Weyl Connections and their Role in Holography
aa r X i v : . [ h e p - t h ] A p r Weyl Connections and their Role in Holography
Luca Ciambelli a,b and Robert G. Leigh b,c a CPHT, CNRS, Ecole Polytechnique, IP Paris, 91128 Palaiseau Cedex, France b Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo ON, N2L 2Y5, Canada c Department of Physics, University of Illinois, 1110 West Green St., Urbana IL 61801, U.S.A.
Abstract
It is a well-known property of holographic theories that diffeomorphism invariance in the bulk space-time implies Weyl invariance of the dual holographic field theory in the sense that the field theory couplesto a conformal class of background metrics. The usual Fefferman-Graham formalism, which provides uswith a holographic dictionary between the two theories, breaks explicitly this symmetry by choosing aspecific boundary metric and a corresponding specific metric ansatz in the bulk. In this paper, we showthat a simple extension of the Fefferman-Graham formalism allows us to sidestep this explicit breaking;one finds that the geometry of the boundary includes an induced metric and an induced connection onthe tangent bundle of the boundary that is a
Weyl connection (rather than the more familiar Levi-Civitaconnection uniquely determined by the induced metric). Properly invoking this boundary geometry hasfar-reaching consequences: the holographic dictionary extends and naturally encodes Weyl-covariantgeometrical data, and, most importantly, the Weyl anomaly gains a clearer geometrical interpretation,cohomologically relating two Weyl-transformed volumes. The boundary theory is enhanced due to thepresence of the Weyl current, which participates with the stress tensor in the boundary Ward identity. Introduction
The basic principle of general relativity is invariance under diffeomorphisms with, as it is usually formulated,a metric playing the role of the dynamical degrees of freedom. Nonetheless, we usually make use of specificchoices of coordinates and parametrizations of the metric, since we are often interested in particularsubregions of the space-time manifold. These parametrizations are not harmless in that they break (orgauge fix) some subset of the diffeomorphisms, and one has a restricted class of diffeomorphisms whichexplicitly preserves the form of a given parametrization. It is most clarifying to choose a parametrizationsuch that the unbroken symmetries act geometrically on the subregion of spacetime. This is particularlyimportant, for example, for hypersurfaces of any type and co-dimension, but even more generally, forsub-bundles (distributions) of the tangent bundle.Fefferman and Graham in their seminal works [1, 2] found a bulk gauge (FG gauge) preserving thestructure of time-like hypersurfaces in AdS d +1 spacetimes. This is useful to discuss the time-like conformalboundary, which in suitable coordinates is located at z = 0, z being the holographic coordinate such that z = const hypersurfaces are time-like. The FG gauge induces on the boundary a metric and its Levi-Civita connection. Although everything is consistent, there exists some leftover freedom in choosing theboundary metric. This comes about because the induced metric on the z = 0 hypersurface is defined, due tocertain bulk diffeomorphisms, up to a rescaling by a non-trivial function of the boundary coordinates. Wetherefore often refer to the boundary as possessing a conformal class of metrics and say that the boundaryenjoys Weyl symmetry. The latter is however often ignored in physical applications, for we usually fix theboundary metric and thus break this symmetry.In an attempt to bring electromagnetism and gravity into a unified framework [3], Weyl introducedthe concept of Weyl transformation, which encapsulates the possibility of rescaling the metric with anarbitrary scalar function. Weyl symmetry is not considered in many physical systems, but it is a keyfeature of holography. For instance, it is a very powerful tool in the fluid/gravity correspondence [4–7],where it is exploited in organizing the boundary theory.The main observation that we focus on here is that the Levi-Civita connection is not Weyl-covariant,the metricity condition being the source of this non-covariance. This problem can be sidestepped byintroducing the notion of a Weyl connection and more generally of Weyl geometry [8, 9]. These conceptshave been mentioned in the literature from time to time with reference to a variety of proposed physicalapplications, mostly in conformal gravitational theory, but also in cosmology and in particle physics, seee.g. [10–22]. In the present paper, we will show that Weyl connections play a role in the holographiccorrespondence, on the field theory side of the duality. Indeed, our first result will be to show that, byslightly generalizing the FG ansatz to what we call the Weyl-Fefferman-Graham gauge (WFG), the
Weyldiffeomorphism responsible for the rescaling of the boundary metric becomes a geometric symmetry. Theconsequences of this modification are simple: this bulk geometry induces on the boundary a metric and aWeyl connection, instead of its Levi-Civita counterpart. In the dual quantum field theory, these objectsact as backgrounds and sources for current operators. Thus, Weyl geometry makes an appearance inholography, not through a modification of the bulk gravitational theory, but in the organization of the dualfield theory.To establish these results, it is important to employ the notion of a (possibly non-integrable) distribution(i.e., a sub-bundle of the tangent bundle), replacing the less general notion of hypersurfaces and foliations.Since this may be unfamiliar to the casual reader, we take some time to review the mathematics, which isinformed by theorems of Frobenius. In this way of thinking, the more relevant object is a tangent space,rather than a space itself.The FG gauge admits an expansion of the metric from the boundary to the bulk in powers of theholographic coordinate z . Solving Einstein equations allows the extraction of the different terms of the For a review on applications of Weyl geometry in physics, see [23] and references therein. z d − which gives the vacuum expectation valueof the energy-momentum tensor operator of the dual field theory, as originally discussed in [24–27]. It is atheorem that, given these two quantities, one can reconstruct, at least order by order, a bulk AdS spacetimein FG gauge — with some caveats due to the Weyl anomaly, which we will discuss shortly. The resolutionof Einstein equations order by order for the WFG gauge on the other hand leads to a modification of thesubleading terms in this expansion. In fact, we will demonstrate that the modifications are such that eachterm is Weyl-covariant; in the FG gauge, the subleading terms transform under Weyl transformations ina very complicated non-linear fashion (which, as we discuss, comes about because they are determined bynon-Weyl-covariant Levi-Civita boundary curvature tensors). We will show how to solve Einstein equationsin the boundary-to-bulk expansion keeping the space-time dimension d + 1 arbitrary.It is a familiar aspect of the FG formalism that the on-shell bulk action diverges as one approaches theboundary. Traditionally, this is dealt with by including local counterterms which are functionals of theinduced geometry, in a solution-independent way [26, 28–30]. There remains one physical subtlety, whichis the appearance of a simple pole in d − k , with k integer. This effect is more appropriately thought ofas an anomaly in the Weyl Ward identity, a basic feature of renormalization theory [31]. This anomalycan be traced back to the fact that holographic renormalization breaks Weyl covariance by fixing a z = ǫ hypersurface to regulate the theory. No Weyl-covariant renormalization procedures exist. Consequently, aWeyl anomaly is present, and contributes in any even-dimensional boundary theory. There is of course ahuge literature on this subject, but an interesting historical account on the Weyl anomaly is [32], with auseful list of relevant references therein, as e.g. [33–35]. Notice also that a more field-theoretical approachto the anomaly, inspired by string theory and based on the non-invariance of the path integral measureunder Weyl transformations can be found in [36, 37]. The Weyl anomaly is an integral over geometricaltensors, the form of which depends on the dimension. In Ref. [38], the geometrical tensors contributing tothe anomaly were found in a scheme dependent way. The classification of the Weyl anomaly based on thecohomology of the BRST differential associated with the Weyl symmetry has been performed in [39–41].We will unravel a different packaging of the Weyl anomaly, through the use of the WFG gauge — the Weylanomaly will in fact become an integral over Weyl-covariant geometrical tensors. This result reorganizesthe theory in a much simpler fashion and opens the door to a relevant direction of investigation, whichis the