Comments on the Algebraic Properties of Dilaton Actions
aa r X i v : . [ h e p - t h ] N ov Comments on the Algebraic Properties of Dilaton Actions ∗ A. Schwimmer a and S. Theisen ba Weizmann Institute of Science, Rehovot 76100, Israel b Max-Planck-Institut f¨ur Gravitationsphysik, Albert-Einstein-Institut, 14476 Golm, Germany
Abstract
We study the relation between the dilaton action and sigma models for theGoldstone bosons of the spontaneous breaking of the conformal group. We arguethat the relation requires that the sigma model is diffeomorphism invariant. Theorigin of the WZW terms for the dilaton is clarified and it is shown that in thisapproach the dilaton WZW term is necessarily accompanied by a Weyl invariantterm proposed before from holographic considerations. ∗ Partially Supported by the Center for Basic Interactions of the Israeli Academy of Sciences. A.S. also ac-knowledges support from the Alexander von Humboldt-Foundation. . Introduction
Chiral anomalies are well understood algebraically [1][2]. Their general form can beobtained by considering the theory in d = 2 n as a boundary of a d + 1 dimensionalmanifold. The action in d + 1 dimensions is the local Chern-Simons action and since thisaction is gauge invariant only up to a boundary term the correct anomaly is reproduced bythis boundary term. In a sense this could be considered as a manifestation of “holography”.When the chiral symmetry is spontaneously broken Goldstone bosons are present. Sincethe Goldstone bosons have to reproduce the chiral anomalies, specific features of theiraction (“sigma-model” in the following) follow from the aforementioned structure: besidesa local term in d = 2 n dimensions which realizes nonlinearly the symmetry there is a secondterm [3] (“WZW term” in the following) which lives in d + 1 dimensions and reproducesthe anomaly through the above mechanism.In the present note we want to study in detail the analogous problems for trace anomalies.Following the explicit calculation of the trace anomalies in the AdS/CFT duality [4] itwas realized [5] that a mechanism rather analogous to the one described above for chiralanomalies is at work: the gravitational action in d +1 dimension plays the role of the Chern-Simons action and a particular subgroup of the d + 1-dimensional diffeomorphism acts asthe analogue of the gauge transformations producing the anomalies at the boundary.When the conformal symmetry is spontaneously broken the Goldstone boson (in thefollowing “the dilaton”) should reproduce the trace anomalies [6][7]. The effective actionwith this property can be constructed and a WZW term appears. Compared with thegeneral properties of the chiral Goldstone bosons action outlined above the dilaton actionhas strange features: the WZW term is local directly in d = 2 n and does not seem to haveany higher dimensional origin.In order to understand this feature we rely on the basic distinguishing property of thedilaton: even though the spontaneous breaking of the conformal symmetry in Euclideansignature is the breaking of the SO ( d +1 ,
1) group to SO ( d ) × T d there is only one Goldstoneboson, the dilaton [8][9][10]. This is of course a consequence of the fact that all theconformal currents can be constructed in terms of the energy momentum tensor, theirconservation being the consequence of tracelessness. The gauging of the SO ( d + 1 , SO ( d +1 , / [ SO ( d ) × T d ] coset. We propose that the reduction from the d + 1 fields parametrizing1he coset to the single dilaton is achieved by a special new, characteristic feature of thesigma-model: diffeomorphism invariance in d = 2 n dimensions for the invariant term and in d +1 dimensions for the WZW term, respectively. By choosing a particular parametrizationthe coset coordinates are reduced to a single field and the WZW term becomes explicitlylocal.We will formulate the sigma-model in a general metric background. Since the input d -dimensional metric should give rise in the sigma-model to a metric depending on d + 1coordinates – the Goldstone boson fields – we are led from the beginning to consider a“holographic” setup. Moreover, the metric in the sigma-model action should admit theaction of a