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W Gravity From Chern–Simons Theory

arXiv:hep-th/9112060v1 20 Dec 1991THU-91/2412/91W Gravity From Chern–Simons TheoryJan de Boer∗and Jacob Goeree†Institute for Theoretical PhysicsUniversity of UtrechtPrincetonplein 5P.O. Box 80.0063508 TA UtrechtAbstractStarting with three dimensional Chern–Simons theory with gauge group Sl(N, IR),we derive an action Scov invariant under both left and right WN transformations.We give an interpretation of Scov in terms of anomalies, and discuss its relation withToda theory.∗e-mail: deboer@ruunts.fys.ruu.nl†e-mail: goeree@ruunts.fys.ruu.nl

1. IntroductionA few years ago Zamolodchikov made a systematic study of the possible exten-sions of the Virasoro algebra [1].

Besides the extensions involving Kac-Moody andsuperconformal currents, he found a new non-linear extension of the Virasoro alge-bra, based on the occurrence of a spin-three field W. This field, together with theusual stress energy tensor T, forms the so-called W3 algebra, which is defined by thefollowing operator product expansions:T(z)T(w)∼c/2(z −w)4 + 2T(w)(z −w)2 + ∂T(w)z −w ,T(z)W(w)∼3W(w)(z −w)2 + ∂W(w)z −w ,W(z)W(w)∼c/3(z −w)6 + 2T(w)(z −w)4 + ∂T(w)(z −w)3+1(z −w)2310∂2T(w) + 2b2Λ(w)+1z −w115∂3T(w) + b2∂Λ(w),(1.1)where the non-linear term Λ(z) is defined asΛ(z) = : T(z)T(z) : −310∂2T(z),(1.2)and the constant b2, determined by associativity, readsb2 =1622 + 5c. (1.3)The above algebra can be generalized to the case of a WN algebra, which containsfields W (k) of spin k, with k running from 2 to N. In [2] it was shown that thisalgebra is intimately related to the affine Kac-Moody algebra associated to su(N),in the sense that the fields W (k) can be constructed from the Kac-Moody currentsby using the higher-order Casimir invariants of the underlying finite dimensionalLie algebra.

Besides this connection with affine Kac-Moody algebras, it is now clearthat these non-linear extensions of the Virasoro algebra play a central role in manyother areas of two dimensional physics. They have been shown to appear in Toda2

theories [3], gauged WZW models [4], reductions of the KP hierarchy [5, 20] and inthe matrix model formulation of two dimensional quantum gravity [7].Despite the relevance of these W algebras for the above mentioned branches oftwo dimensional physics, which makes it clear that they represent some universalstructure, it is fair to say that some aspects of these algebras are still poorly under-stood. Although by now we have many different realizations of these algebras andmost of the algebraic aspects are well sorted out, it is the geometrical interpretationof these algebras which is still missing.

Whereas we know that the Virasoro algebraarises after gauge fixing the two dimensional diffeomorphism invariance (conformalgauge), a similar geometrical interpretation for the W case is lacking.In this paper we will try to uncover some of the mysteries of ‘W geometry’ byconstructing an action Scov which has local W transformations as its symmetries.This action then describes what is commonly denoted as W gravity. Scov can beviewed as the W generalization of the covariant action for pure gravity, first con-structed by Polyakov [8]:S =c96πZZR ✷−1R.

(1.4)We will construct Scov by starting with a topological gauge theory in three dimen-sions, namely Chern–Simons theory [12], on a three manifold of the form M = Σ×IR,where Σ is a two dimensional Riemann surface. What we will end up with is thena theory for W gravity on Σ.

The reason for this approach is that we believe thatthe moduli space of WN gravity is related (if not equal) to the moduli space of flatSl(N, IR) bundles. This latter space appears naturally as the classical phase space ofSl(N, IR) Chern–Simons theory, which hints towards a possible connection betweenthis theory and W gravity.

Indeed, it was shown by H. Verlinde that the aboveaction for pure gravity (1.4) can be obtained from Sl(2, IR) Chern–Simons theory[13].There have been previous constructions of actions which admit local W3 sym-metries, which all amount to gauging of the W3 transformations. This techniquehas led to both a chiral action and to a fully covariant action for W3 gravity [19].As we will show, our construction results in an action closely related to the one of[19].

An important advantage of our method is that we can apply it quite gener-ally to Sl(N, IR), resulting in the covariant action for WN gravity. Specializing tothe case of Sl(3, IR) gives our result for W3 gravity, which was already reported in[14].

Another advantage is that formulating the WN gravity in terms of Sl(N, IR)3

Chern–Simons theory should make it more tractable to study its moduli space.This paper is organized as follows. In section 2 we will review some general-ities about Chern–Simons theory.

We will introduce the concepts needed for theconstruction of the covariant action, such as wavefunctions, polarization, the inner-product, the K¨ahler potential etc. Furthermore, we will illustrate our method byfirst reviewing the construction of the covariant action in the so-called ‘standard po-larization’ [21, 25].

Next, in sections 3 and 4, we will compute the covariant actionScov for a different choice of polarization, namely the choice which relates Sl(N, IR)transformations to WN transformations. It will turn out that this first result for thecovariant action admits a large symmetry group, which can be used to gauge awaysome of the degrees of freedom.

In addition Scov contains a number of auxiliary fieldswhich can be eliminated by replacing them by their equations of motion. Detailswill be given for the case of ordinary and W3 gravity.

In section 5 we will prove theinvariance of Scov under left and right WN transformations, discuss its relation withToda theory, and give an interpretation of Scov in terms of anomalies. Finally, wewill address some open problems and give some concluding remarks in section 6.2.

Chern–Simons theoryChern–Simons theory on a three manifold M is described by the actionS =k4πiZM Tr( ˜A ∧˜d ˜A + 23 ˜A ∧˜A ∧˜A),(2.1)where the connection ˜A is a one form with values in the Lie algebra g of some Liegroup G, and ˜d denotes the exterior derivative on M. In this paper M will be of theform M = Σ×IR, Σ being a Riemann surface, for which ˜A and ˜d can be decomposedinto space and time components, i.e. ˜A = A0dt + A, with A = Azdz + A¯zd¯z, and˜d = dt∂/∂t + d. Rewriting the action asS =k4πiZdtZΣ Tr(A ∧∂tA + 2A0(dA + A ∧A)),(2.2)we recognize that A0 acts as a Lagrange multiplier which implements the constraintF = dA + A ∧A = 0.Furthermore, we deduce from this action the following4

non-vanishing Poisson brackets{Aa¯z(z), Abz(w)} = 2πik ηabδ(z −w),(2.3)where Az = Pa AazT a, with Tr(T aT b) = ηab.Upon quantizing the theory we have to replace the above Poisson bracket bya commutator, and we have to choose a ‘polarization.’This simply means thatwe have to divide the set of variables (Aaz, Aa¯z) into two subsets. One subset willcontain fields Xi and the other subset will consist of derivativesδδXi, in accordancewith (2.3).

The choice of these subsets is called a choice of polarization. We willdenote the subset containing the fields by ΠiAz and ΠkA¯z, where Πi and Πk arecertain projections on subspaces of the Lie algebra g. The standard polarizationcorresponds to the case where Πi = 1 and Πk = 0.Of course we also have to incorporate the Gauss law constraints F(Az, A¯z) = 0.Following [13, 22, 24] we will impose these constraints after quantization.

So we willfirst consider a ‘big’ Hilbert space obtained by quantization of (2.3), and then selectthe physical wavefunctions Ψ by requiring F(Az, A¯z)Ψ = 0.As an example of this construction let us consider the standard polarization, inwhich the set of fields is given by the Aaz, so the Aa¯z act as derivatives −2πkδδAaz . Thisimplies that the physical wavefunctions will be functions of Az.

