⃗f = fµ(x)∂µ →Q⃗f = −Tr fµξµ Λ. (17)Comparing (17) with (12) we obtainPµ(x) = −ZdΩξ ξµuD+m+1(x, θξ).

(18)Moreover from (2) it follows that Q⃗f = FXf with Xf = −fµξµ. ThereforenQ⃗f , Q⃗go(1)GD = Tr J(1)(Xf)Xg= Tr {fµξµ , Λ}⊖gν(x)ξν= Tr {fµξµ , Λ} gν(x)ξν= −Tr {fµξµ , gν(x)ξν} Λ= −Tr (fµ(∂µgν) −gµ(∂µfν)) Λ= Q[⃗f ,⃗g].

(19)This implies, as before, that the first Gel’fand-Dickey brackets of the Pµ defined by (18)reproduce the algebra of generators of infinitesimal diffeomorphisms in D dimensions.Furthermore, we can extend the map defined by (17) to symmetric contravariant tensorsof higher rank. Under diffeomorphisms, a rank r contravariant symmetric tensor transformsinfinitesimally with the Lie derivative T µ1,...,µr →T µ1...µr + ǫ(L ⃗fT)µ1...µr, where(L ⃗fT)µ1...µr = fν (∂νT µ1...µr) −rT ν,(µ1...µr−1 ∂νfµr),(20)and the brackets on the superindices stand for symmetrization.Let us therefore define the associated functional in the form:T →QT = −Tr T µ1...µrξµ1...ξµr Λ.

(21)An easy computation parallel to the one in (19) yieldsnQ⃗f , QTo(1)GD = QL ⃗fT. (22)Moreover, if we define QR via the components of a contravariant tensor of order s,{QR , QT }(1)GD = Q[R , T ]S,(23)– 7 –

where [R, T]S stand for the symmetric Schouten bracket [6], which in a coordinate basisreads[R, T]µ1...µr+s−1S= sRν,(µ1..µs−1∂νT µs..µr+s−1) −rT ν,(µ1..µr−1∂νRµr..µr+s−1)This implies, as before, that if we definePµ1...µr = −ZdΩξ ξµ1 · · · ξµruD+m+r(24).the first Gel’fand-Dickey bracket among the P’s define a closed subalgebra given by{Pµ1...µr(x) , Pµr+1...µr+s(y)}(1)GD = (rXj=1Pµ1..ˆµj..µr..µr+s∂µj+sXj=1∂µr+jPµ1..µr..,ˆµr+j,..,µr+s)x · δ(x −y),(25)where the subindex with a hat is omitted. In particular, the above equation implies thatPµ1...µs transforms under diffeomorphisms as a s-covariant symmetric tensorial one-density.The analysis of the second Poisson structure simplifies considerably if we make use ofthe isomorphism uj →uj−m between the linear part of J(2) and J(1) mentioned in (10).

Onecan see with little effort that the second Gel’fand-Dickey bracket involving the first nonzerofield, uD+1, and any other higher field ui>D+1 is linear. Henceforth, we already know theexpression for all these brackets invoking the above mentioned isomorphism:{uD+m+1 , uD+m+k}(1)GD →{uD+1 , uD+k}(2)GD .Therefore, the key properties (19) and (22) still hold for the second Gel’fand-Dickey brackets,whereas for (23) this is not the case due to the quadratic terms involved.Summarizing, we have unraveled a set of similarities among the higher dimensionalclassical W-algebras constructed in [2] and the standard one-dimensional algebras of the w∞type.

We have shown explicitly how to construct a finitely generated subalgebra isomorphicto the algebra of diffeomorphisms in D-dimensions, therefore these new structures can benaturally understood as extensions of the symmetry algebra for generally covariant theories.Moreover, we have also shown that there is an infinite tower of fields Pµ1...µk transformingas k-covariant one-densities. Nevertheless, as expected, there are some relevant differencesbetween the one and higher dimensional case.The most important one is that, as wasexplicitly shown in [2], there are also fields transforming as infinite dimensional reps. ofthe diffeomorphism subalgebra.

Here, we should clearly distinguish between the first andsecond Gel’fand-Dickey brackets. Whilst in the first the Pµ1···µk form a closed subalgebrawhich naturally generalize w∞to higher dimensions [5], in the second bracket this is not thecase.

It is not clear to us at this point if these extra fields are intrinsically necessary for theclosure of the nonlinear algebra, or if they can be wiped out by some suitable hamiltonianreduction. Another important difference lies in the difficulty in constructing an analog of– 8 –

wn in higher dimensions. This should be reminiscent of the work in [7], where it was shownthat one way to construct consistent theories involving higher spin gauge fields is through theintroduction of an infinite number of fields with all possible spins.

However, this remark mustbe taken with due care, since our algebras represent Hamiltonian structures, and thereforeshould rather be related to the spatial part of diffeomorphism invariant field theories, suchas for example canonical gravity. Yet the fact that these infinite dimensional algebras ariseas hamiltonian structures of integrable systems, points in the direction of an algebraical, orconformal-field-theory-like approach, in the search for the quantization of those theories.AcknowledgmentsWe are grateful to J. M. Figueroa O’Farrill, J. Petersen and C. M. Hull for a carefulreading of the manuscript.

ER takes great pleasure in expressing his thanks to the PhysicsDept. of the University of Santiago for its hospitality.REFERENCES[1] J. M. Figueroa-O’Farrill and E. Ramos, The Classical Limit of W-Algebras, Phys.

Lett.282B (1992) 357,(hep-th/9202040). [2] F.Mart´ınez-Mor´as and E. Ramos, Higher Dimensional Classical W-algebras , Preprint-KUL-TF-92/19 and US-FT/6-92,(hep-th9206040).

[3] V. W. Guillemin, Advances in Math. 10 (1985) 131.

[4] M. Wodzicki,Noncommutative Residue in K-Theory, Arithmetic and Geometry. Ed.Yu.

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[5] C.M. Hull, W-geometry, Preprint-QMW-92-6, (hep-th/9211113).

[6] K. H. Bhaskara and K. Viswanath, Poisson Algebras and Poisson Manifolds, PitmanResearch Notes in Math. Series, vol.

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Fradkin and M.A. Vasil’ev, Ann.

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