group isomorphic to the Weyl group which makes the connection to holographyeven stronger. In spite of that the freedom for dilaton actions constrained by the algebraicapproach is much larger than the one which follows from a strict application of holography.In particular, as we will discuss in detail, there is no relation between the d + 1 dimensionalactions and solutions we are using in the construction.Applying the above mentioned procedure both for the invariant terms of the dilatonaction and the one reproducing the trace anomalies (the “WZW” part) we get an interestingconnection between the two once one imposes the condition that there is no potential forthe dilaton, a necessary condition for the spontaneous breaking of conformal invariance.Our conclusion is that the special action proposed for the dilaton in a holographic setupin [11] has a general algebraic origin being normalized by the “ a ” trace anomaly wheneverconformal invariance is spontaneously broken.The paper is organized as follows:In Section 2 we construct the d dimensional part of the reparametrization invariantsigma-model and we show how it reduces to the Weyl invariant part of the dilaton action.In Section 3 we construct the d +1 dimensional reparametrization invariant WZW term andwe reduce it to the dilaton WZW term. We discuss the relation between invariant termsand the WZW terms following form the requirement of vanishing potential for the dilaton.The relations between WZW terms corresponding to different even dimensions is madeexplicit. In the last section we discuss various applications of the formalism developed andpossible generalizations. In Appendix A we review the holographic calculations of traceanomalies and the realization of Weyl symmetry in holography which motivate the choicesof the explicit metric backgrounds in Sections 2 and 3.Related and complementary discussions of some of the aspects addressed here can befound in [12][13][14][15][16]. While these references rely on supersymmetry, this is notassumed here. 2 . The Weyl invariant part of the dilaton sigma-model We will follow here an algebraic approach though, as we will see, the results have animmediate holographic interpretation.Consider in d dimensions the breaking of the conformal group SO ( d +1 ,
1) to the Poincar´egroup SO ( d ) × T d . The coset of Goldstone bosons can be parametrized by d +1 fields X µ ( x i )where µ = 1 , ..., d + 1 and i = 1 , ..., d . The metric on the space of the X fields is AdS d +1 with isometry SO ( d + 1 ,
1) such that the broken isometries are nonlinearly realized.Since we want to construct the analogue of the “gauged sigma-model” we allow a moregeneral metric G µν ( X τ ) on which the Weyl transformations act. The condition this metrichas to fulfill in order to serve our purposes are:a) It should be a d + 1 metric but a functional of a d dimensional metric g ij , i, j = 1 , ..., dG µν = F µν [ g ij ] . (2 . δ σ G = F [ g ij exp 2 σ ( x )] − F [ g ij ] (2 . δ σ denotes the action on the d + 1 dimensional metric isomorphic to the Weyltransformations.c) For g ij = δ ij it should reduce to the natural metric on the SO ( d +1 , SO ( d ) × T d coset which is AdS d +1 .A class of metrics which satisfy the above requirements are solutions of d + 1 dimensional“bulk” actions which admit AdS d +1 solutions, specified by the boundary metric g ij (whichin the continuation we will denote by g (0) ij ) in the Fefferman-Graham gauge. Obviously thisclass satisfies the above requirements, the group action isomorphic to Weyl transformationsbeing the PBH transformations as explained in the Appendix.The connection to holography is now obvious though we stress that we will use only thealgebraic properties of the solution. In particular the specific action to which the metricis a solution will not play a role. It is an interesting question if there are metrics whichsatisfy the above requirements not arriving from a holographic construction.The natural building blocks for the gauged sigma-model are the induced metric: h ij ( x ) = G µν ( X τ ( x )) ∂ i X µ ( x ) ∂ j X ν ( x ) (2 . δ ij h ij , here we insist on reparametrization invariance in d dimensions which,together with the