determination of the coefficients in any even dimension. Indeed, in this relatively short paper, weshow how to explicitly construct the anomaly coefficient in three and five bulk dimensions only, whileformally deriving its expression in every even boundary dimension, leaving further elaboration to futureinvestigations. We have in fact derived the result in seven and nine dimensions as well, although we donot report the details here, other than to use those results to describe the general structure. Inspiredby [42], we will moreover present a simple cohomological interpretation of the Weyl anomaly, based on thedifference of two Weyl-related bulk top forms.The presence of the anomaly is usually encoded in the fact that the boundary energy-momentumtensor acquires an anomalous trace [43–46]. Indeed in the FG gauge, it is found that it must be a priori traceless. This boundary Ward identity is obtained by considering the boundary background as dictatedby the induced metric only. It is thus natural that there is only one sourced current. However, one findsthat one must typically improve the energy-momentum tensor, as originally found in [47]. We advocatein this paper a different interpretation, corroborated by the WFG extension. Specifically, we interpret theboundary theory as defined on a background metric (again given by the induced-from-the-bulk metric) anda background Weyl connection, given by the leading order of a bulk dual one-form that occurs in the WFGmetric parameterization. In this respect, two different currents can and indeed do both participate in theboundary Ward identity. From this perspective we are gauging the Weyl symmetry in the boundary [48–51],although more properly, we should view it as a local background symmetry. Actually, it is the WFG gaugein the bulk that is promoting this Weyl connection to a background configuration in the boundary. We willin particular show that the holographic dictionary furnishes directly this boundary Ward identity relating3he trace of the energy-momentum with the divergence of the Weyl current. This will be elegantly verifieddirectly from the boundary action, without invoking holography. Consequently, our setup is useful also toanalyze the profound relationship between Weyl invariance and conformal invariance, a subject which hasbeen discussed extensively, for example in [52, 53] and references therein. While the holographic dictionarywill be explored in full detail, the enhancement of the boundary theory will only be briefly described here,with further elaboration left to future works.The paper is organized as follows. Section 2 introduces the Weyl connection, its metricity and torsionproperties and its curvature tensors and associated identities. Emphasis is given to its relationship withthe ordinary Levi-Civita connection. We then analyze in Section 3 the FG gauge and define the Weyl-Fefferman-Graham gauge. We show that the WFG gauge is form-invariant under the Weyl diffeomorphism.We then discuss the important result that we are indeed inducing a Weyl connection on the boundary.The latter makes the (tangent bundle of the) boundary a (generally non-integrable) distribution. Section4 describes the improved holographic dictionary: the boundary Ward identity is derived and it is shownthat every term in the bulk-to-boundary expansion is by construction Weyl-covariant. These results aresupported by Appendix A, to which we delegate useful details for the computation of Einstein equationsorder by order. The next part of this section is devoted to a thorough analysis of the Weyl anomaly, andits cohomological derivation. In this paper, we will confine detailed results to the d = 2 and d = 4 cases.Results for d = 6 and d = 8 will be reported elsewhere, but some of their key structural aspects will bereferred to here. In Section 5, we present some relevant field theoretical results: we re-derive the Wardidentity intrinsically and present examples of simple Weyl-invariant actions. We then conclude and offersome final remarks in Section 6. Recall that given a manifold M with metric g and connection ∇ (on the tangent bundle T M ), we definethe metricity ∇ g and torsion T via ∇ X g ( Y , Z ) = ∇ X ( g ( Y , Z )) − g ( ∇ X Y , Z ) − g ( Y , ∇ X Z ) , (1) T ( X, Y ) = ∇ X Y − ∇ Y X − [ X, Y ] , (2)where X, ... are arbitrary vector fields and [
X, Y ] denotes the Lie bracket. Suppose we have a basis { e a } of vector fields, and define the connection coefficients via ∇ e a e b = Γ cab e c . (3)It is a familiar theorem that requiring both the metricity and torsion of the connection to vanish leadsto a uniquely determined set of connection coefficients, those of the Levi-Civita (LC) connection. Indeed,further defining the rotation coefficients [ e a , e b ] = C abc e c , (4)we find the general result˚Γ dac = g db (cid:16) e a ( g bc ) + e c ( g ab ) − e b ( g ca ) (cid:17) − g db (cid:16) C abf g fc + C caf g fb − C bcf g fa (cid:17) , (5)where g ab ≡ g ( e a , e b ) and we use the circle notation to refer to the LC quantities. This reduces with thechoice of coordinate basis e a = ∂ a to the familiar Christoffel symbols.The vanishing of metricity and torsion are certainly invariant under diffeomorphisms. Therefore, allthe geometrical objects built using the LC connection transform nicely under diffeomorphisms. We note4hough that metricity is not invariant under Weyl transformations g g/ B , instead transforming as ∇ g ( ∇ g −
2d ln
B ⊗ g ) / B . (8)Consequently, if we wish to consider geometric theories in which Weyl transformations play a role, it isinconvenient to choose the usual LC connection. Instead, one attains a connection that is covariant withrespect to both Weyl transformations and diffeomorphisms by introducing a Weyl connection A , [8, 9],which transforms non-linearly under a Weyl transformation g g/ B , A A − d ln B . (9)By design then, the Weyl metricity is covariant ( ∇ g − A ⊗ g ) ( ∇ g − A ⊗ g ) / B , (10)and it makes sense to set it to zero if one wishes. Fortunately, there is a theorem which states that thereis a unique connection (also generally referred to as a Weyl connection) that has zero torsion and Weylmetricity, see [9]. In this case, the connection coefficients are given by the formulaΓ dac = g db (cid:16) e a ( g bc ) + e c ( g ab ) − e b ( g ca ) (cid:17) − g db (cid:16) C abf g fc + C caf g fb − C bcf g fa (cid:17) − (cid:16) A a δ dc + A c δ da − g db A b g ca (cid:17) . (11)We note that these connection coefficients are in fact invariant under Weyl transformations. Conse-quently, the curvature of the Weyl connection has components R abcd = e c (Γ adb ) − e d (Γ acb ) + Γ fdb Γ acf − Γ fcb Γ adf − C cdf Γ afb (13)that are themselves Weyl invariant. This Weyl-Riemann tensor possesses less symmetries than its Levi-Civita counterpart, and indeed the degrees of freedom contained within are in one-to-one correspondencewith the Levi-Civita Riemann tensor, plus a 2-form F , which is the field strength F = d A . To see this, wecan write the Weyl curvature components in terms of the LC curvature components, R abcd = ˚ R abcd + ˚ ∇ d A b δ ac − ˚ ∇ c A b δ ad + (˚ ∇ d A c − ˚ ∇ c A d ) δ ab + ˚ ∇ c A a g bd − ˚ ∇ d A a g bc (14)+ A b ( A d δ ac − A c δ ad ) + A a ( g bd A c − g bc A d ) + A ( g bc δ ad − g bd δ ac ) . (15) The Weyl transformation should not be confused with a conformal transformation, which is a diffeomorphism. They dolook similar in their actions on the components of the metric,
W eyl : g ab ( x ) g ab ( x ) / B ( x ) , (6) conformal : g ab ( x ) g ′ ab ( x ′ ) = g ab ( x ) /ω ( x ) . (7)Here though, B ( x ) is an arbitrary function, while ω ( x ) is a specific function, associated with a special diffeomorphism that isa conformal isometry. To be more specific, what we mean by this notation is( ∇ g − A ⊗ g )( X, Y , Z ) = ∇ X g ( Y , Z ) − A ( X ) g ( Y , Z )The notation A ( X ) used here and throughout the paper refers to the contraction of a 1-form with a vector, A ( X ) ≡ i X A ≡ A a X a . Here we are using the convention R abcd e a ≡ R ( e b , e c , e d ) ≡ ∇ e c ∇ e d e b − ∇ e d ∇ e c e b − ∇ [ e c ,e d ] e b (12) Ric ab = R cacb , is given by Ric ab = ˚ Ric ab − d F ab + ( d − (cid:16) ˚ ∇ ( a A b ) + A a A b (cid:17) + (cid:16) ˚ ∇ · A − ( d − A (cid:17) g ab (16)in space-time dimension d . We then read off that the Weyl-Ricci tensor has an antisymmetric part Ric [ ab ] = − d F ab , (17)while the symmetric part differs from the LC Ricci tensor, Ric ( ab ) = ˚ Ric ab + ( d − (cid:16) ˚ ∇ ( a A b ) + A a A b (cid:17) + (cid:16) ˚ ∇ · A − ( d − A (cid:17) g ab . (18)The corresponding Weyl-Ricci scalar is the trace, R = ˚ R + 2( d − ∇ · A − ( d − d − A . (19)Under a Weyl transformation, R → R B , so we see that the LC Ricci scalar must transform very non-trivially under Weyl, ˚ R
