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Jan de Boer and Jacob Goeree

arXiv:hep-th/9110073v1 30 Oct 1991THU-91/1810/91The Covariant W3 ActionJan de Boer and Jacob GoereeInstitute for Theoretical PhysicsUniversity of UtrechtPrincetonplein 5P.O. Box 80.0063508 TA UtrechtAbstractStarting with Sl(3, IR) Chern–Simons theory we derive the covariant action for W3gravity.

1. IntroductionTwo dimensional gravity has been extensively studied during the last few years.Three different approaches to the subject, namely (i) study of the induced action of2D gravity in both the conformal (where it reduces to the Liouville action) as wellas the light cone gauge, (ii) the discretized approach of the matrix models, and (iii)topological gravity, have all been very powerful (at least for c < 1), giving equivalentresults.Higher spin extensions of 2D gravity can also be studied using the above methods.These theories are commonly denoted as theories of W gravity.

Especially W3 gravityin the light cone gauge has been the subject of many recent research [12, 13, 14,15]. In this paper we will also concern ourselves with the study of W3 gravity, butfrom a different angle.

Believing that the ‘W3 moduli space’ is somehow relatedto the moduli space of flat Sl(3, IR) bundles, we will study W3 gravity startingfrom Sl(3, IR) Chern–Simons theory whose classical phase space is the space of flatSl(3, IR) bundles.Our analysis resembles the one in [1].In this reference H. Verlinde showedhow the physical state condition in Sl(2, IR) Chern–Simons theory can be reducedto the conformal Ward identity, giving as a by-product the fully covariant actionof 2D gravity. We will start with Sl(3, IR) Chern–Simons theory, and derive thecovariant action for W3 gravity.

It will turn out that this action describes Sl(3, IR)Toda theory coupled to a ‘W3 background,’ confirming general beliefs. Although werestrict ourselves to the case of W3 in this paper, we believe that many of our resultscan be generalized.

This will be reported elsewhere [9].2. Chern–Simons theoryChern–Simons theory on a three manifold M is described by the actionS =k4πiZM Tr(A ∧dA + 23A ∧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 the2

form 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, andd = dt∂/∂t + ˜d.

Rewriting the action asS =k4πiZdtZΣ Tr( ˜A ∧∂t ˜A + 2A0( ˜d ˜A + ˜A ∧˜A)),(2.2)we recognize that A0 acts as a Lagrange multiplier which implements the constraint˜F = ˜d ˜A + ˜A ∧˜A = 0.Furthermore, we deduce from this action the followingnon-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 by acommutator, and we have to choose a ‘polarization.’ This simply means that wehave to divide the set of variables ( ˜Aaz, ˜Aa¯z) into two subsets. One subset will containfields Xi and the other subset will consist of derivativesδδXi, in accordance with(2.3).

The choice of these subsets is called a choice of polarization. Of course wealso have to incorporate the Gauss law constraints ˜F( ˜A) = 0.

Following [1, 5, 7]we will impose these constraints after quantization. So we will first consider a ‘big’Hilbert space obtained by quantization of (2.3), and then select the physical statesΨ by requiring ˜F( ˜A)Ψ = 0.In [1] it was shown that these physical state conditions for Sl(2, IR) Chern–Simons theory with a certain choice of polarization are equivalent to the conformalWard identities satisfied by conformal blocks in Conformal Field Theory (CFT).More precisely, it was shown that two of the three constraints in ˜F( ˜A)Ψ = 0 couldbe explicitly solved, leaving one constraint which is equivalent to the conformalWard identity.

In this paper we will generalize these results to the case of Sl(3, IR)with a choice of polarization that leads to the Ward identities of the W3 algebra [2]. (A different choice of polarization leading to the related W 23 algebra was made in[11].) To explain our strategy, we will in the next section first reconsider the case ofSl(2, IR) Chern–Simons theory.3

3. Sl(2, IR)In order to understand how one obtains the Virasoro Ward identity from Sl(2, IR)Chern–Simons theory, let us first recall how one can in general obtain Ward identitiesfrom zero-curvature constraints.