field redefinition invariance present in (2.3), will allow us to project tothe dilaton. Therefore the minimal sigma-model action having these properties is: S = 1 ℓ d Z d d x p det h ij . (2 . G µν to be in the FG gauge, we split the fields X µ into X i ( x ) i = 1 , ..., d and Φ( x ) in the “ ρ ”-direction. Now we can achieve the reduction to thedilaton action by choosing the gauge: X i ( x ) = x i (2 . S = 1 ℓ d Z d d x p det g ij ( x, Φ( x ))Φ d/ ( x ) s ℓ g ij ( x, Φ( x )) ∂ i Φ ∂ j Φ( x )4Φ( x ) . (2 . N = 4 Super Yang-Mills Coulomb branch.In order to exploit the symmetries of the action it is convenient to start with its unfixedform (2.4). Since in the FG gauge the metric is determined by its boundary value g (0) ij ( X ),the action is a functional of g (0) ij and X µ . The action has symmetries of two kinds:a) Field transformations of the X µ fields relating two different background metrics G µν . These transformation make explicit the variation under a change of g (0) ij . Thetransformations are inherited from residual gauge transformations in the FG gauge, i.e.the PBH transformation parametrized by σ ( X j ) and X j -dependent field transformationsparametrized by ζ i ( X j ). We rewrite the PBH transformations of the Appendix, makingit explicit that in the framework of the sigma-model we deal with field transformations atfixed coordinates x i : Φ ′ = Φ(1 + 2 σ ( X j )) (2 . X ′ i = X i − a i ( X j , Φ( x )) − ζ i ( X j ) (2 . a i ( X, Φ( x )) = ℓ Z Φ( x )0 dρ ′ g ij ( x, ρ ′ ) ∂ j σ ( X ) . (2 . g (0) ij by: δg (0) ij ( X k ) = 2 σ ( X k ) g (0) ij ( X k ) + ∇ i ζ j ( X k ) + ∇ j ζ i ( X k ) (2 . g (0) ij and all the functional dependencesare on X j .b) Reparametrizations of the x i variables parametrized by ξ i ( x k ): x ′ i = x i − ξ i ( x k ) (2 . X µ transform as: δX µ ( x k ) = ξ i ∂ i X µ ( x k ) . (2 . g (0) ij and Φ.After the transformation (2.7) the special choice (2.5) is not anymore respected. Inorder to reinstate it we should make a reparametrization with the special choice of theparameters ξ i : ξ i ( x k ) = a i ( X k = x k , Φ( x )) + ζ i ( X k = x k ) (2 . δg (0) ij ( x ) = 2 σ ( x ) + ∇ i ζ j + ∇ j ζ i (2 . δ Φ( x ) = 2 σ ( x )Φ( x ) + [ a i ( X k = x k , Φ( x )) + ζ i ( X k = x k )] ∂ i Φ( x ) (2 . X k was replaced with the variables x k usingthe gauge (2.5).Equations (2.14), (2.15) make explicit the invariance of (2.6) under the Weyl transfor-mations. The transformations are non-anomalous since there is no boundary term whichcould be the source of violation in the above classical argument.Equation (2.15) shows that in order to have the usual transformation of the dilaton field τ under Weyl transformation: τ → τ + σ (2 . For a related discussion see [17]. τ . Using (2.15) the field redefinition can befound iteratively giving an expansion in powers of e τ where the terms of O ( e nτ ) contain2( n −
1) derivatives. The 0-th order solution of (2.15) is:Φ (0) = e τ . (2 . ζ i = 0.)To proceed we need the Φ-expansion of a i which depends on the higher order terms in theFG expansion of the metric. Both can be found in [18]. In this way we find1 ℓ Φ (1) = 12 e τ ( ∇ τ ) + α e τ ˆ R (2 . ℓ Φ (2) = e τ (cid:18) R ij ∇ i τ ∇ j τ d − − R ( ∇ τ ) d − d −
2) + 14 ∇ i τ ∇ j τ ∇ i ∇ j τ + 716 ( ∇ τ ) (cid:19) + α e τ (cid:18)
12 ˆ ∇ i τ ˆ ∇ i ˆ R + ˆ ∇ i τ ˆ ∇ i τ ˆ R (cid:19) + e τ (cid:16) β ˆ ˆ R + β ˆ R + β ˆ R ij ˆ R ij + β ˆ C ijkl ˆ C ijkl (cid:17) (2 . α parametrizes the homogeneous solution of (2.15) at O ( ∂ ) and β , . . . , β param-eterizes the homogeneous solutions at O ( ∂ ). The hatted quantities are built from theinvariant metric ˆ g ij = e − τ g ij and g ij = g (0) ij .Plugging this into (2.6) we obtain the “universal minimal” invariant part of the action(cf. Appendix A): S = 1 ℓ d Z d d x p ˆ g (cid:18) ℓ d −
1) ˆ R + ℓ d − d −
3) ( ˆ E − ˆ C ) + O ( ∂ ) (cid:19) (2 . h ij as well as from the second fundamentalform. Also, we have set α = β i = 0 (cf. (2.18),(2.19)). Note that in d = 2 the O ( ∂ )terms are R √ gR while in d = 4 the O ( ∂ ) terms are R √ g ( E − C ). The discussion here generalizes that of [17] to a curved metric. Such terms were considered, in a somewhat different context, in [19].