For this choice ofpolarization it is well known [21] that the solution of the zero-curvature constraintF Az, 2πkδδAz!Ψ(Az) = −: 2πk ∂δδAz+ ¯∂Az + 2πk [Az,δδAz] : Ψ(Az) = 0,(2.4)is given by:Ψ(Az) = exp S(Az) = exp −SW ZW(g),(2.5)where SW ZW(g) is the Wess-Zumino-Witten action:SW ZW(g) =k4πZd2z Tr(g−1∂gg−1 ¯∂g) −k12πZB Tr(g−1dg)3,(2.6)and Az and g are related via Az = g−1∂g.If we now want to compute transition amplitudes between some initial physicalstate Ψ1 and some final state Ψ2, we have to consider the inner-product between5

these states in Chern–Simons theory. (The evolution operator is simply the identityhere, as the Hamiltonian vanishes for a topological theory.) The expression for suchan inner-product is:⟨Ψ1 | Ψ2⟩=ZDAeK(Az,A¯z) ¯Ψ1(A¯z)Ψ2(Az).

(2.7)This formula should be read as follows. (i) DA is short for DAzDA¯z.

(ii) ¯Ψ(A¯z) isthe solution of the zero-curvature constraint, but now with the role of Az and A¯zinterchanged. So it is given by:¯Ψ(A¯z) = exp ¯S(A¯z) = exp −SW ZW(h),(2.8)with A¯z = h¯∂h−1.

(iii) The K¨ahler term K(Az, A¯z) appears since we want to takethe inner-product between wavefunctions depending on different variables, Az andA¯z, which are conjugate variables. So we should perform a ‘Fourier’ transformation.In (2.7) this is automatically taken care offif one does the integral over A¯z (or Az),provided that the K¨ahler term takes the form:K(Az, A¯z) =k2πZd2z Tr(AzA¯z).

(2.9)The inner-product can now be written as:⟨Ψ1 | Ψ2⟩=ZDA exp Scov(Az, A¯z),(2.10)where Scov(Az, A¯z) = S(Az)+ ¯S(A¯z)+K(Az, A¯z) is a covariant action, i.e. invariantunder both left and right transformations given byδAz=∂η + [Az, η],δA¯z=¯∂η + [A¯z, η].

(2.11)Expressed in terms of the group variables g, h Scov takes the following form:Scov = −SW ZW(g) −SW ZW(h) −k2πZd2z Tr(g−1∂g ¯∂hh−1). (2.12)6

This action is invariant under the transformations g →gf, h →f −1h, which can beeasily proven if one makes of the Polyakov–Wiegman formula [11]:SW ZW(gf) = SW ZW(g) + SW ZW(f) + k2πZd2z Tr(g−1∂g ¯∂ff −1). (2.13)(In fact the invariance of Scov under g →gf, h →f −1h is just the integrated formof (2.11).) From this invariance one suspects that it should be possible to write Scovin terms of the invariant product G = gh.

Indeed, comparing (2.12) with (2.13), itis evident that the covariant action is given by [9, 25]:Scov = −SW ZW(G). (2.14)In this paper we will repeat the above steps for a different polarization, namelyone which relates the Sl(N, IR) gauge transformations to WN transformations.

Forthis polarization we will solve the Gauss law constraint FΨ = 0, compute the inner-product between two wavefunctions Ψ1 = exp S and ¯Ψ2 = exp ¯S, add a K¨ahler termK, to end up with a covariant action Scov = S+ ¯S+K. Scov will depend on two groupvariables g, h, which are elements of a certain subgroup of Sl(N, IR), and on 2(N −1)parameters µi, ¯µi (i = 2, .

. .

, N) which will play the role of conjugate variablesof the Wi, ¯Wi fields of the WN algebra. We will show that for this non-standardpolarization Scov is again invariant under transformations of the form g →gf, h →f −1h, where f is now restricted to a Sl(N −1, IR)×IR subgroup of Sl(N, IR).

Usingthis invariance we will be able to rewrite the action in terms of the invariant productG = gh. The resulting action Scov(G) is our definition for the covariant action forWN gravity.

This action is invariant under both left and right WN transformations.So, in comparison to the standard polarization, we have somehow split the Sl(N, IR)symmetry transformations into Sl(N−1, IR)×IR and WN, ¯WN transformations. Notethat this splitting is in agreement with the following dimension formuladim sl(N, IR) = dim sl(N −1, IR) + 1 + 2(N −1).

(2.15)7

3. The Solution of the Zero-Curvature ConstraintIn this section we will solve the zero-curvature constraints of Chern–Simonstheory for G = Sl(N, IR) in a certain, nonstandard polarization, following closely thestrategy of [14].

Let us first state the main result of this section: in the polarizationwhere we take as fields ΠiAz and ΠkA¯z, and as derivatives with respect to thesefields therefore Π†kAz and Π†iA¯z, the solutions of the zero-curvature constraints arewave functions of the form:Ψ[ΠiAz, ΠkA¯z] = eS(ΠiAz,ΠkA¯z)Ψ[µ2, . .

. , µN],(3.1)where µk denote the (1 −k, 1) differentials that occur naturally in WN-gravity (forinstance, µ2 is just the well-known Beltrami-differential), and Ψ[µ2, .

. .

, µN] solvesthe Ward identities of the classical WN-algebra [20]. The action S is given in equation(3.24).

The nonstandard polarization is given by the projections Πi and Πk. Theseare projections on certain subspaces of the Lie algebra sl(N, IR), that form closedsub-Lie algebras.

The subalgebra Πksl(N, IR) is the abelian subalgebra that consistsof all N × N matrices Mij with Mij = 0 unless i < N and j = N. The subalgebraΠisl(N, IR) consists of all traceless N × N matrices Mij with Mij = 0 if i = N andj < N. More explicitly:Πisl(N, IR) =∗· · ·∗∗.........∗· · ·∗∗0· · ·0∗,Πksl(N, IR) =0· · ·0∗.........0· · ·0∗0· · ·00. (3.2)The projections Π†i and Π†k are defined through Π†i = 1 −Πk and Π†k = 1 −Πi.

Notethat for arbitrary X, Y ∈sl(N, IR) we have Tr(ΠiXΠkY ) = 0, so that the Poissonbracket (2.3) of any two fields is indeed zero. Another important property is thatX ∈Πig, Y ∈Πkg ⇒[X, Y ] ∈Πkg.

(3.3)In order to understand why we need solutions of the classical WN-Ward identitiesin (3.1), we will first investigate the relation between the zero curvature equationF(Az, A¯z) = 0 and the classical WN-Ward identities.8

3.1. The WN-Ward identitiesThe relation between WN-Ward identities and zero-curvature equations is essen-tially due to Drinfel’d and Sokolov [5], who showed that taking a particular formfor the connection Az gives in a natural way (via Hamiltonian reduction) the secondGelfand-Dickii bracket [6].

In turn, it is known [20] that these brackets reproduceexactly the classical form of the operator expansions of WN-algebras. It is thereforea natural starting point to take the same form for Az as Drinfel’d and Sokolov did:A0z =010· · ·00001· · ·00...............000· · ·10000· · ·01WNWN−1WN−2· · ·W20.

(3.4)We will write this also asA0z = Λ + W(3.5)where Λ denotes the matrix with only the one’s next to the diagonal, and W denotesthe piece containing only the fields Wi, so that W ∈Π†kg. We will sometimes alsowrite Λ = PN−1i=1 ei,i+1, where ei,j is the matrix with a one in its (i, j) entry, andzeroes everywhere else.

The zero-curvature equation now readsF = ∂A0¯z −¯∂W + [Λ + W, A0¯z] = 0. (3.6)Since Az and A¯z are conjugate variables with respect to the Poisson bracket (2.3),and A0z still contains arbitrary fields Wi, we will put the fields conjugate to the Wi inA¯z.

These fields are precisely the (1 −k, 1) differentials µk, and we therefore requirethatΠkA0¯z =N−1Xi=1µN+1−iei,N. (3.7)In matrix notation this means that A0¯z has the formA0¯z =∗· · ·∗µN∗· · ·∗µN−1.........∗· · ·∗µ2∗· · ·∗∗.