7→ B (cid:16) ˚ R + 2( d − ∇ ln B − ( d − d − ∂ ln B ) (cid:17) (20)in order to cancel the transformation of the non-Weyl-invariant expression involving the Weyl connection A . Similarly, Ric ab Ric ab implies˚ Ric ab ˚ Ric ab + g ab ˚ ∇ · ∂ ln B + ( d − (cid:16) ˚ ∇ ( a ∂ b ) ln B + ∂ a ln B ∂ b ln B − g ab ( ∂ ln B ) (cid:17) (21)We thus see the important role played by the Weyl connection. Organizing the theory with respect to thelatter is a more natural prescription, whenever this theory includes Weyl transformations.Given a Weyl connection, we can organize tensors in such a way that they have a specific Weyl weightand we use the notation ˆ ∇ X t = ∇ X t + w t A ( X ) t. (22)whereby t
7→ B w t t, ˆ ∇ t
7→ B w t ˆ ∇ t. (23)For the specific case of a scalar field φ , we would then write ˆ ∇ a φ = e a ( φ ) + w φ A a φ . The condition thatWeyl metricity vanishes is translated in this notation as ˆ ∇ g = 0.Finally we remark that the Bianchi identity for the Weyl-Riemann tensor is ∇ a R ebcd + ∇ c R ebda + ∇ d R ebac = 0 (24)Contracting the e, c indices, we get the once-contracted Bianchi identity ∇ a Ric bd − ∇ d Ric ba + ∇ c R cbda = 0 . (25)which given that the Weyl-Riemann and Weyl-Ricci tensors are Weyl invariant (that is, they have weightzero), can also be written as ˆ ∇ a Ric bd − ˆ ∇ d Ric ba + ˆ ∇ c R cbda = 0 . (26)6f we multiply by g ab , we find g ab ˆ ∇ a Ric bd − ˆ ∇ d R + ˆ ∇ c ( g ab R cbda ) = 0 . (27)This can be simplified further by noting that g ab R cbda = g cb (cid:16) Ric bd + 2 F bd (cid:17) (28)and hence the twice contracted Bianchi identity can be simplified to g ab ˆ ∇ a ( G bc + F bc ) = 0 (29)where G ab = Ric ab − Rg ab is the Weyl-Einstein tensor. Since G and F have Weyl weight zero, this canalso be written as g ab ∇ a ( G bc + F bc ) = 0 (30)This is the analogue of the familiar result in Riemannian geometry, ˚ ∇ a ˚ G ac = 0. The Fefferman-Graham theorem [1, 2] says that the metric of a locally asymptotically AdS d +1 (LaAdS)geometry can always be put in the formd s = L d z z + h µν ( z ; x )d x µ d x ν . (31)The conformal boundary is a constant- z hypersurface at z = 0 in these coordinates. To obtain thisform, one has used up all of the diffeomorphism invariance, apart from residual transformations of the x µ x ′ µ ( x ), which of course would change the components of h µν in general.Near z = 0, h µν ( z ; x ) may be expanded h µν ( z ; x ) = L z (cid:20) γ (0) µν ( x ) + z L γ (2) µν ( x ) + z L γ (4) µν ( x ) + ... (cid:21) + z d − L d − (cid:20) π (0) µν ( x ) + z L π (2) µν ( x ) + ... (cid:21) + .... (32)Here, we are regarding the boundary dimension d as variable (in fact, we will regard d ∈ C formally asneeded. This is discussed further later in the paper.) Given this expansion, γ (0) µν ( x ) has an interpretationas an induced boundary metric: z L d s −→ z → γ (0) µν ( x )d x µ d x ν = d s bdy . (33)It is this object that sources the stress energy tensor in the dual field theory, with π (0) µν ( x ) its vev. All ofthe other terms in the expansion are determined in terms of γ (0) µν ( x ) , π (0) µν ( x ) by the bulk classical equationsof motion.Equation (33) defines the induced boundary metric up to a Weyl transformation. We see indeed thatthere is an ambiguity in the construction of this metric which amounts in defining the latter up to a scalarfunction of the boundary coordinates. Although it is often stated, this ambiguity is usually disregardedand a specific choice is made. This holographic dimensional regularization avoids the necessary introduction of logarithms (as in e.g. [30]) that occurwhen d is an even integer. Weyl diffeomorphism ) z z ′ = z/ B ( x ) , x µ x ′ µ = x µ (34)plays an important role. It has the effect of inducing a Weyl transformation of the boundary metriccomponents: using (33) with holographic coordinate now z ′ we obtaind s bdy = γ (0) µν ( x ) B ( x ) d x µ d x ν . (35)However, this diffeomorphism does not leave the bulk metric in the Fefferman-Graham gauge, but rathertransforms it to d s = L (cid:18) d z ′ z ′ + ∂ µ ln B ( x ) d x µ (cid:19) + h µν ( z ′ B ( x ); x )d x µ d x ν (36)where h µν ( z ′ B ( x ); x ) = L z ′ " γ (0) µν ( x ) B ( x ) + z ′ L γ (2) µν ( x ) + z ′ L B ( x ) γ (4) µν ( x ) + ... (37)+ z ′ d − L d − (cid:20) B ( x ) d − π (0) µν ( x ) + z ′ L B ( x ) d π (2) µν ( x ) + ... (cid:21) + .... (38)Thus, this diffeomorphism takes us out of FG gauge (as it is one of the diffs that was fixed in going tothat gauge), and acts on the boundary tensors γ ( k ) µν ( x ) and π ( k ) µν ( x ) by a local Weyl rescaling with specific k -dependent weights.The standard way to deal with the fact that we have been taken out of FG gauge is to employ anadditional diffeomorphism acting on the x µ x µ + ξ µ ( z ; x ) which becomes trivial at the conformal bound-ary in such a way that γ (0) µν ( x ) is left unchanged, but the cross term in (36) is cancelled (see e.g. [42]).However, this diffeomorphism unfortunately has a complicated effect on all of the subleading terms inthe metric — they no longer transform linearly as in (37), but instead transform non-linearly under thecombined transformations and, we claim, this obscures the geometric significance of the sub-leading terms.There is nothing inconsistent here: in FG gauge, the subleading terms are given on-shell by expressionsinvolving the Levi-Civita curvature of the induced metric, which themselves transform non-linearly underWeyl transformations.The fact that the Weyl diffeomorphism has taken us out of the Fefferman-Graham gauge, and inparticular acts on a piece of the bulk metric other than h µν as in eq. (36), motivates replacing theFefferman-Graham gauge byd s = L (cid:18) d zz − a µ ( z ; x )d x µ (cid:19) + h µν ( z ; x )d x µ d x ν . (39)which we refer to as Weyl-Fefferman-Graham (WFG) gauge. In this form, the bulk metric is given in termsof two tensor fields, h µν and a µ , and the Weyl diffeomorphism acts asd s = L (cid:18) d z ′ z ′ − a µ ( z ′ B ( x ); x )d x µ + ∂ µ ln B ( x ) d x µ (cid:19) + h µν ( z ′ B ( x ); x )d x µ d x ν . (40)Thus, the Weyl diffeomorphism can be interpreted as acting on the fields h µν and a µ , preserving the formof the WFG gauge without the need for a compensating diffeomorphism; the action on h µν is as beforewhile a µ shifts non-linearly and so ultimately will be interpreted as a connection or gauge field. We willbe precise about the details of these transformations below.8n WFG gauge, the constant- z hypersurface Σ at z = 0 remains the conformal boundary with inducedmetric γ (0) , as z L d s −→ z → γ (0) µν ( x )d x µ d x ν . (41)Thus the presence of a µ in the ansatz does not modify the induced metric at z = 0. However, as we willsee, this does not mean that a µ does not appear at the conformal boundary. This is surprising since a µ ispure gauge in the bulk, but we will see that a µ has a clear geometric interpretation in the boundary theory.To understand this claim, we first note that the metric is no longer diagonal in the z, x µ coordinates, andso we must take greater care in interpreting how we approach the conformal boundary. We now describethat process carefully.It is natural, given the metric ansatz (39), to introduce the 1-form e ≡ Ω( z ; x ) − (cid:18) d zz − a µ ( z ; x )d x µ (cid:19) (42)This form defines a distribution C e ⊂ T M defined as C e = ker ( e ) = span n X ∈ Γ( T M ) (cid:12)(cid:12)(cid:12) i X e = 0 o . (43)Note that there is an ambiguity in multiplying e (or equivalently the X ’s) by a function on M , and wehave represented this ambiguity by introducing the function Ω in (42).We remark that if a µ were zero, then C e would be the span of the vectors ∂ µ , which form a basis forthe tangent spaces to constant- z hypersurfaces. In the present context though, this more general notion ofa distribution is the appropriate geometrical structure. In the general case, it is convenient to introduce abasis for C e as the set of vectors D µ ≡ ∂ µ + a µ ( z ; x ) z∂ z . (44)This implies that we can regard a µ as providing a lift from T Σ (with basis { ∂ µ } ) to C e , that is, it can bethought of as an Ehresmann connection. By the Frobenius theorem, C e is an integrable distribution if (cid:2) D µ , D ν (cid:3) ∈ C e . (45)To understand this condition, it is convenient to introduce a vector dual to e , e ≡ Ω( z ; x ) z∂ z (46)which has been normalized to i e e = e ( e ) = 1, and we regard { e, D µ } as a basis for T ( z ; x ) M . We thencompute (cid:2) D µ , D ν (cid:3) = Ω( z ; x ) − f µν ( z ; x ) e, f µν ( z ; x ) ≡ D µ a ν ( z ; x ) − D ν a µ ( z ; x ) . (47)So we find that integrability is the condition f µν = 0, and thus by Frobenius, the distribution C e woulddefine under that circumstance a foliation of M by co-dimension one hypersurfaces. We will not find itnecessary to assume that the distribution is in fact integrable, and thus we will not assume a µ to be flat. Here, we are regarding Σ as an isolated hypersurface in M . We can thus regard M as a fibre bundle π : M → Σ. AnEhresmann connection provides a splitting of the tangent bundle
T M = H ⊕ V , and the D µ vectors form a basis of H ,identified with C e , at the point ( z, x µ ).