Given operator valued connections A and ¯A∗, thezero curvature condition reads:F Ψ = : ∂¯A −¯∂A + [A, ¯A] : Ψ = 0. (3.1)Here the dots denote normal ordering, which simply amounts to putting all δ/δXto the right.

(Here X is some arbitrary field.) If we take†:A = 01δδµ0!,(3.2)and put ¯A12 = µ, we can solve for the remaining components of ¯A, if we requirethat the curvature operator must have the formF = 00∗0!.

(3.3)Equation (3.1) reduces to one equation which is precisely the Virasoro Ward identityfor c = 6k"(¯∂−µ∂−2(∂µ)) δδµ + 12∂3µ#Ψ = 0. (3.4)If we start with Sl(2, IR) Chern–Simons theory and pick a polarization, then A and¯A consist of three fields and their variational derivatives.

On the other hand, theVirasoro Ward identity contains just one field. So in order to obtain the VirasoroWard identity as zero curvature constraint of Chern–Simons theory, we must in-troduce two extra degrees of freedom without changing the contents of the zerocurvature constraints.

We can in this case simply introduce extra degrees of free-dom by performing a gauge transformation. Under such a gauge transformationthe curvature transforms homogeneously: F →g−1Fg, and F = 0 will still give the∗Note that from now on ˜A will be denoted as A.†In the followingδδX should in fact read 2πkδδX , for all fields X.4

Virasoro Ward identity. If we require F to remain of type (3.3), g must be of theformg = g110g21g−111!.

(3.5)We can parametrize g via a Gauss decomposition:g = 10χ1! eφ00e−φ!,(3.6)and we find that F →Fe2φ.

The gauge transformed A, ¯A areAg= χ + ∂φe−2φe2φ(∂χ −χ2 +δδµ)−χ −∂φ!,(3.7)¯Ag= 12∂µ + χµ + ¯∂φµe−2φe2φ(µ δδµ −12∂2µ −µχ2 −χ∂µ + ¯∂χ)−12∂µ −χµ −¯∂φ!. (3.8)If we pick the following polarization:A= A11A12δδ ¯A12−A11!,(3.9)¯A= −12δδA11¯A12−δδA1212δδA11!,(3.10)and let F(A, ¯A) act on a wavefunction eSΨ[µ], where S still has to be determined,we find that F(A, ¯A)eSΨ = 0 ⇔eSF(A′, ¯A′)Ψ = 0, with‡A′= A11A12δSδ ¯A12 +δδ ¯A12−A11!,(3.11)¯A′= −12δSδA11 −12δδA11¯A12−δSδA12 −δδA1212δSδA11 + 12δδA11!.

(3.12)To proceed, suppose that we want to match (3.11) and (3.12) with (3.7) and(3.8). This givesA11=χ + ∂φ,A12=e−2φ,(3.13)¯A12=µe−2φ.‡Here we ignored terms of the typeδ2SδXδX′ , that involve delta-function type singularities thathave to be regularized somehow.

Therefore the validity of our discussion will be limited to thesemi-classical level. In the full quantum theory we expect corrections to the expressions given inthis paper.5

Using these expressions, we can express the variational derivatives with respect toA and ¯A in terms of those with respect to χ, φ and µ. One finds:δδA11=δδχ,δδA12=−12e2φ ∂δδχ + δδφ −2µ δδµ!,(3.14)δδ ¯A12=e2φ δδµ.Quite remarkably, the terms containingδδµ in (3.11) and (3.12) agree precisely withthose in (3.7) and (3.8).

As Ψ depends only on µ, we can omit theδδφ andδδχ termsin (3.12). In conclusion, we see that (3.11) and (3.12) are exactly identical to (3.7)and (3.8) if the following relations hold:δSδA11=−∂µ −2χµ −2¯∂φ,δSδA12=e2φ 12∂2µ + µχ2 + χ∂µ −¯∂χ,(3.15)δSδ ¯A12=e2φ(∂χ −χ2).Before continuing, we will first rewrite these expressions in a form that is more suit-able for generalizations to other cases.