(3.8)9

It is now possible to solve the equation ΠiF = 0 and to determine all other entriesin A0¯z. All other entries in A0¯z are polynomials in µi, Wi, and their derivatives.

Whatremains are the N−1 equations Π†kF = 0, and these are precisely the Ward-identitiesof the WN-algebra, which follows from [5]. The explicit form of A0¯z is given in theappendix.

For example, in the case of Sl(2, IR) one findsA0¯z = 12∂µµµT −12∂2µ−12∂µ!,(3.9)and the remaining zero-curvature equation reads0 = F21 = −12∂3µ + ∂(µT) + (∂µ)T −¯∂T. (3.10)Here we used the standard notation W2 = T.In Chern–Simons theory we have, however, an arbitrary connection A, and nota special one like (3.4).

It turns out that we need gauge transformations to obtainWN-Ward identities from arbitrary connections A.3.2. The Role of Gauge TransformationsIf we impose zero-curvature constraints on wave-functions Ψ, then we must spec-ify (as in ordinary quantum mechanics) how we quantize the expression F(Az, A¯z) =∂A¯z −¯∂Az + [Az, A¯z], if we replace Π†iA¯z and Π†kAz by functional derivatives withrespect to ΠiAz and ΠkA¯z.

We will simply put all derivatives to the right of thefields∗. Looking at the expression for the Poisson bracket (2.3), we see that, whenacting on wave functions Ψ[µ2, .

. .

, µN], Wi should be identified with 2πkδδµi . Thestatement that Ψ[µ2, .

. .

, µN] solves the WN-Ward identities, is equivalent toF(A0z, A0¯z)Ψ[µ2, . .

. , µN] = 0.

(3.11)At this point we make the crucial observation, that if g is an arbitrary Sl(N, IR)-valued function, independent of the µi, equation (3.11) impliesF((A0z)g, (A0¯z)g)Ψ = g−1F(A0z, A0¯z)gΨ = 0,(3.12)∗A different choice would differ from ours by terms that are of higher order in 1/c; becauseour approach is valid only up to the lowest order in 1/c, in which case the quantum WN-algebrareduces to the classical one, we can completely neglect such differences.10

where (A0z)g and (A0¯z)g denote the gauge transformed connections g−1A0zg + g−1∂gand g−1A0¯zg + g−1¯∂g. We will assume that this is the most general curvature onecan write down, which annihilates those Ψ that solve the WN-Ward identities.3.3.

The SolutionLet us now go back to the original problem, that is, solving the zero-curvatureconstraint Fψ = 0, whereΠ†kAz = 2πkδδΠkA¯z(3.13)andΠ†iA¯z = −2πkδδΠiAz,(3.14)and let us look for solutions of type (3.1), i.e. ψ = eSΨ.

Multiplying the zero-curvature equation with e−S, it reads (e−SFeS)Ψ = 0. This equation can also bewritten asF Az + 2πkδSδΠkA¯z, A¯z −2πkδSδΠiAz!Ψ = 0.

(3.15)F contains, in general, double derivatives, giving rise to terms2πk2δ2SδA1δA2 whenworking out e−SFeS.However, these terms are of higher order in 1/c, and, asdiscussed previously, we will ignore these. Now we want to show that any Ψ thatsolves the WN-Ward identities, is a solution of (3.15).

Because we assumed that themost general curvature which annihilates such wave functions Ψ is given by (3.12),we find, upon comparing (3.12) and (3.15), that solutions of the form (3.1) exist ifand only if we can find an S such thatAz + 2πkδSδΠkA¯z=(A0z)g + derivatives not containingδδµi,(3.16)A¯z −2πkδSδΠiAz=(A0¯z)g + derivatives not containingδδµi. (3.17)Restricting (3.16) to the part in Πig, we find that ΠiAz = Πi(g−1∂g + g−1Λg +g−1Wg).

The matrix W contains derivatives with respect to the µi, and we do notwant derivatives in the parametrization of our fields. Therefore, g should satisfyΠi(g−1Wg) = 0.

One may easily verify that this restricts g to be an element ofGP = exp(Π†ig), the group which has Π†ig as its Lie algebra. This shows that we11

must parametrize ΠiAz viaΠiAz = Πi(g−1∂g + g−1Λg) ≡ΠiΛg,(3.18)where we defined Λg = g−1Λg + g−1∂g. Similarly, taking the Πk of (3.17), we findthe parametrization for ΠkA¯z:ΠkA¯z = Πk(g−1A0¯zg + g−1 ¯∂g) = Πk(g−1A0¯zg).

(3.19)Due to the fact that Πk(g−1A¯zg) = Πk(g−1(ΠkA0¯z)g), no derivatives will enter inthe definition of ΠkA¯z either, and we conclude that we have a parametrization ofthe N2 −1 fields ΠiAz and ΠkA¯z in terms of N2 −1 independent variables, theN2 −N components of g ∈GP, and the N −1 variables µi. The equations, whichwe still have to solve, are the components of (3.16) in Π†kg, and the components of(3.17) in Π†ig.

They read, when separated into pieces which do and do not containderivatives,2πkδSδΠkA¯z=Π†kΛg,(3.20)−2πkδSδΠiAz=Π†i(g−1A0¯z,fg + g−1 ¯∂g),(3.21)2πkδδΠkA¯z=Π†k(g−1Wg) + derivatives not containingδδµi,(3.22)−2πkδδΠiAz=Π†i(g−1A0¯z,dg) + derivatives not containingδδµi,(3.23)where A0¯z = A0¯z,f + A0¯z,d, and A0¯z,d is the part of A0¯z containing the derivatives. Asis shown in the appendix, the last two equations are automatically satisfied in theparametrization (3.18) and (3.19).

Thus it remains to solve S from (3.20) and (3.21).This can be done, resulting inS =k2πZd2z Tr(Πk(g−1A0¯zg)Π†kΛg) −k2πZd2z Tr(Λ¯∂gg−1) −SW ZW(g),(3.24)where SW ZW is the Wess-Zumino-Witten action defined in (2.6). Equation (3.20)follows straightforwardly from (3.24), because varying ΠkA¯z while keeping ΠiAzconstant, means we only need to vary A0¯z, or, equivalently, the µi.

Under such avariation,δS =k2πZd2z Tr(δ(ΠkA¯z)Π†kΛg),(3.25)12

showing (3.20). To demonstrate (3.21), we have to vary ΠiAz while keeping ΠkA¯zconstant.

If we denote the corresponding variation of g by δg, we findδS =k2πZd2z Tr(Πk(g−1A0¯zg)δ(Π†kΛg)) −k2πZd2z Tr(Λδ(¯∂gg−1)) −δSW ZW(g). (3.26)Using Πk = 1 −Π†i the first term in (3.26) can be written ask2πZd2z Tr((g−1A0¯z,fg + g−1 ¯∂g)δ(Λg) −Π†i(g−1A0¯z,fg + g−1 ¯∂g)δ(ΠiAz)).

(3.27)The second term of the last expression is already what we want, so we would likethe remainder of δS to vanish. Using straightforward algebra, this remainder canbe written asδS=−k2πZd2z Tr(F(Λg, g−1A0¯z,fg + g−1¯∂g)(g−1δg))=−k2πZd2z Tr(F(Λ, A0¯z,f)(δgg−1)).

(3.28)Recall that we constructed A0¯z in such a way that ΠiF(Λ + W, A0¯z,f + A0¯z,d) = 0;restricting this to the piece containing no derivatives, gives ΠiF(Λ, A0¯z,f) = 0. Asδgg−1 ∈Π†ig, it follows immediately that (3.28) vanishes, proving the validity of(3.21).The wave functions eSΨ obtained here are the analogue of (2.5) in a different po-larization.

In fact, they should be seen as the Fourier transform of (2.5) with respectto Π†kA¯z. The first term of (3.24) can be more or less understood as arising fromthis Fourier transform.

Furthermore, we see that in (3.24) part of the WZW actionhas survived in the form SW ZW(g). The whole action (3.24) bears an interestingsimilarity with the wave functions introduced in [24].