6f one keeps the homogeneous terms in Φ, one finds the additional terms∆ S = 1 ℓ d Z d d x p ˆ g ( − d ℓ α ˆ R + ℓ d ( d + 2) α ˆ R + ℓ d − d − α ˆ R − d ℓ (cid:16) β ˆ ˆ R + β ˆ R + β ˆ R ij ˆ R ij + β ˆ C ijkl ˆ C ijkl (cid:17)) . (2 . τ is universal. A unique feature of theminimal action which we will use in the following is that it is the only term written interms of the Φ field which contains a “potential” of the dilaton field τ . In flat space thisis simply d/ , while after the field redefinition it gives rises to the √ ˆ g term.Another application of our formalism is the study of the symmetries of (2.6) for thespecial case g (0) ij = δ ij , i.e. AdS background metric. The AdS metric is invariant underspecial conformal transformations accompanied by an appropriate Weyl transformation,i.e. in the notation of (2.7): ζ i = 12 ǫ i x − x i ( ǫ · x ) (2 . ǫ i = const. and σ ( x ) = − d ( ∂ · ζ ) = ǫ · x . (2 . a i have a very simple form: a i = ℓ x ) ∂ i σ ( x ) = ℓ ǫ i Φ( x ) . (2 . g (0) ij is invariant under a reparametrization transformation: x ′ i = x i − ǫ i x + x i ( ǫ · x ) − ℓ ǫ i Φ( x ) (2 . ′ ( x ′ ) = Φ( x ) + 2 ǫ · x Φ( x ) (2 . . The WZW term We want to construct a sigma-model term which reproduces the trace anomalies. Thisshould be an action on a d + 1 dimensional manifold with boundary and should share withthe invariant term the property of reducing to a functional just of the dilaton field andthe boundary metric. We will achieve that by requiring that the action is invariant underreparametrizations in d + 1 dimensions. We define f αβ = G µν ( X ) ∂ α X µ ∂ β X ν (3 . G µν is the bulk metric in FG gauge and α, β = 1 , ..., d + 1. The fields X µ dependnow on d + 1 coordinates which we split from the beginning into x i , i = 1 , ..., d and ρ . Theboundary of the manifold is at ρ = 0. The WZW action is then S WZW = 1 ℓ d Z d d x dρ p det f αβ . (3 . X µ into X i ( x, ρ ) and Φ( x, ρ ).The symmetries of the action are again:a) Field transformations relating backgrounds defined by different g (0) ij . These transfor-mations, which involve fields at fixed coordinates, will have exactly the same form as inthe previous section, i.e. (2.7) and (2.9).b) Reparametrizations in d + 1 dimensions.We choose again a special set of coordinates by: X i ( x, ρ ) = x i . (3 . S WZW = 12 ℓ d Z d d x dρ ∂ ρ Φ p det g ij ( x, Φ( x, ρ ))Φ( x, ρ ) d/ . (3 . ρ and Φ( x, ρ ) is possible in the integral at eachfixed x such that the action (3.4) depends on Φ( x, ρ ) just through its boundary valueΦ( x, ρ = 0). 8he symmetry transformations which leave the gauge condition (3.3) unchanged areagain PBH transformations accompanied by reparametrizations: δg (0) ij ( x ) = 2 σ ( x ) + ∇ i ζ j ( x ) + ∇ j ζ i ( x ) (3 . δ Φ( x, ρ ) = 2 σ ( x )Φ( x, ρ ) + [ a i ( X k = x k , Φ( x, ρ )) + ζ i ( X k = x k )] ∂ i Φ( x, ρ ) (3 . σ ( x ) and ζ i ( x ) being the parameters of the transformation. We remark that as in theprevious section a i are determined by the PBH transformations such that Φ is treated justas an expansion parameter, its x -dependence not being acted upon.At the boundary ρ = 0, Φ( x, ρ = 0) has exactly the same Weyl transformation propertyas Φ( x ) in the previous section and the field redefinition relating it to the dilaton field τ ( x )is identical.As long as Φ( x,