Let G be the subgroup of Sl(2, IR) consistingof all g of type (3.5), and let Λ = 0100!, which is equal to (3.2) up to termscontainingδδµ. Furthermore, let ( ¯A0)12 = µ and let ¯A0 be such that F(Λ, ¯A0) isof type (3.3).

For this ¯A0 one finds F(Λ, ¯A0) = 00−12∂3µ0!. To rewrite thepolarizations, define projections Πk and Πi on the Lie algebra sl(2, IR) viaΠk bac−b!= 0a00!,Πi bac−b!= ba0−b!.The polarization (3.9) and (3.10) is such that the fields are in ΠiA and Πk ¯A, andthat the derivatives are in Π†i ¯A and Π†kA, where Π†i = 1 −Πk and Π†k = 1 −Πi.Now (3.7) and (3.8) read, up to terms containingδδµ, A = g−1Λg + g−1∂g and¯A = g−1 ¯A0g + g−1 ¯∂g.

Therefore, (3.13) can be compactly formulated asΠiA=Πi(g−1Λg + g−1∂g),(3.16)Πk ¯A=Πk(g−1 ¯A0g + g−1 ¯∂g),(3.17)6

and equations (3.15) become−δSδΠiA=Π†i(g−1 ¯A0g + g−1 ¯∂g),(3.18)δSδΠk ¯A=Π†k(g−1Λg + g−1∂g). (3.19)Surprisingly, these equations can be integrated§ to giveS =k2πZd2z Tr(Π†kAΠk ¯A) −k2πZd2z Tr(Λ¯∂gg−1) −ΓW ZW[g],(3.20)where ΓW ZW is the Wess-Zumino-Witten actionΓW ZW[g] =k4πZd2z Tr(g−1∂gg−1 ¯∂g) −k12πZB Tr(g−1dg)3.

(3.21)For Sl(2, IR) we find thatS =k2πZd2zhµ∂χ −µχ2 −2¯∂φχ −∂φ¯∂φi,(3.22)which indeed solves (3.15). To make contact with the results of [1] we have to redefineφ →−12ϕ, and χ →12ω + 12∂ϕ.

Then the action becomes S = S0(ω, ϕ, µ)+SL(ϕ, µ),whereS0 = −k4πZd2zh12µω2 −ω(¯∂ϕ −∂µ −µ∂ϕ)i,(3.23)and SL is a chiral version of the Liouville action:SL =k4πZd2zh12∂ϕ¯∂ϕ + µ(∂2ϕ −12(∂ϕ)2)i. (3.24)The actions (3.23), (3.24) are precisely the same as the ones found in [1].

Alto-gether we have now shown how the Virasoro Ward identity follows from Sl(2, IR)Chern–Simons theory. In general, the procedure consists of three steps: (i) pick apolarization and parametrization of the components of A and ¯A, (ii) move A and¯A through a term of the form eS and (iii), perform a gauge transformation.

In thenext section we will apply these steps to the case of Sl(3, IR) Chern–Simons theory.§The proof of this fact will be given elsewhere [9].7

4. Sl(3, IR)The Ward identities of the W3-algebra can obtained in the same way as theVirasoro Ward identity was obtained in the previous section.

We start withA =010001δδνδδµ0,(4.1)and put ¯A13 = ν and ¯A23 = µ. The remaining components of ¯A are fixed by requiringF to be of the formF =000000F31F320.

(4.2)As was shown in e.g. [12], F31 and F32 are directly related to the W3-Ward iden-tities.

The subgroup G of Sl(3, IR) that preserves this form of F consists of allg ∈Sl(3, IR) satisfying g13 = g23 = 0. To parametrize these g, we will again use aGauss decomposition:g =100φ110φ3φ21eα000eβ−α000e−β1χ0010001.