9y taking e in the form (46), we have fixed some of the diffeomorphism invariance; the residualdiffeomorphisms that preserve the form of e in (46) are given by z ′ = z ′ ( z ; x ) , x ′ µ = x ′ µ ( x ).This setof diffeomorphisms includes but is larger than the Weyl diffeomorphisms. Given the interpretation ofholography in terms of renormalization, we expect that these diffeomorphisms correspond to generic local (in x ) coarse grainings. This is ultimately the reason for our construction, which retains a clear covariantgeometric interpretation for this more general notion of renormalization. These residual diffeomorphismsact on the form e as e e ′ = Ω ′ ( z ′ ( z ; x ); x ′ ( x )) − (cid:18) d z ′ ( z ; x ) z ′ ( z ; x ) − a ′ µ ( z ′ ( z ; x ); x ′ ( x ))d x ′ µ (cid:19) (49)and thus leave it invariant if ∂x ′ ν ( x ) ∂x µ a ′ ν ( z ′ ; x ′ ) = ∂ ln z ′ ( z ; x ) ∂ ln z a µ ( z ; x ) + ∂ ln z ′ ( z ; x ) ∂x µ , Ω ′ ( z ′ ; x ′ ) = ∂ ln z ′ ( z ; x ) ∂ ln z Ω( z ; x ) . (50)The first equation is consistent with the interpretation of a as an Ehresmann connection. The second equa-tion implies that the inherent ambiguity in the definition of the distribution C e represented by Ω( z ; x ) canbe thought of as the (local) reparametrization invariance of z . We can for example use this reparametriza-tion invariance to set Ω( z ; x ) → L − if we wish. The diffeomorphisms that preserve this choice (or, moregenerally preserve any specific Ω( z ; x )) are a subset of the aforementioned residual diffeomorphisms andare of the form z ′ = z/ B ( x ) , x ′ µ = x ′ µ ( x ), which are precisely the Weyl diffeomorphisms (together with anarbitrary reparameterization x ′ = x ′ ( x )) that preserve the form of the metric (39). In this case, the firstequation in (50) reduces to ∂x ′ ν ( x ) ∂x µ a ′ ν ( z ′ ; x ′ ) = a µ ( z ; x ) − ∂ µ ln B ( x ) , (51)and so we are to interpret the a µ ( z ; x ) as a connection for the Weyl diffeomorphisms (34). This transfor-mation differs from what appears in eq. (40) only because here we are allowing for an arbitrary transversediffeomorphism x ′ = x ′ ( x ) as well. Given this result, it may not come as a surprise that a µ ( z ; x ) will inducea Weyl connection on the conformal boundary, and we will establish precisely that below.To recap, we have been led to the following (non-coordinate) basis for the tangent space { e, D µ } = n L − z∂ z , ∂ µ + a µ z∂ z o (52)which have the following Lie brackets (cid:2) D µ , D ν (cid:3) = Lf µν e, (cid:2) D µ , e (cid:3) = − Le ( a µ ) e (53)To proceed further, we Fourier analyze a µ ( z ; x ) and h µν ( z ; x ) in the sense that we will expand them ineigenfunctions of e . Such eigenfunctions are of course just the monomials in z ∈ R + . For h µν ( z ; x ) weobtain then the same expansion as before, eq. (32), and for a µ ( z ; x ) we write a µ ( z ; x ) = (cid:20) a (0) µ ( x ) + z L a (2) µ ( x ) + ... (cid:21) + z d − L d − (cid:20) p (0) µ ( x ) + z L p (2) µ ( x ) + ... (cid:21) + ..., (54) Indeed, the vector field e could more generally be of the form e → e ′ = e + θ µ ( z ; x ) D µ (48)which satisfies e ( e ) = 1 for any θ µ . (In the language of footnote 6 the e of (46) is special in that e ∈ V .) In the generalcase, we have (cid:2) D µ , D ν (cid:3) = f µν e ′ − f µν θ λ D λ and thus integrability remains the condition f µν = 0. The second diffeomorphism,discussed earlier, that returns the metric to the FG ansatz after a boundary Weyl transformation corresponds on the contraryto setting a µ → θ µ = 0. In most field theory contexts, such local rescalings are not considered. However, it is clear that they are of general interest.For example, it is widely appreciated that ultraviolet divergences that occur in calculations of entanglement observables arisethrough the pile-up of modes near the entanglement cut, and renormalization towards the cut is the natural procedure. Weshould also mention that similar structure is known to arise in holographic fluids. a µ is a pure gauge part of the bulk metric, it should not source a (vector)operator in the boundary theory. However, what we will show is that a (0) µ is not part of the boundarymetric but will appear instead as part of the induced boundary connection. It thus represents a choicethat we should have of selecting a connection which is not the Levi-Civita connection determined entirelyby a choice of the induced boundary metric.More precisely, what we will show is that for the WFG ansatz, the induced connection is not the Levi-Civita connection of the induced metric, but instead a Weyl connection. Given the expansions (32,54), wesee that the Weyl diffeomorphism (34) acts as γ ( k ) µν ( x ) γ ( k ) µν ( x ) B ( x ) k − , π ( k ) µν ( x ) π ( k ) µν ( x ) B ( x ) d − k (55) a ( k ) µ ( x ) a ( k ) µ ( x ) B ( x ) k − δ k, ∂ µ ln B ( x ) , p ( k ) µ ( x ) p ( k ) µ ( x ) B ( x ) d − k (56)and so in particular γ (0) µν ( x ) γ (0) µν ( x ) / B ( x ) , a (0) µ ( x ) a (0) µ ( x ) − ∂ µ ln B ( x ) (57)and thus we may anticipate that a (0) µ will play the role of a boundary Weyl connection. All of the othersubleading functions in the expansions (32,54) are interpreted to have, `a la (55–56), definite Weyl weights,that is they are Weyl tensors. It is then natural to expect that they will be determined in terms of theWeyl curvature, which we discussed in the last section.We introduced the concept of the distribution C e precisely in order to properly discuss the notion ofan induced connection, as C e is a sub-bundle of T M . That is, given a bulk connection ∇ on T M (whichwe will take to be the LC connection), we can apply it to vectors in C e , which will be of the general form ∇ D µ D ν = Γ λµν D λ + Γ eµν e. (58)The coefficients of the induced connection on C e are by definition the Γ λµν appearing in (58). Noticethat these connection coefficients should not be confused with the usual Christoffel symbols, which areassociated with coordinate bases. By direct computation, we findΓ λµν = γ λµν ≡ h λρ (cid:16) D µ h ρν + D ν h µρ − D ρ h νµ (cid:17) (59)and furthermore if we evaluate this expression at z = 0, we find γ (0) λµν = γ λρ (0) (cid:16) ( ∂ µ − a (0) µ ) γ (0) νρ + ( ∂ ν − a (0) ν ) γ (0) µρ − ( ∂ ρ − a (0) ρ ) γ (0) µν (cid:17) . (60)This result can be compared to (11), from which we conclude that the induced connection on the boundaryis in fact a Weyl connection, with the role of the geometric data g ab and A a in (11) being played hereby γ (0) µν and a (0) µ . In comparing, we make use of the fact that here the intrinsic rotation coefficients are C µν λ = 0, as in (47). We will use the notation ∇ (0) for the corresponding Weyl connection (whoseWeyl-Christoffel symbols are given by (60)), and the curvature as R (0) λµρν . A tensor with components t µ ...µ n ( x ) that has Weyl weight w t transforms as t µ ...µ n ( x )
7→ B ( x ) w t t µ ...µ n ( x ), while ˆ ∇ (0) ν t µ ...µ n ( x ) ≡∇ (0) ν t µ ...µ n ( x ) + w t a (0) ν t µ ...µ n ( x ) transforms covariantly with the same weight. As noted above, all of thecomponent fields aside from a (0) µ transform covariantly with respect to arbitrary Weyl transformations,and the Weyl weights of the various component fields are given above in (55). In the next section, we willbriefly study some aspects of the holographic dictionary, and we will find that every equation is covariantwith respect to arbitrary Weyl transformations — it is a bona fide (background) symmetry of the dualfield theory. In particular, we will find that the appearance of a (0) µ ( x ), since it transforms non-linearly11nder Weyl transformations, is through Weyl-covariant derivatives of other fields, or through expressionsinvolving the Weyl-invariant field strength f (0) µν . Before moving on, we would like to stress again the mainresult of this section: the usual bulk LC connection built using the bulk metric in the enhanced WFGgauge induces on the boundary a Weyl connection and therefore a boundary Weyl-covariant geometry. In this section, we will explore some details of the holographic dictionary corresponding to the WFG ansatz.The LC connection in the bulk has the form ∇ D µ D ν = γ λµν D λ − h νλ ψ λµ e (61) ∇ D µ e = ψ λµ D λ (62) ∇ e D µ = ψ λµ D λ + Lϕ µ e (63) ∇ e e = − Lh λρ ϕ ρ D λ (64)where ψ µν = ρ µν + L h µλ f λν , ρ µν = h µλ e ( h λν ) , ϕ µ = e ( a µ ) , f µν = D µ a ν − D ν a µ (65)and we note that ϕ µ is proportional to the rotation coefficient C eµe , i.e., (cid:2) e, D µ (cid:3) = Lϕ µ e . In addition, wewill use the notation θ = trρ = e (ln √− det h ) and ζ µν = ρ µν − d θδ µν . In Appendix A, we record someadditional details, including the Weyl-Riemann curvature components.As is the case in the FG gauge, γ (0) µν ( x ) defines a background boundary metric and acts as a source forthe stress energy tensor of the dual field theory, with π (0) µν ( x ) its vev. We have seen that in WFG gauge, a (0) µ ( x ) has the interpretation of a Weyl connection in the dual field theory; in addition, we will show belowthat p (0) µ ( x ) appears in the Weyl Ward identity as if it were the vev for the Weyl current. We will discussthese operators further in Section 5.As usual [30], one finds that the bulk equations of motion determine the subleading component fieldsin terms of γ (0) µν ( x ) , a (0) µ ( x ) , π (0) µν ( x ) and p (0) µ ( x ). In this paper, we will confine our attention to vacuumsolutions of Einstein gravity that are asymptotically locally anti-de Sitter. For example, the ee -componentof the vacuum Einstein equations is0 = G ee + Λ g ee = − tr ( ρρ ) − L tr ( f f ) − R + θ + Λ (66)where Λ = − d ( d − L is the cosmological constant of AdS d +1 and we define for the sake of brevity R λµρν = D ρ γ λνµ − D ν γ λρµ + γ δνµ γ λρδ − γ δρµ γ λνδ (67)with R = h µν R ρµρν the corresponding Ricci scalar. Expanding (66) we find0 = (cid:20) Λ + d ( d − L (cid:21) − z L h d − L − X (1) + R (0) i + ... − ( d − z d L d (cid:20) d L − Y (1) + ˆ ∇ (0) · p (0) (cid:21) + ... (68)where R (0) is the boundary Weyl-Ricci scalar and X (1) = γ µν (0) γ (2) µν , Y (1) = γ µν (0) π (0) µν . (69) The notation used here can be interpreted in terms of expansion ( θ ), shear ( ζ ), vorticity ( f ) and acceleration ( ϕ ) of theradial congruence e .