0) and g (0) ij are transformed simultaneously the action will be invariant.This is the analogue in this set up of the fact that whatever the generating functionalof a CFT W ( g ) is, if we define ˆ g ij = e − τ g ij then W (ˆ g ) will be invariant under a jointtransformation of the metric and the dilaton. The expression which is local and reproducesthe anomalies i.e. the WZW term, is the difference W ( g ) − W (ˆ g ).From (3.6) it is clear that in flat space τ = 0 corresponds to Φ( x, ρ = 0) = 1. Therefore,using the change of variable from ρ to Φ( x, ρ ) we define the WZW action by S WZW = 12 ℓ d Φ( x, Z d Φ d d x p det g ij ( x, Φ)Φ d/ . (3 . d → n .We discuss now the way (3.7) reproduces the trace anomalies. Under (3.5),(3.6) themetric g (0) ij transforms by a Weyl transformation. If this transformation is accompaniedby the appropriate transformation of Φ or equivalently of τ , this will be an invariance.Therefore the variation will not get a contribution from the upper limit of integration. Onthe other hand for the lower limit of integration the expression would be invariant if thelower limit which corresponds to Φ( x,
0) = 1 transformed as follows from (3.6), i.e. by theamount δ (Φ = 1) = 2 σ ( x ) . (3 . δS WZW = − ℓ d Z d d x σ ( x ) q det g ij ( x, Φ = 1) . (3 . d = 2 n are to befound among the terms with no ℓ dependence. With this identification it is clear that thetype A anomalies [20] following from (3.9) are the ones calculated a long time ago [4],[18].One can now explicitly work out the WZW part in the dilaton action in an externalmetric following from (3.7). Using the FG expansions for g ij ( x, ρ ) in [4] and [18] we findin d = 4: S WZW = Z √ g d x ℓ (cid:16) − e − τ (cid:17) − ℓ R (cid:16) − e − τ (cid:17) + 14 ℓ ( ∂τ ) e − τ + 164 (cid:18) τ E + 4 (cid:16) R ij − Rg ij (cid:17) ∇ i τ ∇ j τ − ∇ τ ) τ + 2( ∇ τ ) (cid:19) + (cid:16) γ − (cid:17) τ C . (3 . γ is a combination of the two parameters which specify the homogeneous solutionof the PBH transformation for g (2) [18]. They could be fixed by explicitly solving theequations of motion of the bulk gravitational action. For the Einstein-Hilbert action withcosmological constant, which is dual do N SYM theory, γ = 0 and one finds that the twoanomaly coefficients a and c are the same. In deriving (3.10) we have set to zero the freeparameters which appear in the solution of (3.6), i.e the coefficients α and β i in (2.18) and(2.19).An alternative way of writing (3.10) is S WZW = Z d x ( − ℓ (cid:16)p ˆ g − √ g (cid:17) + 124 ℓ (cid:16)p ˆ g ˆ R − √ gR (cid:17) + 164 (cid:18) τ E + 4 (cid:16) R ij − Rg ij (cid:17) ∇ i τ ∇ j τ − ∇ τ ) τ + 2( ∇ τ ) (cid:19) + (cid:16) γ − (cid:17) τ C ) . (3 . √ ˆ g whose normalization is completely fixed by the “ a ” anomaly. InAppendix A this fact is analysed in detail showing that it has an algebraic origin and it is10ndependent both on the metric used and a possible replacement of the “minimal action”(3.7) by an action having additional curvature terms.As shown in the previous section the action (2.6) is the only one among the invariantactions which has a potential term for the dilaton. Since the total effective action forthe dilaton cannot have a potential term for the dilaton this term has to be cancelledby the one appearing in the WZW part (3.11). We conclude therefore that (2.6) with anormalization “ − a ” should necessarily accompany (3.11). This unexpected conclusion is aconsequence of our embedding the dilaton in the space of the d + 1 Goldstone bosons withthe symmetry structure assumed. In the normalization of the type A anomaly coefficients.t. δ σ W = a R σE + type B, the total dilaton action is16 a (cid:0) S + 4 S WZW (cid:1) . (3 . AdS .Alternative arguments for the special role of (2.6) were recently put forward in [16].If we want to calculate the WZW terms S (0)WZW , i.e. the dilaton self-interaction termsin flat external metric from (3.7) with g (0) ij = δ ij , we obtain a deceptively simple lookinguniversal expression: S (0)WZW = − d ℓ d Z d d x (cid:18) x, d/ − (cid:19) . (3 . ℓ . In principle this calcula-tion should check the relative normalizations of the terms in different dimensions followingfrom the general relation between type A anomalies [18],[5].
4. Discussion.
The special feature we used for constructing the sigma-model actions which reduced tothe dilaton action was their diffeomorphism invariance. Using this invariance we couldgauge away some of the “would be Goldstone bosons” ending with the relevant one, thedilaton. We believe that this is a general feature whenever a space-time symmetry isspontaneously broken. It would be interesting to have a general treatment of this pattern. This is similar in spirit to e.g. [9][10], but in these references backgrounds with isometries areconsidered while here no such assumption is made. a anomaly whichshould accompany the WZW term. This feature appeared when we formulated the actionsin term of the Φ field, the coordinate left unfixed after using diffeomorphism invariance.The change of variable to the τ field, which realizes additively the Weyl transformation,obscures this connection. It is an open question if one can reformulate in an invariantfashion the “minimal” action in terms of the τ field.We used extensively the “gauging” of the sigma-model in order to study its symmetries.The gauging used was a coupling to a general d dimensional metric. We used the fact thatthe natural AdS d +1 metric on the Goldstone boson space can be related in the Poincar´epatch to a d dimensional flat metric. Our “gauging” was a natural deformation of thisrelation leading to a natural appearance of the holographic set up. Alternative paths forcoupling the sigma-model to a metric, still respecting diffeomorphism invariance should beexplored. Once the above structure was picked we were led to consider metrics which weresolutions of equations of motion corresponding to a bulk action. There is a large freedomin the choice of the bulk action and in particular there is no need that the action pickedfor the sigma-model has any relation to the action which provided the metric solution. Inthis sense the treatment seems to be purely algebraic. Nevertheless one should investigateif this very relaxed holographic framework is really necessary or the requirements we listedfor the metrics can be achieved in a different way.At a more basic level the gauging we used via the coupling to a metric might not be themost natural one. Following the analogy to the chiral situation a coupling of the sigma-model to SO ( d + 1 ,
1) gauge fields would be more natural. This would require however theunderstanding of the relation (if any) between the a Weyl anomaly and descent equationsof the conformal group.