The wave functions Ψ thatsolve the classical WN-Ward identities can be obtained from a constrained WZWmodel as in [21, 28]. Having solved the zero-curvature equations, we can in the nextsection proceed with the computation of inner-products in this polarization.4.

The Covariant ActionAs explained in section 2 we have to compute the inner-product between twowavefunctions Ψ1 and ¯Ψ2 to obtain our first result for the covariant action for WN13

gravity. The wavefunction ¯Ψ can be constructed in a similar way as Ψ was con-structed in the previous section.

Introducing gauge fields Bz, B¯z, where the fieldsare now in Π†kBz and Π†iB¯z, the wavefunction ¯Ψ takes the form:¯Ψ[Π†kBz, Π†iB¯z] = e¯S(Π†kBz,Π†i B¯z) ¯Ψ[¯µ2, . .

. , ¯µN].

(4.1)Here Π†kBz and Π†iB¯z are parametrized byΠ†kBz=Π†k(hB0zh−1 −∂hh−1),Π†iB¯z=Π†i(h¯Λh−1 −¯∂hh−1) ≡Π†i ¯Λh,(4.2)with ¯Λ = Λt = PN−1i=1 ei+1,i,Π†kB0z =N−1Xi=1¯µN+1−ieN,i,(4.3)and the other components of B0z can be computed from the condition Π†iF(B0z, ¯Λ) =0. In (4.2) h ∈exp(Πisl(N, IR)), and ¯S appearing in (4.1) is given by:¯S =k2πZd2z Tr(Π†k(hB0zh−1)Πk ¯Λh) + k2πZd2z Tr(¯Λh−1∂h) −SW ZW(h).

(4.4)The last ingredient we need in our construction of the covariant action is the K¨ahlerform. Since this K¨ahler form should establish the Fourier transformation from ΠiAzto Π†iB¯z and from Π†kBz to ΠkA¯z (or vice versa), it is given by:K(A, B) =k2πZd2z Tr(ΠiAzΠ†iB¯z −ΠkA¯zΠ†kBz),(4.5)(where the minus sign follows from standard Fourier theory:if one uses eipx asintegration kernel to transform from x to p, one should use e−ipx to go from p to x).Using the explicit form of the inner-product⟨Ψ1 | Ψ2⟩=ZD(ΠiAz)D(ΠkA¯z)D(Π†iB¯z)D(Π†kBz) eS+ ¯S+K ¯Ψ1[¯µ]Ψ2[µ]≡ZD(ΠiAz)D(ΠkA¯z)D(Π†iB¯z)D(Π†kBz) eScov(A,B),(4.6)we can now read offthe covariant action for WN gravity.

WritingΨ[µ2, . .

. , µN] = exp −SWN(µ),(4.7)14

this result reads:Scov=k2πZd2z Tr(Πk(g−1A0¯zg)Π†kΛg) −k2πZd2z Tr(Λ¯∂gg−1) −SW ZW(g)+k2πZd2z Tr(Π†k(hBzh−1)Πk ¯Λh) + k2πZd2z Tr(¯Λh−1∂h) −SW ZW(h)+k2πZd2z Tr(ΠiAzΠ†iB¯z −ΠkA¯zΠ†kBz) −SWN(µ) −¯SWN(¯µ). (4.8)This action depends on g, h, µi, ¯µi which amounts to a total number of variablesgiven by: 2N(N −1) + 2(N −1) = 2(N2 −1).

On the other hand, one expectsWN gravity to be described by the fields gµν, dµνρ, . .

., which are symmetric tensorsof rank 2, 3, . .

. , N, resulting in a total number of degrees of freedom given by:3+4+· · ·+(N +1) = 12(N −1)(N +4).

The discrepancy between the total numberof degrees of freedom in Scov as given in (4.8), and the total number of degrees offreedom in WN gravity can be partially resolved as follows: (i) Scov is invariant undera Sl(N −1, IR) × IR symmetry group, which can be used to gauge away (N −1)2degrees of freedom, (ii) in Scov there are auxiliary fields (i.e. fields which appear onlyalgebraically in Scov), which can be eliminated by replacing them by their equationsof motion.4.1.

Symmetries of the ActionTo investigate the symmetries of the above action (4.8) we define another pro-jection operator Πy = ΠiΠ†i, by decomposing an arbitrary element of sl(N, IR) inthe following way:sl(N, IR) = Πksl(N, IR) ⊕Πysl(N, IR) ⊕Π†ksl(N, IR),(4.9)so an element f ∈exp(Πysl(N, IR)) is of the formf =∗· · ·∗0.........∗· · ·∗00· · ·0∗,(4.10)and they form a Sl(N −1, IR) × IR subgroup. We expect that Scov will be invariantunder transformations of the form g →gf and h →f −1h.

(This is motivated by15

the fact that for the standard polarization (where Πk = 0 so Πy = 1) f would bean arbitrary element of Sl(N, IR), and, as we saw in section 2, for that case thesetransformations indeed leave the covariant action invariant.) So let us study how Schanges under the transformation g →gf and h →f −1h, with f as in (4.10).Note that for any element f ∈exp(Πysl(N, IR)) we haveAdf(Πi,ksl(N, IR)) ⊂Πi,ksl(N, IR),(4.11)(and the same holds for Π†i,k), where Adf(g) = f −1gf.One easily checks thatthis implies, that for any X ∈sl(N, IR) we have Πi,kAdf(X) = Πi,kAdf(Πi,kX) =Adf(Πi,kX), i.e.Πi,k ◦Adf = Adf ◦Πi,k(4.12)(and again the same holds for the Π†i,k).Using (4.12) and the fact that Π†kf −1∂f = 0 we see that the first terms in thefirst and second line of (4.8) are invariant, whereas the rest of the action changes as:δScov=−k2πZd2z Tr(g−1Λg ¯∂ff −1) −SW ZW(f) −k2πZd2z Tr(g−1∂g ¯∂ff −1)−k2πZd2z Tr(h¯Λh−1∂ff −1) −SW ZW(f −1) + k2πZd2z Tr(∂ff −1 ¯∂hh−1)+ k2πZd2z Tr(∂ff −1¯Λh + ¯∂ff −1Λg + f −1∂ff −1 ¯∂f),(4.13)where we used the Polyakov-Wiegman formula (2.13) and the fact that Tr(XΠi,kY ) =Tr(Y Π†i,kX).

SinceSW ZW(f) + SW ZW(f −1) =k2πZd2z Tr(f −1∂ff −1 ¯∂f),(4.14)it follows that Scov is invariant under the above transformations.In a similar way as was done in section 2 for the standard polarization it is nowstraightforward to write down the action in terms of the invariant product G = gh.We find Scov = ∆S −SWN(µ) −¯SWN(¯µ), with∆S=k2πZd2z Tr(ΛG¯ΛG−1) + k2πZd2z Tr(¯ΛG−1∂G) −k2πZd2z Tr(Λ¯∂GG−1)−k2πZd2z Tr(Πk(A0¯z −¯ΛG)ΠyGΠ†k(B0z −ΛG)ΠyG−1) −SW ZW(G),(4.15)16

where we used the following decomposition for G: G = Π†kGΠyGΠkG, which interms of matrices looks like:G =1...1∗· · ·∗1∗· · ·∗0.........∗· · ·∗00· · ·0∗1· · ·∗......1∗1. (4.16)Scov = ∆S −SWN(µ) −¯SWN(¯µ) is our final result for the covariant action for WNgravity.

Note that in Scov only the Πk part of A0¯z and the Π†k part of B0z appear.Thus specifying A0¯z, B0z as in (3.7), (4.3) is sufficient to compute the action. As wewill see in the next subsection it turns out that Scov still contains redundant degreesof freedom.4.2.

Auxiliary FieldsAt this stage our action depends on G, µi, ¯µi, so we have reduced the number ofdegrees of freedom to: N2 −1 + 2(N −1) = (N + 3)(N −1), which is still morethan one would naively expect for WN gravity. Below we will further reduce thisnumber due to the observation that some fields only appear algebraically in the ac-tion, i.e.

auxiliary fields, which can in principle be eliminated by replacing them bytheir equations of motion. Unfortunately, the general expression (4.15) valid for allSl(N, IR) is not suitable for the determination of auxiliary fields.