12n (68), the order one equation is trivially satisfied while the z contribution gives X (1) entirely in termsof the Weyl-Ricci scalar curvature: X (1) = − L d − R (0) . (70)This result looks exactly the same as is obtained in the usual FG calculation, but we stress that the righthand side involves now the Weyl covariant Weyl-Ricci scalar.For later use, we also note the subleading term in the expansion p − det h ( z ; x ) = (cid:18) Lz (cid:19) d q − det γ (0) ( x ) (cid:20) z L X (1) + 12 z L X (2) + ... + 12 z d L d Y (1) + ... (cid:21) , (71)with X (2) given in (A.19). Using (A.51) one obtains X (2) = − L d − (cid:20) Ric (0) µν Ric (0) µν − d d − R (0)2 − ( d − tr ( f (0)2 ) (cid:21) − L ∇ (0) ν a (2) ν . (72)As in the FG story, we must be careful with the O ( z d ) terms here because of divergences in theevaluation of the on-shell action — those divergences are responsible for the Weyl anomaly in the dualfield theory [31], the structure of which we will discuss in detail below. Nevertheless, we may read off the‘left-hand-side’ of the Weyl Ward identity from eq. (68),ˆ ∇ (0) · p (0) + d L γ µν (0) π (0) µν . (73)We will see later that this is the expected form given the interpretation of π (0) µν and p (0) µ in terms of currentsin the dual field theory. We will also study the form of the anomalous right-hand-side later, in particularin d = 2 and d = 4.Similarly, one finds that the leading O ( z ) term in G eµ is proportional to γ λν (0) ∇ (0) ν (cid:16) G (0) λµ + f (0) λµ (cid:17) = 0 , (74)the vanishing of which is the twice-contracted Bianchi identity of the Weyl connection, as was discussedabove (see eq. (30)).The leading non-trivial terms in the µν -components of the Einstein equations determine γ (2) µν = − L d − (cid:16) Ric (0)( µν ) − d − R (0) γ (0) µν (cid:17) = − L d − L (0)( µν ) , (75)where L (0) is the Weyl-Schouten tensor. Its trace (69) correctly reproduces (70). We take each of theseresults as representative of the fact that the subleading terms in the expansion of the metric are determinedby the Weyl curvature, analogous to what happens in the usual FG gauge in which they are determinedby the LC curvature of the induced metric. As we mentioned previously, the difference is that now all ofthe subleading terms in the bulk fields are Weyl-covariant.The holographic dictionary for WFG will be taken to be the obvious generalization of the usual rela-tionship [24], i.e., Z bulk [ g ; γ (0) , a (0) ] = exp ( − S o.s. [ h, a ; γ (0) , a (0) ]) = Z F T [ γ (0) , a (0) ] (76)where on the left we have the on-shell action of the bulk classical theory whose metric is given by h, a withasymptotic configurations γ (0) , a (0) , while the right-hand-side is the generating functional of correlation13unctions of operators sourced by γ (0) , a (0) . Although this is expressed in terms of the ‘bare’ sources, itis implicit that a regularization scheme for the left-hand-side is employed and that the boundary counter-terms are introduced to absorb power divergences that arise in the evaluation of the on-shell action, [30].Here, we will organize the discussion by taking the space-time dimension d to be formally complex; theon-shell action is convergent for sufficiently small d , and as we move d up along the real axis, we encounteradditional divergences as d approaches an even integer. It is well-known in the context of Fefferman-Graham that as a byproduct this divergence induces the Weyl anomaly of the dual field theory, and isassociated with the appearance of logarithms in the field expansions when d is precisely an even integer,as discussed in [31, 42]. Here we will review this bit of physics, as the existence of the Weyl connection, aswe will see, organizes the Weyl anomaly in a much more symmetric fashion than it is usually described.It is taken for granted that Z bulk is diffeomorphism invariant. Under the holographic map this implies,among other things, that the dual field theory can be regulated in a diffeomorphism-invariant fashion [30].However, the bulk calculation is classical, and thus, in principle, is a functional of the bulk metric g as wellas the boundary values. We therefore suppose that Z bulk h g ′ ; γ ′ (0) , a ′ (0) , ... (cid:12)(cid:12)(cid:12) z ′ , x ′ i Z bulk h g ; γ (0) , a (0) , ... (cid:12)(cid:12)(cid:12) z, x i = 1 , (77)where the notation refers to the fact that we are computing the partition function in different coordinatesystems. Here of course we are particularly interested in the Weyl diffeomorphism ( z ′ , x ′ ) = ( z/ B ( x ) , x )which relates the boundary values γ ′ (0) = γ (0) / B , a ′ (0) = a (0) − d ln B . Z bulk is given in the classical limitby evaluating the (renormalized) on-shell action, Z bulk = e − S o.s. [ g ; γ (0) ,a (0) ,... | z,x ] . We then ask, is it also truethat this cleanly induces a Weyl transformation on the boundary? That is, is it true that Z bdy [ x ; γ ′ (0) , a ′ (0) , ... ] Z bdy [ x ; γ (0) , a (0) , ... ] ? = 1 , (78)where Z bdy is the generating functional in the given background. As is well-established (see e.g. [42]), whathappens is that there is an anomaly Z bulk h g ′ ; γ ′ (0) , a ′ (0) , ... (cid:12)(cid:12)(cid:12) z ′ , x i Z bulk h g ; γ (0) , a (0) , ... (cid:12)(cid:12)(cid:12) z, x i = e A k Z bdy [ x ; γ ′ (0) , a ′ (0) , ... ] Z bdy [ x ; γ (0) , a (0) , ... ] (79)in dimension d = 2 k . Recall that we are employing the specific Weyl diffeomorphism, which is inducinga Weyl transformation on the boundary, but no boundary diffeomorphism. If we take the log of theseexpressions, the result is that0 = S bulk [ g ′ ; γ ′ (0) , ... | z ′ , x ] − S bulk [ g ; γ (0) , ... | z, x ] = S bdy [ x ; γ ′ (0) , a ′ (0) , ... ] − S bdy [ x ; γ (0) , a (0) , ... ] + A k . (80)That is, when we compare the evaluation of the bulk on-shell action in different coordinate systems, theresult appears as the difference of boundary actions in Weyl-equivalent backgrounds, up to an anomalousterm, which is not the difference of two such actions . The only source for such a term is a pole at d = 2 k in the evaluation of the bulk action, which arises because the on-shell action is not a boundary term, butcontains pieces that must be integrated over z . The bulk action for Einstein gravity is generally given by( vol S = √− det h d d x ) S bulk [ g ; γ (0) , ... | z, x ] = 116 πG Z M e ∧ vol S ( R − . (81)14n shell, it evaluates to S bulk [ g ; γ (0) , ... | z, x ] = − d πGL Z M e ∧ vol S = − d πGL Z M d zz ∧ d d x √− det h, (82)where we recall that d is the boundary dimension. We then expand √− det h in powers of z , as given in(71) to obtain S bulk [ g ; γ (0) , ... | z, x ] = − d πGL Z M d z ∧ d d x (cid:16) Lz (cid:17) d +1 q − det γ (0) (cid:16) z L X (1) + 12 z L X (2) + . . . (cid:17) . (83)Consider now the difference of Weyl-transformed bulk actions as in (80) and define vol Σ = p − det γ (0) d d x .The idea is to start with S bulk [ g ′ ; γ ′ (0) , ... | z ′ , x ], use the explicit Weyl transformation of the different quan-tities in the expansion (see (55)) and then change the name of the integration variable from z ′ to z . Wewill demonstrate this for the first two poles, which occur at d = 2 and d = 4. Using (82), we obtain fromthe left hand side of (80)0 = d πGL Z M d B − d d (cid:18) Lz (cid:19) d ! ∧ vol Σ − d πGL Z M d d (cid:18) Lz (cid:19) d ! ∧ vol Σ + d πGL Z M d B − ( d − d − (cid:18) Lz (cid:19) d − ! ∧ G (1)Σ − d πGL Z M d d − (cid:18) Lz (cid:19) d − ! ∧ G (1)Σ + d πGL Z M d B − ( d − d − (cid:18) Lz (cid:19) d − ! ∧ G (2)Σ − d πGL Z M d d − (cid:18) Lz (cid:19) d − ! ∧ G (2)Σ + . . . , (84)with G (1)Σ = X (1) vol Σ (Weyl weight − ( d − G (2)Σ = X (2) vol Σ (Weyl weight − ( d − d = 2. We observe that the offending term in d → − is d πGL Z M d B − ( d − d − (cid:18) Lz (cid:19) d − ! ∧ G (1)Σ − d πGL Z M d d − (cid:18) Lz (cid:19) d − ! ∧ G (1)Σ = 18 πGL Z Σ ln B G (1)Σ . (85)The equality in this equation is obtained expanding B around 1 and eventually imposing d = 2. Forconcreteness we rewrite this final result using the holographic value of X (1) , (70). Then, we read from (80): A = 18 