Acknowledgements:
Useful discussions with O. Aharony and S. Yankielowicz are grate-fully acknowledged. We in particular thank Z. Komargodski for his collaboration at theearly stages of this project and for his helpful comments throughout. S.T. thanks S.Kuzenko for the invitation, hospitality and discussions at Western Australian Universityin Perth during the final stages. 12 ppendix A. Review of Trace Anomalies in Holography
We will adopt the point of view that while AdS/CFT duality gives a physical realizationin which the trace anomalies of a CFT appear, one can abstract from it general, algebraicproperties rather similar to the “descent treatment” [1] of chiral anomalies.The basic setup involves a d + 1 (where d = 2 n ) dimensional manifold with the topologyof the Poincar´e patch of AdS. For the d dimensional boundary we will take a manifoldwith Euclidean signature and the topology of R d (taking the topology to be e.g. S d isunimportant as long as we discuss local anomalies).The metric G µν on the d + 1 dimensional manifold can be brought to the “Fefferman-Graham (FG in the following) gauge” ideally suited for the problem: ds = G µν dX µ dX ν = ℓ (cid:18) dρρ (cid:19) + 1 ρ g ij ( x, ρ ) dx i dx j . (A.1)Here µ, ν = 1 , . . . , d + 1 and i, j = 1 , . . . , d . The coordinates are chosen such that ρ = 0corresponds to the boundary. We will assume that g ij is regular at ρ = 0.We will consider G µν , which are solutions of a gravitational equation of motion deter-mined through the FG expansion in terms of the boundary value g (0) ij ( x ) = g ij ( x, ρ = 0).The “FG ambiguities” in the expansion determining G µν will not influence our calculationssince they appear one order higher than the expressions we use.If we use as gravitational action the minimal one, i.e. Einstein term and a negativecosmological constant, then on the solution this action becomes: S = 1 ℓ d Z d d +1 X p det G µν . (A.2)As we will discuss in the continuation, for the type A anomalies related to the WZW termwe are interested in, (A.2) gives the general expression up to an overall normalization.The FG gauge is extremely useful since the symmetries playing an essential role for traceanomalies appear as its residual gauge freedom [18] :a) diffeomorphism transformations which act just on the x variables, inducing thereforethe ordinary d -dimensional diffeomorphisms on the boundary Here our curvature convention are the opposite to those of [18], i.e. [ ∇ i , ∇ j ] V k = R ijkl V l .
13) an additional subgroup of the d + 1-dimensional diffeomorphisms depending on onefunction σ ( x ). The transformation (called PBH in the following) leaves the metric in theFG gauge, i.e. L ξ G ρρ = L ξ G ρi = 0, with solution: ρ ′ = ρ e σ ( x ) ≃ ρ (1 + 2 σ ( x )) ,x ′ i = x i − a i ( x, ρ ) (A.3)where a i ( x, ρ ) = ℓ Z ρ dρ ′ g ij ( x, ρ ′ ) ∂ j σ ( x ) . (A.4)Correspondingly, the metric g ij ( x, ρ ) transforms as: δg ij ( x, ρ ) = 2 σ (1 − ρ∂ ρ ) g ij ( x, ρ ) + ∇ i a j ( x, ρ ) + ∇ j a i ( x, ρ ) . (A.5)The covariant derivatives are with respect to g ij ( x, ρ ) with ρ fixed.In particular the boundary value of the metric g (0) ij transforms by an ordinary Weyltransformation: δg (0) ij ( x ) = 2 σ ( x ) g (0) ij ( x ) . (A.6)Therefore the PBH transformation lifts the Weyl transformations into an isomorphic sub-group of diffeomorphisms in d + 1 dimensions. The change between two metrics corre-sponding to boundary values related by (A.6) can be represented as a d + 1 dimensionalPBH transformation.A diffeomorphism invariant action S on a manifold with boundary transforms under adiffeomorphism with parameters ζ µ as: δS = Z d d +1 X ∂ µ ( ζ µ L ) = 2 σρL | ρ =0 (A.7)where L is the gravitational Lagrangian density and we used the PBH diffeomorphism inthe ρ -direction. Therefore the anomalous Weyl variation of the action is