Instead, we haveto use an explicit parametrization for G in (4.15) in order to isolate the auxiliaryfields. We will work this out for the case of Sl(2, IR) and Sl(3, IR).Sl(2, IR):For this case we parametrize our G as follows:G = 10ω1!

eφ00e−φ! 1−¯ω01!.

(4.17)Labeling A0¯z, B0z as in (3.7), (4.3), respectively, we find Scov = ∆S −SW2[µ] −¯SW2[¯µ]with,17

∆S = −k2πZd2zh∂φ¯∂φ + ω(2¯∂φ + ∂µ) + ¯ω(2∂φ + ¯∂¯µ)+µω2 + ¯µ¯ω2 + 2ω¯ω −(1 −µ¯µ)e−2φi,(4.18)and SW2[µ] is the solution of the Ward-identity(¯∂−µ∂−2∂µ)δSW2[µ]δµ= −k4π∂3µ. (4.19)From (4.18) we recognize that ω, ¯ω are auxiliary fields.

Replacing these fields bytheir equations of motion gives:Scov = SL[φ, µ, ¯µ] + K[µ, ¯µ] −SW2[µ] −¯SW2[¯µ]. (4.20)HereSL =k4πZd2zq−ˆgˆgab∂aφ∂bφ + 4e−2φ + φ ˆR,(4.21)is the well-known Liouville action, the metric ˆg is defined by ds2 = |dz + µd¯z|2, andK[µ, ¯µ] readsK[µ, ¯µ] =k4πZd2z (1 −µ¯µ)−1 ∂µ¯∂µ −12µ(¯∂¯µ)2 −12 ¯µ(∂µ)2.

(4.22)So, finally, we have reduced our set of fields to the three basic ones, namely thethree components by which we label the metric g = e−2φˆg. Scov as given in (4.20) isour final result for the case of Sl(2, IR), and is in fact almost equivalent to Polyakov’saction for induced 2D gravity:Scov =c96πZZR ✷−1R,(4.23)written out in components for the metric g = e−2φˆg.

The only difference is thecosmological termR d2z √−ˆge−2φ, which one usually adds to the induced action. Theabsolute value of the coefficient in front of the cosmological term is not important,as it can be arbitrarily rescaled by adding a constant to φ.Note that the term in the Liouville action (4.21) that is linear in φ, is propor-tional to the curvature R. This is due to the fact that the trace or Weyl anomaly18

is also proportional to the curvature. Actually, this fact can already be seen from(4.18), where we did not yet eliminate ω and ¯ω.

The term linear in φ in (4.18) isproportional toR φdω, where ω is the one form ω dz −¯ω d¯z. This shows that ωshould be interpreted as being the spin connection [13], since R is the curvature ofthe spin connection.Sl(3, IR)This case was already extensively discussed in [14].There it was shown that ifwe take the following Gauss decomposition for G:G =100ω110ω3ω21eϕ1000eϕ2−ϕ1000e−ϕ21−¯ω1−¯ω301−¯ω2001,(4.24)and parametrize A0¯z and B0z again as in (3.7) and (4.3) (with µ ≡µ2, ν ≡µ3), theaction contains ω3, ¯ω3 as auxiliary fields.

Substituting their equations of motionresults in the following action: Scov = ∆S −SW3[µ, ν] −¯SW3[¯µ, ¯ν], with∆S =k2πZd2zn12Aij∂ϕi ¯∂ϕj +Xie−Aijϕj −Aij(ωi + ∂ϕi)(¯ωj + ¯∂ϕj)(4.25)−eϕ1−2ϕ2(µ −12∂ν −νω1)(¯µ + 12 ¯∂¯ν + ¯ν¯ω1) −e−ϕ1−ϕ2ν¯ν−eϕ2−2ϕ1(µ + 12∂ν + νω2)(¯µ −12 ¯∂¯ν −¯ν¯ω2) + µT + νW + ¯µ ¯T + ¯ν ¯Wo,where Aij is the Cartan matrix of Sl(3, IR) Aij = 2−1−12!. T, W, ¯T, ¯W aredefined through the following Fateev-Lyukanov [29] construction:(∂−ω2)(∂−ω1 + ω2)(∂+ ω1)=∂3 + T∂−W + 12∂T,(¯∂−¯ω2)(¯∂−¯ω1 + ¯ω2)(¯∂+ ¯ω1)=¯∂3 + ¯T ¯∂+ ¯W + 12 ¯∂¯T,(4.26)and we shifted µ →µ −12∂ν, ¯µ →¯µ+ 12∂¯ν.

The first part of ∆S is precisely a chiralSl(3, IR) Toda action, confirming the suspected relation between W3-gravity andToda theory, see also section 5.2. Actually, one would expect that in a ”conformalgauge”, the covariant W3-action will reduce to a Toda action.

Indeed, if we putν = ¯ν = 0 in ∆S, then also ω1, ω2, ¯ω1, ¯ω2 become auxiliary fields. Substituting theirequations of motion as well, we find that∆S =k4πZd2zq−ˆg 12ˆgab∂aϕi∂bϕjAij + 4Xie−Aijϕj + R⃗ξ · ⃗ϕ!+ 4K[µ, ¯µ], (4.27)19

where K[µ, ¯µ] is the same expression as for the Sl(2, IR) case (4.22), and ˆg is againgiven by ds2 = |dz +µd¯z|2. In the case of Sl(3, IR), ⃗ξ · ⃗ϕ, with ⃗ξ being one half timesthe sum of the positive roots, is just given by ϕ1 +ϕ2.

The action (4.27) is the sameToda action that was originally present in ∆S in a chiral form, and the integrationover ω1, ¯ω1, ω2, ¯ω2 has the effect of coupling it to a background metric ˆg.Of course, the most interesting part of the action is the part containing ν, ¯ν.Unfortunately, if we do not put ν = ¯ν = 0, we can integrate over either ρ1, ¯ρ1 or overρ2, ¯ρ2, but not over both at the same time, due to the presence of third order termsin ∆S. Another clue regarding the contents of the action (4.25) can be obtainedby treating the second and third line in (4.25) as perturbations of the first line of(4.25).

This means that we try to make an expansion in terms of µ, ¯µ, ν, ¯ν. Thesaddlepoint of the ω-terms is at ωi = −∂ϕi and ¯ωi = −¯∂ϕi.

From (4.26) we can nowsee that T, W, ¯T, ¯W are, when evaluated in this saddle point, the (anti)holomorphicenergy momentum tensor and W3-field that are present in a chiral Toda theoryT=−12Aij∂ϕi∂ϕj −⃗ξ · ∂2⃗ϕ,W=−∂ϕ1((∂ϕ2)2 + 12∂2ϕ2 −∂2ϕ1) + 12∂3ϕ1 −(1 ↔2),(4.28)and similar expressions for ¯T, ¯W.This suggests that the full action ∆S contains the generating functional forthe correlators of the energy-momentum tensor and the W3-field of a Toda theory,”covariantly” coupled to W3-gravity. The presence of the third order terms in W, ¯Win (4.25) prevents us from computing the action of this covariantly coupled Todatheory.

The same structure is also present in the action for WN gravity, as we willdiscuss in section 5.2.In a similar way as for SL(2, IR), the terms linear in ϕi in (4.25) are expected tobe related to the Weyl and W3-Weyl anomaly. The term linear in ϕi in (4.27) showsthat the Weyl anomaly is related to shifts in ⃗ξ · ⃗ϕ = ϕ1 + ϕ2.