πGL Z Σ ln B G (1)Σ = − L πG Z Σ ln B R (0) vol Σ . (86)This numerical coefficient is the correct one, leading to the central charge c = 3 L G [31, 54]. We will shortlycomment on the implications, but notice already that R (0) is not the Levi-Civita curvature, as usuallyfound, but rather the Weyl curvature, which depends on both γ (0) and a (0) . As such, it is a Weyl-covariantscalar.We move to d = 4 (the on-shell action itself must be supplemented by boundary counterterms to movepast d = 2, but these do not contribute to the current computation). Here the pole for d → − gives d πGL Z M d B − ( d − d − (cid:18) Lz (cid:19) d − ! ∧ G (2)Σ − d πGL Z M d d − (cid:18) Lz (cid:19) d − ! ∧ G (2)Σ = 14 πGL Z Σ ln B G (2)Σ . (87) To evaluate these expressions, a regulator is required. The last step of renaming the integration variable has a correspondingeffect on the cutoff and thus is not innocuous in the renormalization procedure. Such a regulator is not Weyl-covariant, whichis consistent with the fact that an anomaly arises. Most of the details of the renormalization occur in expressions that are thedifference of two Weyl-equivalent actions, whereas the anomaly is not and has been cleanly extracted. X (2) , we explicitly compute A = − L πG Z Σ ln B (cid:18) L (cid:18) Ric (0) µν Ric (0) µν − R (0)2 − tr ( f (0)2 ) (cid:19) + ˆ ∇ (0) ν a (2) ν (cid:19) vol Σ . (88)This result has the same form as familiar expressions given in the literature [31] if a µ is set to zero, atwhich point the last two terms would drop and the first two would then involve the LC curvature of theinduced metric (giving the well-known “ a = c ” result for Einstein gravity). There are several differences;first, the curvature tensors appearing in the anomaly as before are Weyl covariant. In addition, we see theappearance of a term of the form tr ( f (0)2 ), where f (0) µν is the field strength of the Weyl connection a (0) .We note that in a formalism where a (0) has been set to zero, f (0) is by that assumption flat, but here weare not requiring this. It would be interesting to understand a physical boundary field theory situationwhere f (0) makes an appearance (note that in holographic fluid states, f (0) is related to vorticity of thefluid state). Finally, notice also that the subleading term a (2) makes an appearance in the anomaly, butonly through a total derivative.In a general dimension, all of the subleading modes a ( j ) µ will make an appearance. It is important tonote though that all of the higher modes a ( j> µ are not in fact determined by the equations of motion.This is in keeping with the fact that they represent pure gauge degrees of freedom in the bulk, and herein the d = 4 Weyl anomaly a (2) makes an appearance as a total derivative ambiguity; as far as the Wardidentity is concerned, it can be absorbed into the vev of the Weyl current. One finds that in d = 2 k dimensions, the mode a ( k ) appears entirely in just the same way, as a contribution to the anomaly of theform ˆ ∇ (0) · a ( k ) . Such total derivatives are often simply dropped in discussions of the anomalies, and herewe see that they are in fact ambiguities. In higher dimensions, one finds that the modes a (0 2) ˜ A (cid:17) η µν . (90)where ˜ A = a (0) − dσ . So we see that indeed, the Levi-Civita curvature is recovered from the Weyl-Riccicurvature by setting a (0) to zero, while the Weyl-Ricci curvature is Weyl invariant because it dependson the St¨uckelberg-like field ˜ A . In the usual formalism, conformally flat metrics generally give a non-zero Ricci curvature that depends on the conformal factor, while in the Weyl connection formalism, thecurvature vanishes for all conformally flat metrics if we simultaneously choose a (0) = dσ instead of zero ,that is, ˜ A = 0. The Weyl-Ricci curvature is Weyl-invariant since ˜ A is Weyl invariant. That is, for all data In fact, as we will show elsewhere, the anomaly can be written in terms of the Euler character (of the Lorentz connection)and the Weyl tensor squared, so the appearance of f (0) µν does not represent a new central charge. We thank Weizhen Jia forpointing this out to us. γ (0) = e σ η, a (0) = ∂σ ) in the Weyl orbit of ( η, η, d = 4 anomaly given in (88), we see that allterms are Weyl invariant. Of course, it is the anomaly coefficients that are of most interest rather thanthe values that a given curvature polynomial takes in some particular geometry.As we mentioned, there are terms in the anomaly ( trf (0)2 in d = 4) that have no analogue in theusual formalism (as they are assumed to vanish). It is not clear to us what interpretation there might befor a non-zero f (0) background in a given field theory. We hope to return to this question in a separatepublication. It is possible though that Weyl gauge field configurations may play an important role in fieldtheories on space-times of non-trivial topology or with boundaries.To recap, the Weyl anomaly in d = 2 k is associated with the difference of two bulk volumes (cid:16) e ∧ G ( k )Σ (cid:17) ′ − (cid:16) e ∧ G ( k )Σ (cid:17) ∝ d(ln B X ( k ) vol Σ ) , (91)Each term on the left is a closed form (since they are top forms in the bulk), with the difference being anexact form, the exterior derivative of the local Weyl anomaly form.The anomalies in the Levi-Civita framework are classifiable and are known explicitly at least through d = 8 in the usual formalism, and their topological origin apparently understood through BRST methods(see e.g. [39]). We expect that the inclusion of the Weyl connection in the latter story would reorganize itin a useful way, and we expect to return to this in a future publication. In this section, we will make some preliminary remarks about the dual field theory, with more detail leftfor future work. The holographic analysis implies that we should now consider a field theory coupled toa background metric and Weyl connection, with action S [ γ (0) , a (0) ; Φ] where Φ denotes some collection ofdynamical fields to which we will assign definite Weyl weights. As we will explain, this is perfectly naturalfrom the field theory perspective as well, but constitutes a new organization of such field theories (which inthe usual formulation are coupled only to a background metric, as thoroughly reviewed for instance in [55]).The quantum theory possesses a partition function Z [ γ (0) , a (0) ], obtained by doing the functional integralover the dynamical fields, that depends on the background, both through explicit dependence in the actionand in the definition of the functional integral measure. A background Ward identity is generated bychanging integration variables Φ( x ) 7→ B ( x ) w Φ Φ( x ) giving Z [ γ (0) , a (0) ] = e A [ B ] Z [ B ( x ) − γ (0) , a (0) − d ln B ( x )] (92)with A a possible anomalous contribution. Thus the Weyl Ward identity is a relationship between different theories, that is, field theories in different backgrounds and so, more properly, we refer to the above equationas a background Ward identity. It is of interest then to consider classical actions that are background Weylinvariant, satisfying S [ γ (0) , a (0) ; B ( x ) w Φ Φ] = S [ B ( x ) − γ (0) , a (0) − d ln B ( x ); Φ]. An example is a free scalartheory, whereby w Φ = ( d − 2) is the engineering dimension, and the action is S [ γ (0) , a (0) ; Φ] = − Z d d x q − det γ (0) γ µν (0) ˆ ∇ (0) µ Φ ˆ ∇ (0) ν Φ (93)where ˆ ∇ (0) µ Φ = ∂ µ Φ + w Φ a (0) µ Φ is background Weyl-covariant by itself in the above sense. In the usualformalism without the Weyl connection, the corresponding action is not Weyl invariant, but its Weyltransformation can be cancelled, up to a total derivative, by the addition of a specific additional termproportional to R d d x p − det γ (0) ˚ R Φ . Here, (93) is itself background Weyl invariant, and we may add The formalism can be extended to include sources for any operators, but we restrict our focus here. R d d x p − det γ (0) R (0) Φ , where R (0) is the Weyl-Ricci scalar, with any coefficient. With the presence of the Weyl connection, it is a trivial matter to write a variety of actionswith