expressed directlythrough the boundary value and the trace anomalies can be read off directly from (A.7).To make the discussion more concrete, we give the explicit expressions needed for thecalculation of anomalies in d = 4. The coefficients of the FG-expansion of the metric g i,j ( x, ρ ) = (0) g ij ( x ) + ρ (1) g ij ( x ) + ρ (2) g ij ( x ) + . . . (A.8)14re largely fixed by PBH transformations (A.5): (1) g ij = − ℓ d − (cid:18) R ij − d − Rg ij (cid:19) (2) g ij = c ℓ C g ij + c ℓ ( C ) ij + ℓ d − n d − ∇ i ∇ j R − d − R ij + 18( d − d − R g ij − d − R kl R ikjl + d − d − R ik R jk + 1( d − d − RR ij + 14( d − R kl R kl g ij − d d − ( d − R g ij o . (A.9)Here and below g = (0) g and likewise for the covariant derivatives and the curvatures. Thecoefficients c and c parametrize the homogeneous solutions of (A.5) at O ( ρ ). Insertingthis into the expansion p det g ( x, ρ ) = p det g (cid:26) ρ tr( (1) g ) + ρ h
12 tr (2) g −
14 tr (cid:0) (1) g (cid:1) + 18 (cid:0) tr (1) g (cid:1) i + . . . (cid:27) (A.10)one finds p det g ( x, ρ ) = p det g ( − ℓ d − ρ R + ℓ ρ h γ C + 132( D − d − E i + . . . ) (A.11)where PBH leaves γ arbitrary . E is the dimensionally continued four-dimensional Eulerdensity E = R ijkl R ijkl − R ij R ij + R . The contribution to the anomalies can be readoff immediately from (A.11).We remark that the type A anomaly is completely fixed by the PBH relation moduloan over all normalization of the action. The same normalization fixes in an unambiguousfashion the ρ and ρ terms. This universality of the type A contribution with the accom-panying two terms is actually much more general: any modification of the bulk action byhigher curvature terms will lead to the same grouping of the three terms, only modifyingthe overall normalization. This is, in fact, easy to prove.Consider an arbitrary (local) scalar Ψ( x, ρ ) built from the bulk curvature and its covari-ant derivatives. It will have a FG expansion of the formΨ = Ψ (0) + ρ ℓ Ψ (1) + ρ ℓ Ψ (2) + . . . (A.12) E.g. for the minimal action γ = − d − d − . ρ n is accompanied by a curvature scalar of the boundary metric with 2 n derivatives,we see that necessarily Ψ (1) ∝ R and Ψ (2) must be a linear combination of R, R , R ij R ij and C . Now we use the fact that Ψ was a bulk scalar, i.e. under PBH it transforms as δ Ψ = ξ µ ∂ µ Ψ = − σ ρ ∂ ρ Ψ + a i ∂ i Ψ= − ℓ ρ σ Ψ (1) + ρ (cid:16) − ℓ σ Ψ (2) + ℓ (1) a i ∂ i Ψ (1) (cid:17) + O ( ρ ) (A.13)where ∂ i Ψ (0) = 0 was used. At O ( ρ ) this says that δ Ψ (1) = − σ Ψ (1) , but there is no suchcurvature scalar. So we find at O ( ρ ) that δ Ψ (2) = − σ Ψ (2) , from where we conclude thatΨ (2) ∝ C and the term proportional to E is unaffected.This argument can be generalized to any even dimension.Therefore the type A anomaly and the related √ ˆ g term are a result of the algebraicstructure independent of the bulk action used. For convenience we can use therefore (A.2)with an arbitrary overall normalization. For the type A anomaly in any even dimension(A.2) plays a role analogous to the Chern-Simons action for chiral anomalies. The impli-cations of the “coupling” between the type A anomaly and the √ ˆ g term are discussed inthe main part of the paper.The limiting process ρ → d − n when treated in dimensional regularization.In addition there are a finite number of terms with inverse powers of ρ reflecting thepowerlike ultraviolet divergences present formally in the regularization which should behowever absent in the effective action of a CFT. If the variation of the action is consideredas in (A.7) the pole term in d − n is not anymore there and therefore the limit d → n can be safely taken, however negative powers of ρ are still present. Therefore before takingthe limit in (A.7) we should simply discard these terms. 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