This suggests that theW3-Weyl anomaly is related to shifts in directions orthogonal to ⃗ξ, that is, to shiftsof ϕ1 −ϕ2. Writing the terms in (4.25) linear in ϕi ask4πZ(ϕ1 + ϕ2)dω+ + 3(ϕ1 −ϕ2)dω−,(4.29)where the one forms ω+ and ω−are given byω± = (ω1 ± ω2)dz −(¯ω1 ± ¯ω2)d¯z,(4.30)20

we see that ω+ plays the role of the spin connection, whereas ω−is some kind of W3-spin connection, whose curvature is presumably related to the W3-Weyl anomaly.These statements may acquire a more precise meaning when comparing (4.25) with aone-loop computation for the induced action of W3-gravity, starting with the actionin [19].5. Properties of the Covariant ActionAt this stage the reader may well wonder, what all that we have done so farhas to do with covariant WN-gravity.

This is probably best explained by looking atthe case of ordinary two-dimensional quantum gravity, and before proceeding withgeneral WN-gravity, we will first discuss this much simpler and better understoodcase.5.1. 2D Quantum GravityGiven any action S(gab, ϕ), where ϕ denotes some set of matter fields, that is bothinvariant under Weyl transformations gab →eρgab and under general co-ordinatetransformations, one can define an induced action for 2D gravitye−Sind(gab) =ZDϕe−S(gab,ϕ),(5.1)by integrating out the matter fields ϕ.

If the theory had no anomalies, Sind wouldreduce to an action defined on the moduli space of Riemann surfaces. However,it is well known that there are anomalies, resulting in a non-trivial g-dependenceof Sind.

The precise form of these anomalies depends, of course, on the choice ofregularization scheme used in performing the path integral over the matter fields. Ascheme often used in conformal field theory is ζ-function regularization [15].

This is adiffeomorphism, but not Weyl invariant regularization method. In [16] it was shownthat if S is the action for a b −c system of spin j, and one parametrizes g via ds2 =eρ|dz + µd¯z|2, one can compute δSindδρ , which is proportional to the Liouville action,and δ2Sindδµδ¯µ , which is also non-vanishing, due to the lack of holomorphic factorizationof Sind.

We will call this the holomorphic anomaly.21

In Chern-Simons theory, the holomorphic wave-functions contain Ψ[µ], and theanti-holomorphic wave-functions contain ¯Ψ[¯µ], which are both solutions of the Vi-rasoro Ward-identities. These wave functions can only carry a holomorphic or anti-holomorphic diffeomorphism anomaly, but not the two anomalies of the type men-tioned before.

Therefore, these wave-functions do not fit naturally in the ζ-functionregularization scheme, but in another where the only anomalies are the holomorphicand anti-holomorphic diffeomorphism anomaly. However, it is known that these twoschemes are related to each other via a local counterterm △Γ [17].

This countert-erm consists of two pieces, a Liouville action and another term K, to cancel theholomorphic anomaly, which is proportional to (4.22). Thuse−Sind−△Γ = e−S(µ)−¯S(¯µ).

(5.2)All of this strongly suggests that the action ∆S = S + ¯S +K occurring in the inner-product (4.6) is, in the case of SL(2, IR), nothing but the local counterterm △Γ.That this is indeed the case was shown in section 4.1 (see [13]): upon integratingover the nonpropagating fields ω, ¯ω, the covariant action becomes precisely equalto the local counterterm △Γ. To make the connection more precise, if we writeΨ(µ) = e−S(µ), then (5.2) can be rewritten ase−Sind = e△ΓΨ(µ)¯Ψ(¯µ),(5.3)and we see that the partition function of induced gravity is just the inner product ascomputed in Chern-Simons theory.

This also explains the name ‘covariant’, becauseclearly we have not fixed any gauge in (5.3). For WN, we expect that a similarpicture exists, although we have no concrete realization of it.

What we do have isthe covariant action, and before discussing possible implications this action has forWN-gravity, we will first study this action in some more detail.5.2. Relation to Toda TheoryIn the case of ordinary gravity, the covariant action contains the Liouville ac-tion.

The natural generalization of the Liouville action is the Toda action basedon sl(N, IR), which is known to be deeply related to WN-algebras [3]. Indeed, wewill now show that the covariant action is closely related to Toda theory.

The cases22

N = 2, 3 were already dealt with in section 4, and we will now consider the generalcase. The relation to Toda theory is most easily established by putting µi = ¯µi = 0.The action is then given by (see (4.15))S=k2πZd2z Tr(ΛG¯ΛG−1) + k2πZd2z Tr(¯ΛG−1∂G) −k2πZd2z Tr(Λ¯∂GG−1)−SW ZW(G) −k2πZd2z Tr(Πk ¯ΛhΠ†kΛg),(5.4)where G = gh, as explained in section 4.

Under a variation of hδS = −k2πZd2z Tr((h−1δh)F(ΛG −h−1Π†k(Λg)h, ¯Λ)),(5.5)and under a variation of gδS = −k2πZd2z Tr((δgg−1)F(Λ, ¯ΛG −gΠk(¯Λh)g−1)). (5.6)Instead of G = gh we will now use a Gauss decomposition G = n1bn2 for G, wheren1 is lower triangular, b is diagonal, and n2 is upper triangular, and restrict ourselvesto variations of n1 and n2, for which (5.5) and (5.6) are still valid.

In terms of thisGauss decomposition, they can be written asδS = −k2πZd2z Tr((n−12 δn2)F(n−12 (b−1∂b)n2 + n−12 b−1Πi(Λn1)bn2, ¯Λ)),(5.7)andδS = −k2πZd2z Tr((δn1n−11 )F(Λ, −n1(¯∂bb−1)n−11−n1bΠ†i(¯Λn2)b−1n−11 )). (5.8)The equations of motion for n2 and n1 read thereforeΠF(Λ, −n1(¯∂bb−1)n−11−n1bΠ†i(¯Λn2)b−1n−11 ) = 0,(5.10)where Π< and Π> denote the projections on the space of lower and upper tri-angular matrices respectively.The first equation (5.9) can be solved as follows:Π

if and only if n2Xn−12is. Applying this to F in (5.9), we see that the equations ofmotion for n2 are solved if Πi(Λn1b) is upper triangular.

This is the case if thereexists an element A ∈Π†kg such thatn−11 ∂n1 + n−11 Λn1 = A −∂bb−1 + Λ. (5.11)In a similar way, (5.10) is solved if there is an element B ∈Πkg such that−¯∂n2n−12+ n2¯Λn−12= B + b−1 ¯∂b + ¯Λ.

(5.12)If we now replace A by −n−11 An1 and B by −n2Bn−12 , we see that n1 is a gaugetransformation relating the connections Λ −∂bb−1 and Λ + A, and n2 is a gaugetransformation relating ¯Λ+b−1 ¯∂b and ¯Λ+B. These transformations are well known:they are the Miura transformations that have been used [29] to produce free fieldexpressions for WN-algebras.

Here, the matrices A and B will contain these freefield representations. Let us now substitute the equations of motion for n1 and n2back into the full covariant action (4.15).

After some manipulations, it readsS=k4πZd2z Tr(b−1∂bb−1 ¯∂b) + k2πZd2z Tr(b¯Λb−1Λ) −k2πZd2z Tr(A0¯zA)−k2πZd2z Tr(B0zB) −k2πZd2z Tr(Πk(n−11 A0¯zn1)bΠ†k(n2B0zn−12 )b−1). (5.13)The first two terms are just an expression for a Toda theory in a flat backgroundmetric; the more conventional Toda action follows immediately by substituting b =exp diag(φ1, φ2 −φ1, .

. .

, φN−1 −φN−2, −φN−1).The first two terms read, whenexpressed in terms of φi,k4πZd2z Aij∂φi ¯∂φj + k2πZd2zXie−Aijφj,(5.14)where Aij denotes the Cartan matrix of sl(N, IR). The third and fourth term of(5.13) show that, to lowest order, µi and ¯µi couple simply the fields Wi and ¯Wi, asone would construct them from the Toda theory.