background Weyl invariance.The background fields, as usual, are interpreted as sources for current operators. For example, thestress tensor of the free theory has the formT µνγ (0) ,a (0) ( x ) = √ − det γ (0) δS [ γ (0) , a (0) ; Φ] δγ (0) µν ( x ) = ˆ ∇ µ (0) Φ( x ) ˆ ∇ ν (0) Φ( x ) − γ (0) µν ( x ) γ (0) αβ ( x ) ˆ ∇ (0) α Φ( x ) ˆ ∇ (0) β Φ( x ) . (94)Here we have used pedantic notation to emphasize that the definition of the operator depends on thebackground fields. This operator is Weyl-covariant, by which we meanT µν B ( x ) − γ (0) ,a (0) − d ln B ( x ) ( x ) = B ( x ) d T µνγ (0) ,a (0) ( x ) . (95)That is, if we compare correlation functions of the stress tensor in two Weyl-related backgrounds, therewill be a relative factor of B ( x ) d for each instance of the stress tensor; for brevity, we refer to this as thestress tensor (with two upper indices) having Weyl weight w T = d . Similarly, we have the Weyl currentJ µγ (0) ,a (0) ( x ) = 1 p − det γ (0) δS [ γ (0) , a (0) ; Φ] δa (0) µ ( x ) = w Φ Φ( x ) ˆ ∇ µ (0) Φ( x ) . (96)This operator is also Weyl-covariant in the same sense as the stress tensor and is of weight d . Thus T µν and J µ have the properties of operators that couple to γ (0) µν and a (0) µ and appear in the Weyl anomaly inthe holographic WFG theory. In a holographic theory, we would not have the free field discussion givenhere, but we can still discuss sourcing these operators (in a given background).Earlier, we saw that the classical Weyl Ward identity involved a linear combination of the trace of thestress tensor and the divergence of the Weyl current. This is in fact easily established in general terms.Here we will use classical language, but the argument easily extends to the quantum case by making useof (92). As mentioned above, what we mean by Weyl being a background symmetry is that, classically, S [ γ (0) , a (0) ; B w Φ Φ] = S [ γ (0) / B , a (0) − d ln B ; Φ] . (97)By expanding both sides for small ln B and going on-shell, we find0 = Z d d x δSδa (0) µ ( x ) ∂ µ ln B ( x ) + Z d d x δSδγ (0) µν ( x ) (cid:16) − B ( x ) γ (0) µν ( x ) (cid:17) . (98)We recognize that this may be written as0 = Z d d x q − det γ (0) J µ ( x ) ∂ µ ln B ( x ) + Z d d x q − det γ (0) T µν ( x ) (cid:16) − ln B ( x ) γ (0) µν ( x ) (cid:17) (99)and, by integrating by parts, we have0 = − Z d d x q − det γ (0) (cid:16) ˆ ∇ (0) µ J µ ( x ) + T µν ( x ) γ (0) µν ( x ) (cid:17) ln B ( x ) . (100)This result serves to identify the relative normalization of π (0) µν and p (0) µ (see (73)) and their relation withthe currents defined here. Incidentally, the Weyl-covariant derivative appears in (100) precisely becausethe current J µ (with raised index) has Weyl weight d .We remark that typical discussions of related topics are rife with ‘improvements’ to operators such asthe stress tensor, including mixing with a so-called ‘virial current’ [46, 47]. The operators that we have18efined here have the advantage of transforming linearly, and in particular do not mix with each other,under Weyl transformations. Indeed we note the familiar result that, given (94), we have γ (0) µν T µνγ (0) ,a (0) ( x ) = 2 − d γ (0) µν ˆ ∇ µ (0) Φ( x ) ˆ ∇ ν (0) Φ( x ) (101)and thus given (96),ˆ ∇ (0) µ J µ ( x ) + T µν ( x ) γ (0) µν ( x ) = w Φ Φ ˆ ∇ Φ + (cid:16) w Φ + 2 − d (cid:17) γ (0) µν ˆ ∇ µ (0) Φ( x ) ˆ ∇ ν (0) Φ( x ) = 0 (102)So including the Weyl current automatically yields the correct (here classical) Weyl Ward identity on-shell.One can interpret the usual improvement of the stress tensor (to be traceless) as the absorption of theWeyl current into a redefinition of the stress tensor.The added value of our construction is the freedom to source the Weyl current and stress tensor inde-pendently, because we interpret the background Weyl connection a (0) as independent from the backgroundmetric γ (0) . In a CFT setting, general diffeomorphisms and Weyl transformations are not symmetries, butinstead a conformal diffeomorphism of the boundary metric components can be reabsorbed by a specificWeyl transformation, resulting in a symmetry; that is, the combined transformations give a relationshipbetween backgrounds of the same field theory. Here, we allow arbitrary Weyl transformations and diffeo-morphisms, which relate theories in different backgrounds. Given that in holography we should considerconformal classes of boundary metrics, Weyl transformations should in general be independent from bound-ary diffeomorphisms. One might then expect that this boundary Weyl current is physical, with associatednon-trivial charges. Here we have only touched on a few rudimentary aspects of field theories coupled toWeyl geometry, and will return to further elaboration elsewhere. In this work, we have discussed the consequences of bringing a Weyl connection into the formulation ofholography. In order to address this, we first intrinsically analyzed such connections and their associatedgeometrical tensors. The need for a Weyl connection arises in theories that, in addition to diffeomorphisms,admit a local rescaling of the metric by an arbitrary local function. The vanishing of the metricity requiredfor the familiar Levi-Civita connection is indeed not maintained under such rescalings, and the Weylconnection is defined as the unique torsionless connection with vanishing Weyl metricity, a Weyl-covariantstatement [8, 9]. Although richer than its Levi-Civita counterparts, the geometrical tensors built out ofthis connection turn out to be quite tractable.It has long been understood that holographic field theories possess a Weyl invariance, in the sense thatthey couple not to a metric, but to a conformal class of metrics [1, 2]. The introduction of a (background)Weyl connection in holographic field theories is a suitable reformulation in which local Weyl transfor-mations relate such theories in different backgrounds. In our account, the bulk gravitational theory isunmodified, but the gauge-fixing is relaxed (to what we called Weyl-Fefferman-Graham gauge) in such away that the Weyl diffeomorphisms act geometrically on tensors parametrizing the bulk metric. The Weyldiffeomorphisms correspond to rescaling the holographic coordinate by functions of the transverse coordi-nates while leaving the latter unchanged. While the FG expansion induces the LC connection associatedto the induced boundary metric [1], we have proven that the WFG expansion induces on the boundary aWeyl connection. This result indicates that the WFG gauge is the proper bulk parametrization that leavesthe bulk diffeomorphisms corresponding to the boundary Weyl transformations unfixed. This leads to theinterpretation of the Weyl connection in the boundary as a background field together with the boundarymetric; essentially, the pair ( γ (0) , a (0) ) replaces [ γ (0) ]. An interesting consequence of the WFG gauge isthat the boundary hypersurface is generally not part of a foliation, the distribution that is involved being19enerally non-integrable. We expect that the details of holographic renormalization require a slightly moresophisticated regulator than is usually employed, but the results of this paper do not rely on such details.The WFG gauge involves an expansion in powers of the holographic coordinate in which every coefficientis Weyl-covariant by construction. This result is a powerful reorganization of the holographic dictionary.The Weyl connection sources a Weyl current which explicitly appears in the subleading expansion of thebulk geometry. Subleading orders of the bulk Einstein equations unravel the boundary Weyl geometricaltensors and relationships between boundary expectation values of the sourced operators. In particular wefind the boundary Ward identity relating the trace of the energy-momentum tensor with the divergence ofthe Weyl current, and in the last section have shown that this is the expected result.We then scrutinized the implications of our setup for the Weyl anomaly. Not surprisingly, we foundthe latter to be given now in terms of Weyl-covariant geometrical objects, instead of the correspondingLevi-Civita objects. We