(4.3)Under a gauge transformation F →g−1Fg we findF ′31=eα+βF31 + φ1eα+βF32,(4.4)F ′32=χeα+βF31 + (χφ1eα+β + e2β−α)F32,(4.5)which clearly shows that FΨ[µ, ν] = 0 ⇔(g−1Fg)Ψ[µ, ν] = 0. The polarization wechoose is such that it is invariant under the subgroup G of Sl(3, IR):A=A+ + A−A12A13A21−2A+A23δδ ¯A13δδ ¯A23A+ −A−,(4.6)¯A=−16δδA+ −12δδA−−δδA21¯A13−δδA1213δδA+¯A23−δδA13−δδA23−16δδA+ + 12δδA−,(4.7)8

and the projections Πi and Πk are given byΠka + bcde−2afgha −b=00d00f000,(4.8)Πia + bcde−2afgha −b=a + bcde−2af00a −b. (4.9)The matrices Λ and ¯A0 are also found completely analogously to the Sl(2, IR)-case,one simply requires F(Λ, ¯A0) to be of type (4.2), to find:Λ=010001000,(4.10)¯A0=∂µ + 23∂2νµ + ∂νν−∂2µ −23∂3ν−13∂2νµ∂3µ + 23∂4ν−∂2µ −13∂3ν−∂µ −13∂2ν.

(4.11)It is straightforward to read of the explicit expressions for ΠiA and Πk ¯A from (3.16)and (3.17). They are:A+=12φ1 −φ2 + (φ21 −φ3)χe2α−β −χ∂φ1e2α−β + ∂(α −β)A−=12φ1 + φ2 + (φ21 −φ3)χe2α−β −χ∂φ1e2α−β + ∂(α + β)A12=(2φ1 −φ2)χ + eβ−2α + χ2(φ21 −φ3)e2α−β −χ2∂φ1e2α−β + χ∂(2α −β) + ∂χA13=−χe2α−βA21=(φ3 −φ21 + ∂φ1)e2α−βA23=eα−2β(4.12)¯A13=χ(νφ1 −µ)eα−2β + νe−α−β¯A23=(µ −νφ1)eα−2βAgain, we want to construct an action S, such that A′ and ¯A′, defined throughF(A, ¯A)eSΨ = 0 ⇔eSF(A′, ¯A′)Ψ = 0, are equal to the connections Ag and ¯Ag (thegeneralizations of (3.7) and (3.8)) that are the gauge transforms with g as in (4.3) ofthe connections A and ¯A mentioned in and below (4.1).

If Ag = A′ and ¯Ag = ¯A′ aresatisfied, F(A′, ¯A′)Ψ = 0 is equivalent to the statement that Ψ satisfies the W3-Wardidentities. The connections A′ and ¯A′ can be obtained from (3.9) and (3.10) by firstreplacingδδXi byδδXi + δSδXi everywhere, followed by putting all terms inδδXi that do9

not containδδµ orδδν equal to zero, as Ψ depends only on µ and ν. Comparing theseA′ and ¯A′ with Ag and ¯Ag yields a set of equations forδSδXi, analogous to (3.15).These equations are necessary, but not sufficient, because theδδµ andδδν dependenceof A′ and ¯A′ must also be equal to theδδµ andδδν dependence of Ag and ¯Ag.

Whetherthis is the case has been verified by explicitly computing theδδXi in (4.6) and (4.7) interms ofδδµ,δδν , and the functional derivatives with respect to the fields in the Gaussdecomposition (4.3). It turns out that this dependence is indeed precisely the same.The computations involved here are rather cumbersome, whether there is a moredirect way to see this, is under current investigation [9].

Due to this remarkablefact, we know that if we now solve (3.18) and (3.19) for this case, FeSΨ = 0 willbe satisfied (with this choice of parametrization and polarization) if and only if Ψsatisfies the W3-Ward identities. Again, S is given by (3.20), and reads:S=k2πZd2zh−12Aij∂αi ¯∂αj −φiAij ¯∂αj −¯∂χ(∂φ1 + φ21 −φ3)e2α1−α2+µ((∂−φ2)φ2 + φ2φ1 −φ3) + ν(∂−φ2)(φ3 −φ2φ1)i,(4.13)where α1 = α, α2 = β and Aij is the Cartan matrix of Sl(3, IR), Aij = 2−1−12!.This action is important if one wants to compute inner products of wave functionsΨ in Sl(3, IR) Chern–Simons theory.