The last, mixing term in (5.13)has no simple interpretation in the Toda theory.One can in principle go through the same exercise with µ2 and ¯µ2 unequal tozero; this was worked out in [14] for the case of W3, see also sect. 4.2., and theresult in that case is that the Toda theory gets coupled to a non-trivial background24

metric, determined by µ2 and ¯µ2. We expect that the same thing is true for generalWN algebras, although we have not tried to do the computation.

If one also putsother µi or ¯µi unequal to zero, it is much more difficult to solve the equations ofmotion for n1 and n2 in full generality, and we have been unable to do so.From the previous calculations, one might be tempted to conclude that the co-variant action is just the generating function for the correlators of the Wi and ¯Wifields in a Toda theory. This is, however, not true, because most of the fields inn1 and n2 are not just auxiliary fields, and one cannot therefore in general justsubstitute their equation of motion back into the action.

We will make a few morecomments about this in of the next section.5.3. W, ¯W-Invariance of the Covariant ActionThere is an interesting analogy between the covariant action for WN-gravity, andthe covariant action given in (2.10).

If we start with an action S =R d2z ¯ψγα(∂α +Aα)ψ, then we can repeat the story in section 5.1, and define and induced action byintegrating over the fermions [9]. Now △Γ = K(Az, A¯z) is a kind of ‘chiral anomaly’,with K given by (2.9).

However, from another point of view, K is needed to restorethe gauge invariances (2.11). If we adopt this point of view here, the covariant actionwould arise as an action needed to restore W and ¯W invariance∗.

Thus it seemsnatural to look for the action of the W and ¯W algebra on the covariant action, andto check whether the covariant action indeed restores W and ¯W invariance.It turns out that this invariance is indeed present, although the expressions in-volved are rather cumbersome. We will, therefore, only describe the W-transformations,and omit the tedious proof that these leave the full covariant action invariant.

Wealso will not give the ¯W-transformations, but they can be easily written down, oncethe W-transformations are given.First consider the wave-functions Ψ(µ2, . .

. , µN).

As we discussed in section 3.1,the Ward identities that annihilate these wave functions are F(Λ + W, A0¯z)Ψ = 0.When acting on Ψ, Wi is given by 2πkδΨδµi. If we substitute these expressions for Wiback into Λ + W and A0¯z, we will denote the resulting expressions by Λ + Wψ and∗The precise form of the covariant action does, however, not seem to be completely fixed bythis requirement alone.25

A0¯z,ψ. The Ward identities are then simplyF(Λ + Wψ, A0¯z,ψ) = 0.

(5.15)On the other hand, the counterterm S + ¯S + K = △S induces also certain fieldsWi, obtained by differentiating it with respect to µi. The corresponding matrix Wcontaining these Wi will be denoted by Wind and is easily found from (4.8)Wind = −2πk gΠ†k(g−1∂g + g−1Λg −hB0zh−1)g−1.

(5.16)In the same way as we constructed A0¯z in section 3.1, we can construct an X ∈gsuch thatΠiF(Λ + Wind, X) = 0,(5.17)once we specify ΠkX. IfΠkX =0· · ·0ǫN0· · ·0ǫN−1.........0· · ·0ǫ20· · ·00,(5.18)we will denote the corresponding solution X of (5.17) by X(ǫ).

We now define thefollowing transformation rules for G and the µi:δǫG=X(ǫ)G,(5.19)δǫ(ΠkA0¯z,ψ)=−Πk(¯∂X(ǫ) + [A0¯z,ψ, X(ǫ)]),(5.20)where the ǫi are the parameters of the Wi-transformations. One can prove that (i)these transformations form a WN algebra, and (ii) that these transformations leavethe inner product invariant.

The transformation rules (5.19) and (5.20) look likeordinary gauge transformations, although in this case X is field dependent. All thisis closely related to the well-known fact that W-transformations can be realized asfield-dependent sl(N, IR) gauge transformations [21, 10]; more precisely, they arethe gauge transformations that preserve the form (3.4) of a connection.

Here, thesame mechanism is working, although in a different setting. For instance, one of thecurious features of the transformation rule (5.20) is that δǫµi can contain δΨδµj .

Only26

for W2 this is not the case. In that case the relatively simple transformation rules,leaving invariant the covariant W2 action, see (4.18) and (4.19), are given byδǫµ=−¯∂ǫ −ǫ∂µ + µ∂ǫ,δǫφ=12∂ǫ + ǫω,δǫ¯ω=−ǫe−2φ,δǫω=−12∂2ǫ + ¯µǫe−2φ −ǫ∂ω −ω∂ǫ.

(5.21)In δǫµ one recognizes the transformation rule for a Beltrami differential under aninfinitesimal co-ordinate transformation. In fact, after substituting the equations ofmotion for ω, ¯ω, we haveδǫµ=−¯∂ǫ −ǫ∂µ + µ∂ǫ,δǫφ=12∂ǫ −ǫ1 −µ¯µ(∂φ + 12 ¯∂¯µ) +¯µǫ1 −µ¯µ(¯∂φ + 12∂µ),(5.22)which we expect to be the ordinary transformation rules for the components of themetric g, defined by ds2 = e−2φ|dz + µd¯z|2, under general co-ordinate transforma-tions.

Under the transformation xµ →x′µ = xµ −ξµ the metric changes as:δgab = −gbc∂aξc −gac∂bξc −ξc∂cgab. (5.23)Writing ξa = (ξ, ¯ξ), this implies the following transformation rules for the compo-nents of the metric defined by ds2 = e−2φ|dz + µd¯z|2:δξ,¯ξφ=12∂ξ + 12 ¯∂¯ξ + 12µ∂ξ + 12 ¯µ¯∂¯ξ −ξ∂φ −¯ξ ¯∂φ,δξ,¯ξµ=−¯∂ξ + µ∂ξ −µ¯∂¯ξ −ξ∂µ −¯ξ ¯∂µ + µ2∂¯ξ,δξ,¯ξ¯µ=−∂¯ξ + ¯µ¯∂¯ξ −¯µ∂ξ −ξ∂¯µ −¯ξ ¯∂¯µ + ¯µ2 ¯∂ξ.

(5.24)Redefining our parameters as follows: ǫ = ξ +µ¯ξ, ¯ǫ = ¯ξ+ ¯µξ, the above rules becomeδǫ,¯ǫφ=12∂ǫ + 12 ¯∂¯ǫ −ǫ −µ¯ǫ1 −µ¯µ(∂φ + 12 ¯∂¯µ) −¯ǫ −¯µǫ1 −µ¯µ(¯∂φ + 12∂µ),δǫ,¯ǫµ=−¯∂ǫ + µ∂ǫ −ǫ∂µ,δǫ,¯ǫ¯µ=−∂¯ǫ + ¯µ¯∂¯ǫ −¯ǫ¯∂¯µ,(5.25)which are the same as in (5.22) for the chiral case ¯ǫ = 0. To find the general caseone should of course look at the ¯W analogue of (5.19) and (5.20).27

Having established some of the basic properties of the covariant action, we willnow discuss the implications the covariant action has for W-gravity.6. DiscussionOne of the main gaps in our knowledge of W-gravity, is that we do not knowwhat the proper set of fields is, on which the W-algebra acts.

In other words, whatis the counterpart of the metric in the case of WN-gravity? Naively, one might thinkthat one just has to add symmetric tensor fields of rank 3, .

. .

, N, as we alreadymentioned in section 4, to produce fields with the right spin. One has, however, notbeen able to perform this construction in full detail, and it is still quite a mysteryhow such a construction should work, in which presumably the conformal factorsof the tensor fields should play the role of Toda fields, quantization gives rise toanomalies, the generalized Weyl anomaly gives rise to the Toda action, etc.

It isalso conceivable that certain auxiliary fields are needed to form a full ‘W-multiplet’,and that part of these auxiliary fields become propagating on the quantum level.This shows that it is difficult to count a priori the number of degrees of freedom ofW-gravity.It is possible to give an upper limit for the number of degrees of freedom ofW-gravity, because the covariant action certainly has all degrees of freedom in it.Although we worked only up to lowest order in 1/c, we expect that the higher ordercorrections will essentially only give a field and coupling constant renormalization,as is the case for W2 [23] and seems to be the case for W3 as well [26]. Therefore,we can just count the number of degrees of freedom in the covariant action, and itis given by (N + 3)(N −1).