expect that this outcome will have implications for the study and characterizationof the anomaly in higher even boundary dimensions. The presence of Weyl geometrical tensors allowedfor a cohomological description of the anomaly as a difference of Weyl-related bulk volumes, which offersa clear geometrical interpretation of the anomaly. As mentioned several times in the body of the paper,there are a number of clear followups, particularly on the field theory side, which present themselves, andwe look forward to exploring such implications in the future. Acknowledgements We would like to thank Costas Bachas, Guillaume Bossard, Laurent Freidel, Charles Marteau, Eric Mefford,Tassos Petkou, Marios Petropoulos, Rafael Sorkin and Antony Speranza for interesting discussions relatedto the results of this paper. This research was supported in part by the US Department of Energy undercontract DE-SC0015655, the ANR-16-CE31-0004 contract Black-dS-String and the Perimeter Institute forTheoretical Physics. Research at Perimeter Institute is supported by the Government of Canada throughthe Department of Innovation, Science, and Economic Development Canada and by the Province of Ontariothrough the Ministry of Research, Innovation and Science. A Details of Bulk Expansions We recapitulate here our geometrical setup both in the bulk and in the boundary, and compute the leadingorders of the expansion toward z = 0 of the main quantities involved. These are useful to evaluate Einsteinequations order by order, and hence solve for the various geometrical objects. Concretely, we work in thenon-coordinate basis d s = e ⊗ e + h µν d x µ ⊗ d x ν , e = L (cid:16) d zz − a µ d x µ (cid:17) . (A.1)The dual vectors are e = L − z∂ z , D µ = ∂ µ + za µ ∂ z , (A.2)and they form an orthonormal basis e ( e ) = 1 , e ( D µ ) = 0 , d x µ ( D ν ) = δ µν , d x µ ( e ) = 0 . (A.3)The vector commutators give[ e, D µ ] = Le ( a µ ) e = Lϕ µ e, [ D µ , D ν ] = L (cid:16) D µ a ν − D ν a µ (cid:17) e = Lf µν e, (A.4)20rom which we read C eµe = Lϕ µ , C µνe = Lf µν , C µνα = 0 . (A.5)Throughout this Appendix, we refer for brevity to generalized bulk indices as M = ( e, µ ) and thus vectors e M = ( e, D µ ) and metric components g MN = g ( e M , e N ), the most general non-coordinatized Levi-Civitaconnection is thenΓ PMN = g P Q (cid:16) e M ( g NQ ) + e N ( g QM ) − e Q ( g MN ) (cid:17) − g P Q (cid:16) C MQR g RN + C NM R g RQ − C QN R g RM (cid:17) . (A.6)The metric and its inverse are given in components by g µν = h µν , g eµ = 0 , g ee = 1 , g µν = h µν , g µe = 0 , g ee = 1 . (A.7)Then, calling θ = trρ with ρ µν = h µα e ( h αν ), the Christoffel symbols evaluate toΓ eee = 0 (A.8)Γ eeµ = C eµe = Lϕ µ (A.9)Γ eµe = 0 (A.10)Γ eµν = − e ( h µν ) + L f µν (A.11)Γ µee = h µν C νee = − Lh µν ϕ ν (A.12)Γ µeν = ρ µν + L f µν (A.13)Γ µνe = ρ µν + L f µν (A.14)Γ µµe = θ (A.15)Γ µαβ = h µν (cid:16) D α h βν + D β h αν − D ν h αβ (cid:17) ≡ γ µαβ . (A.16)These connections are explicitly reported in (61), (62), (63) and (64). We additionally define m ( k ) µν ≡ ( γ − γ ( k ) ) µν , n ( k ) µν ≡ ( γ − π ( k ) ) µν , (A.17)and the scalars X (1) = tr ( m (2) ) , (A.18) X (2) = tr ( m (4) ) − tr ( m ) + (cid:16) tr ( m (2) ) (cid:17) , (A.19) Y (1) = tr ( n (0) ) . (A.20)Starting from the metric (32) and the Weyl connection (54) expansions, we compute the inverse metric, thedeterminant and the various connection components appearing in (65). We expand the two series enough21o be able to capture the two leading orders. The result is: h µλ ( z ; x ) = z L (cid:20) γ − − z L m (2) γ − − z L ( m (4) − m ) γ − + ... (cid:21) µλ − z d +2 L d +2 h n (0) γ − + ... i µλ (A.21) p − det h ( z ; x ) = (cid:18) Lz (cid:19) d q − det γ (0) ( x ) (cid:20) z L X (1) + 12 z L X (2) + ... + 12 z d L d Y (1) + ... (cid:21) (A.22) ρ µν ( z ; x ) = L − (cid:20) − δ µν + z L m (2) µν + z L (2 m (4) − m ) µν + ... + d z d L d n (0) µν + ... (cid:21) (A.23) θ ( z ; x ) = L − (cid:20) − d + z L X (1) + z L X (2) − 14 ( X (1) ) ) + ... + d z d L d Y (1) + ... (cid:21) (A.24) ϕ µ ( z ; x ) = L − (cid:20) z L a (2) µ + ... + z d − L d − ( d − p (0) µ + ... (cid:21) (A.25) f µν ( z ; x ) = f (0) µν ( x ) + z L ( ˆ ∇ (0) µ a (2) ν − ˆ ∇ (0) ν a (2) µ ) + ... + z d − L d − ( ˆ ∇ (0) µ p (0) ν − ˆ ∇ (0) ν p (0) µ ) + ... (A.26)with f (0) µν = ∂ µ a (0) ν − ∂ ν a (0) µ . In the expression for f µν we used the boundary derivative introduced in (22),which is the Weyl derivative shifted with the Weyl weight of the object it acts upon. For instance, lookingat (56), a (2) µ and p (0) µ are Weyl-covariant with weights 2 and d − ∇ (0) µ a (2) ν = ∇ (0) µ a (2) ν + 2 a (0) µ a (2) ν , (A.27)ˆ ∇ (0) µ p (0) ν = ∇ (0) µ p (0) ν + ( d − a (0) µ p (0) ν , (A.28)with ∇ (0) the boundary Weyl connection (its connection coefficients are explicitly given in (60)).The expansion of the geometrical objects constructed from (67) is also reported γ λµν = γ (0) µν λ + z L h γ λξ (0) (cid:16) ˆ ∇ (0) ν γ (2) µξ + ˆ ∇ (0) µ γ (2) ξν − ˆ ∇ (0) ξ γ (2) µν (cid:17) − (cid:16) a (2) µ δ λν + a (2) ν δ λµ − a (2) ξ γ λξ (0) γ (0) µν (cid:17)i + ... − z d − L d − h p (0) µ δ λν + p (0) ν δ λµ − p (0) ρ γ λρ (0) γ (0) µν i + ... (A.29) Ric µν = Ric (0) µν + z L h ˆ ∇ (0) λ (cid:16) γ λξ (0) (cid:16) ˆ ∇ (0) ν γ (2) µξ + ˆ ∇ (0) µ γ (2) ξν − ˆ ∇ (0) ξ γ (2) µν (cid:17)(cid:17) +( d − 1) ˆ ∇ (0) ν a (2) µ − ˆ ∇ (0) µ a (2) ν + γ (0) µν ˆ ∇ (0) · a (2) − ˆ ∇ (0) ν ˆ ∇ (0) µ X (1) i + ... + z d − L d − h ( d − 1) ˆ ∇ (0) ν p (0) µ − ˆ ∇ (0) µ p (0) ν + γ (0) µν ˆ ∇ (0) · p (0) i + ... (A.30) R = z L R (0) + z L h γ λν (0) ˆ ∇ (0) λ ˆ ∇ (0) µ (cid:16) m (2) µν − tr ( m (2) ) δ µν (cid:17) + 2( d − 1) ˆ ∇ (0) · a (2) − tr ( m (2) γ − Ric (0) ) i + ... + 2( d − z d L d ˆ ∇ (0) · p (0) + ... (A.31) G µν = G (0) µν + z L h ˆ ∇ (0) λ (cid:16) γ λξ (0) (cid:16) ˆ ∇ (0) ν γ (2) ξµ + ˆ ∇ (0) µ γ (2) ξν − ˆ ∇ (0) ξ γ (2) µν (cid:17)(cid:17) + ( d − 1) ˆ ∇ (0) ν a (2) µ − ˆ ∇ (0) µ a (2) ν − ( d − γ (0) µν ˆ ∇ (0) · a (2) − ˆ ∇ (0) ν ˆ ∇ (0) µ X (1) − γ (2) µν R (0) − γ (0) µν ˆ ∇ (0) λ ˆ ∇ (0) φ (cid:16) ( γ − γ (2) γ − ) φλ − X (1) γ φλ (0) (cid:17) + γ (0) µν tr ( m (2) γ − Ric (0) ) i + ... + z d − L d − h ( d − 1) ˆ ∇ (0) ν p (0) µ − ˆ ∇ (0) µ p (0) ν − ( d − 2) ˆ ∇ (0) · p (0) γ (0) µν i + ... (A.32)These quantities appear explicitly in the Einstein tensor. We then compute the bulk Ricci tensor: Ric MN = R P MP N = e P (Γ PNM ) − e N (Γ PP M ) + Γ QNM Γ PP Q − Γ QP M Γ PNQ − C P N Q Γ PQM , (A.33)22nd so Ric ee = − L ∇ µ ϕ µ − L ϕ − e ( θ ) − tr ( ρρ ) − L tr ( f f ) (A.34) Ric eµ = ∇ α (cid:16) ρ αµ + L f αµ (cid:17) − D µ θ + L ϕ α f αµ (A.35) Ric µe = ∇ α (cid:16) ρ αµ + L f αµ (cid:17) − D µ θ + L ϕ α f αµ (A.36) Ric µν = Ric µν − L ∇ ν ϕ µ − ( e + θ ) (cid:16) ρ µν + L f µν (cid:17) − L ϕ µ ϕ ν + 2 ρ αµ ρ αν + L f να f αµ . (A.37)Notice that Ric eµ = Ric µe . The trace of the Ricci tensor gives the scalar curvature R = g MN (cid:16) e P (Γ PNM ) − e N (Γ PP M ) + Γ QNM Γ PP Q − Γ QP M Γ PNQ − C P N Q Γ PQM (cid:17) . (A.38)It evaluates to R = − e ( θ ) + L tr ( f f ) − tr ( ρρ ) − Lh µν ∇ µ ϕ ν + R − θ − L ϕ µ ϕ ν h µν . (A.39)Therefore the various components of the Einstein tensor read G ee = − tr ( ρρ ) − L tr ( f f ) − R + θ (A.40) G eµ = ∇ α (cid:16) ρ αµ + L f αµ (cid:17) − D µ θ + L ϕ α f αµ (A.41) G µe = ∇ α (cid:16) ρ αµ + L f αµ (cid:17) − D µ θ + L ϕ α f αµ (A.42) G µν = G µν − L ∇ ν ϕ µ − ( e + θ ) (cid:16) ρ µν + L f µν (cid:17) − L ϕ µ ϕ ν + 2 ρ αµ ρ αν + L f να f αµ (A.43)+ h µν (cid:16) e ( θ ) − L tr ( f f ) + tr ( ρρ ) + L ∇ α ϕ α + θ + L ϕ (cid:17) . (A.44)Finally, vacuum Einstein equations are given by G MN + Λ g MN = 0 . (A.45)They become 0 = − tr ( ρρ ) − L tr ( f f ) − R + θ + Λ (A.46)0 = ∇ α (cid:16) ρ αµ + L f αµ (cid:17) − D µ θ + L ϕ α f αµ (A.47)0 = ∇ α (cid:16) ρ αµ + L f αµ (cid:17) − D µ θ + L ϕ α f αµ (A.48)0 = G µν − L ∇ ν ϕ µ − ( e + θ ) (cid:16) ρ µν + L f µν (cid:17) − L ϕ µ ϕ ν + 2 ρ αµ ρ αν + L f να f αµ (A.49)+ h µν (cid:16) e ( θ ) − L tr ( f f ) + tr ( ρρ ) + L ∇ α ϕ α + θ + L ϕ + Λ (cid:17) . 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