This will be the topic of the next section.5. The Inner Product and W3 GravityThe wave functions Ψ[µ, ν] = exp SW[µ, ν] that solve the W3-Ward identities, canbe obtained from a constrained Wess-Zumino-Witten model [3, 12].

This means thatat this stage we know the complete wavefunction eSΨ. These wave functions solvethe holomorphic W3-Ward identities, and can therefore be seen as effective actionsof chiral W3-gravity.

However, in ordinary gravity there is a nontrivial couplingbetween the holomorphic and the anti-holomorphic sectors, and this is where partof the geometry of two-dimensional quantum gravity comes in. In this section wewill consider the coupling between the holomorphic and anti-holomorphic sectors ofW3-gravity, hoping that it will lead to an understanding of the geometry underlyingthe W3-algebra.

This nontrivial coupling appears when computing inner products10

of wave functions in SL(3, IR) Chern–Simons theory. The expression for such aninner product is⟨Ψ1 | Ψ2⟩=ZD(ΠiA)D(Πk ¯A)D(Π†kB)D(Π†i ¯B)eV +S+ ¯SΨ1[µ, ν]¯Ψ2[¯µ, ¯ν].

(5.1)The nontrivial coupling is due to the K¨ahler potential V , which is associated to thesymplectic form defined by (2.3). To find an expression for this K¨ahler potential, wefirst give the definitions of B and ¯B, the variables on which the anti-holomorphicwave function ¯Ψ2 depends.Let H be the subgroup of Sl(3, IR) consisting of allelements h ∈Sl(3, IR) satisfying h31 = h32 = 0; H can be conveniently parametrizedby a Gauss decomposition.

Define the connection B0 by requiring that (B0)31 = ¯νand (B0)32 = ¯µ, and that Π†iF(B0, Λt) = 0, where Λt is the transpose of Λ. Then:B=hB0h−1 −∂hh−1,(5.2)¯B=hΛth−1 −¯∂hh−1,(5.3)and the anti-holomorphic action ¯S is given by¯S =k2πZd2z Tr(Π†kBΠk ¯B) + k2πZd2z Tr(Λth−1∂h) −ΓW ZW[h].

(5.4)In terms of A and B, the K¨ahler potential is given byV =k2πZd2z Tr(ΠiAΠ†i ¯B −Πk ¯AΠ†kB). (5.5)The total exponent K = V + S + ¯S occurring in the inner product (5.1) is now afunction of g, h, and {µ, ν, ¯µ, ¯ν}.

This ”action” K is part of the covariant action ofW3-gravity. The complete covariant action is given by K+SW[µ, ν]+ ¯SW[¯µ, ¯ν], whereSW[µ, ν] and ¯SW[¯µ, ¯ν] are the chiral actions for W3-gravity that were constructedin [12].In the case of Sl(2, IR), K is equal to the Liouville action in a certainbackground metric, plus an extra term depending on µ, ¯µ only.

K represents theW3-analogon of the Quillen-Belavin-Knizhnik anomaly. Clearly, it will be interestingto have an explicit expression for it.

One can work out such an explicit expression,by simply substituting all the expressions given above. The result of all this is alarge, intransparant expression.

However, upon further inspection, it turns out thatK is invariant under local Sl(2, IR) × IR symmetry transformations, which can beused to gauge away four degrees of freedom. Actually, one can proof (see [9]) that K11

only depends on the product gh. Using this the action can be greatly simplified byintroducing a new Gauss decomposition for gh, which is now an arbitrary elementof Sl(3, IR).