From this we should certainly subtract the number ofauxiliary fields in the covariant action. We do not know what this number is for thegeneral case, but for W2 it is two, and in the case of W3 it is at least two (see [14] andsection 4.2).

However, integrating out more than two fields results in this case in anonpolynomial action, whose precise meaning is rather obscure. What is certainlynot true, is that all the off-diagonal elements of G are auxiliary fields for N > 2, sothat it is in general impossible to reduce the covariant action to a Toda-like action.This happens only if we put the off-diagonal elements of G on-shell and µi = ¯µi = 0,as explained in section 5.2.28

If we compare the W3-action of [14] with the action for W3 strings given in[18], we see that the two are partially identical, except that the action in [18] doesnot have terms which couple µi and ¯µi, nor terms involving exponentials, whereasour action does have both such terms. This is due to the fact that their action isdescribing a matter system with W3 symmetry, whereas our action describes theinduced action for W3-gravity.

This relation is similar to the relation between theaction for a free boson on the one hand, and the Liouville action on the other hand.Regarding the question ‘what is the moduli space related to W-algebras’, thisapproach strongly suggests it is just (a component of) the space of flat Sl(N, IR)-bundles. We know that in the standard polarization the partition function of Chern-Simons theory can be written as the integral of some density over the moduli spaceof flat Sl(N, IR)-bundles.

The partition function does, of course, not depend onthe choice of polarization chosen, and we can therefore in principle also rewrite thepartition function of WN-gravity as the integral of some density over the moduli-space of flat Sl(N, IR)-bundles. This moduli-space should arise by looking at thespace of differentials {µi} modulo W-transformations, but it is difficult to make thisrelation more precise.Clearly, there are many problems left in this field, and we hope we will comeback to some of those in the near future.AcknowledgementsWe would like to thank P. van Nieuwenhuizen, B. de Wit and E. Bergshoeffforstimulating discussions and helpful comments.This work was financially supported by the Stichting voor Fundamenteel Onder-zoek der Materie (FOM).A.

AppendixIn this appendix we will give a derivation of (3.22) and (3.23):2πkδδΠkA¯z=Π†k(g−1Wg) + derivatives not containingδδµi,(A.1)29

−2πkδδΠiAz=Π†i(g−1A0¯z,dg) + derivatives not containingδδµi. (A.2)First of all, we will derive an expression for A0¯z, which was defined in section 3.1.

Animportant object here is the linear operator L : g →g defined as follows: one easilyverifies that X →Πi(adΛ(X)) defines an invertible linear operator Π†ig →Πig; Lis the inverse of this map, extended by 0 to an operator g →g. We also define anelement F of g byF =NXi=2µiΛi−1.

(A.3)ThenA0¯z,f =11 + L∂F,(A.4)andA0¯z,d =11 + L∂L[F, W],(A.5)where (1 + L∂)−1 meansPi≥0(−L∂)i. To see this, substitute (A.4) and (A.5) into(3.6):F=−¯∂W + ∂(1 + L∂)−1F + ∂(1 + L∂)−1L[F, W]+hΛ + W, (1 + L∂)−1L[F, W]i.

(A.6)We have to show that ΠiF = 0. First consider the term quadratic in W: [W, (1 +L∂)−1L[F, W]]; by definition, ImL ∈Π†ig, hence (1 + L∂)−1L[F, W] ∈Π†ig.

Thisimplies [W, (1+L∂)−1L[F, W]] ∈Π†kg, (cf. (3.3)), and this term does not contributeto ΠiF.

By similar reasoning it follows thatΠi[W, (1 + L∂)−1F] = Πi[W, F]. (A.7)What remains isΠiF = Πi(∂+ adΛ)(1 + L∂)−1(F + L[F, W]) + [W, F].

(A.8)The definition of L shows thatΠi(adΛ(L(X))) = ΠiX,(A.9)30

and from this we derive thatΠi(∂+ adΛ)(1 + L∂)−1X=ΠiXi≥0(−1)iLi∂i+1X +Xi≥0(−1)i+1Li∂i+1X + adΛ(X)=Πi([Λ, X]). (A.10)We can use this to evaluate (A.8):ΠiF=Πi([Λ, F + L[F, W]] + [W, F])=Πi([Λ, F]) = 0,(A.11)where the last line is a trivial consequence of the fact that F defined in (A.3)commutes with Λ.

Finally, observe that A0¯z as defined here is of the required form(3.8). One can also try to compute an expression for A0¯z by explicitly computing allthe entries of the matrix A0¯z [27], but this is less suitable for general computationsas performed in this appendix, and certainly more complicated.Armed with the expressions (A.4) and (A.5), we can compute the right handsides of (A.1) and (A.2).

Our next task will be to compute the left hand sides of(A.1) and (A.2). Recall thatΠiAz=Πi(g−1∂g + g−1Λg),(A.12)ΠkA¯z=Πk(g−1A0¯zg + g−1 ¯∂g) = Πk(g−1Fg).

(A.13)The last line follows easily from the fact that for X ∈Π†ig, Πk(g−1Xg) = 0, so that,in the expression (A.13), we can always redefine A0¯z with an arbitrary X ∈Π†ig.From (A.12) and (A.13), we find that the variations of ΠiAz and ΠkA¯z in terms ofδg and δF are given byδ(ΠiAz)=Πi(g−1∂(δgg−1)g + g−1[Λ, δgg−1]g),(A.14)δ(ΠkA¯z)=Πk(g−1δFg + g−1[F, δgg−1]g). (A.15)Now, in general, given fields fα in terms of other fields φβ, we can expressδδfαin terms ofδδφβ , by taking the transpose of the inverse of the matrix (δfα/δφβ).Here, we must first invert (A.14) and (A.15), and then take the transpose of theresulting expressions.

Starting with (A.14), and using once more that for X ∈Π†kg,Πi(g−1Xg) = 0, (A.14) can be written asΠi((∂+ adΛ)(δgg−1)) = Πi(gδ(ΠiAz)g−1). (A.16)31

From (A.9) and (A.10) we see that Πi((∂+ adΛ)(1 + L∂)−1LX) = ΠiX. This showsthatδgg−1 = (1 + L∂)−1L(gδ(ΠiAz)g−1).

(A.17)Proceeding in the same way with (A.15), one findsδ(ΠkF) = Πk(gδ(ΠkA¯z)g−1 −adF((1 + L∂)−1L(gδ(ΠiAz)g−1))). (A.18)For equations (A.1) and (A.2), we only need theδδµi, or, equivalently, theδδF behaviorofδδΠiAz andδδΠkA¯z , i.e.

we only need to look at (A.18). The transpose AT of anoperator A is in our case defined by requiring that for arbitrary g valued functionsX and Y , the following identity holds:Zd2z Tr(X(AY )) =Zd2z Tr((ATX)Y ).

(A.19)Among other things, this implies that ∂and L are anti-symmetric, ∂T = −∂andLT = −L. The computation of the transpose of the operators in the right hand sideof (A.18) is now straightforward:Zd2z TrXΠk(gδ(ΠkA¯z)g−1 −adF((1 + L∂)−1L(gδ(ΠiAz)g−1)))=Zd2z Trδ(ΠkA¯z)(g−1Π†kXg)−Zd2z Trδ(ΠiAz)(g−1(L(1 + L∂)−1[F, Π†kX])g).

(A.20)UsingδδΠkA¯z =k2πW we read offfrom (A.20) that, up to terms containing derivativeswith respect to g,δδΠkA¯z=k2πΠ†k(g−1Wg),(A.21)δδΠiAz=−k2πΠ†ig−1(L(1 + L∂)−1[F, W])g.(A.22)The first equation is just (A.1), and the second one is (A.2), as we see from (A.5).Therefore (A.1) and (A.2) are automatically satisfied in the parametrization (A.12)and (A.13), as claimed.32

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