More specific, we takegh =100ρ110ρ3ρ21eϕ1000eϕ2−ϕ1000e−ϕ21−¯ρ1−¯ρ301−¯ρ2001. (5.6)Substituting this, one sees that the action depends simply quadratically on ρ3, ¯ρ3.Therefore, ignoring subtleties arising from the measure when changing variablesfrom A and B to gh and µ, ν, ¯µ, ¯ν, we can perform theR Dρ3D¯ρ3 integration.

Theresulting action can be written in the following form:K =k2πZd2zn12Aij∂ϕi ¯∂ϕj +Xie−Aijϕj −Aij(ρi + ∂ϕi)(¯ρj + ¯∂ϕj)(5.7)−eϕ1−2ϕ2(µ −12∂ν −νρ1)(¯µ + 12 ¯∂¯ν + ¯ν¯ρ1) −e−ϕ1−ϕ2ν¯ν−eϕ2−2ϕ1(µ + 12∂ν + νρ2)(¯µ −12 ¯∂¯ν −¯ν¯ρ2) + µT + νW + ¯µ ¯T + ¯ν ¯Wo,where we defined T, W, ¯T, ¯W through the following Fateev-Lyukanov [16] construc-tion:(∂−ρ2)(∂−ρ1 + ρ2)(∂+ ρ1)=∂3 + T∂−W + 12∂T,(¯∂−¯ρ2)(¯∂−¯ρ1 + ¯ρ2)(¯∂+ ¯ρ1)=¯∂3 + ¯T ¯∂+ ¯W + 12 ¯∂¯T,(5.8)and we shifted µ →µ −12∂ν and ¯µ →¯µ + 12 ¯∂¯ν. The first part of K is precisely achiral Sl(3) Toda action, confirming the suspected relation between W3-gravity andToda theory.

Actually, one would expect that in a ”conformal gauge”, the covariantW3-action will reduce to a Toda action. Indeed, if we put ν = ¯ν = 0 in K, thenwe find that K is also purely quadratic in ρ1, ρ2, ¯ρ1, ¯ρ2.

Performing the integrationsover these variables as well, we find thatK[ϕ1, ϕ2, µ, ¯µ] =k4πZd2zq−ˆg 12ˆgab∂aϕi∂bϕjAij + 4Xie−Aijϕj + R⃗ξ · ⃗ϕ!+K[µ, ¯µ],(5.9)where K[µ, ¯µ] is the same expression as was derived in [1], namelyK[µ, ¯µ] = kπZd2z (1 −µ¯µ)−1(∂µ¯∂¯µ −12µ(¯∂¯µ)2 −12 ¯µ(∂µ)2),(5.10)12

and ˆg is the metric given by ds2 = |dz + µd¯z|2. In the case of Sl(3, IR), ⃗ξ · ⃗ϕ, with⃗ξ being one half times the sum of the positive roots, is just given by ϕ1 + ϕ2.

Theaction (5.9) is the same Toda action that was originally present in K in a chiral form,and the integration over ρ1, ¯ρ1, ρ2, ¯ρ2 has the effect of coupling it to a backgroundmetric 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 orover ρ2, ¯ρ2, but not over both at the same time, due to the presence of third orderterms in K. Another clue regarding the contents of the action (5.8) can be obtainedby treating the second and third line in (5.8) as perturbations of the first line of(5.8). This means that we try to make an expansion in terms of µ, ¯µ, ν, ¯ν.

Thesaddlepoint of the ρ-terms is at ρi = −∂ϕi and ¯ρi = −¯∂ϕi. From (5.8) 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),(5.11)and similar expressions for ¯T, ¯W.This suggests that the full action K contains the generating functional for thecorrelators of the energy-momentum tensor and the W3-field of a Toda theory, ”co-variantly” coupled to W3-gravity.

The presence of the third order terms in W, ¯W in(5.8) prevents us from computing the action of this covariantly coupled Toda theory.Detailed proofs, that were omitted here, as well as generalizations to other W-algebras, will be the subjects of a future publication [9].This work was financially supported by the Stichting voor Fundamenteel Onder-zoek der Materie (FOM).13

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