A Shape Dynamical Approach to Holographic Renormalization

We provide a bottom-up argument to derive some known results from holographic renormalization using the classical bulk-bulk equivalence of General Relativity and Shape Dynamics, a theory with spatial conformal (Weyl) invariance. The purpose of this paper is twofold: 1) to advertise the simple classical mechanism: trading of gauge symmetries, that underlies the bulk-bulk equivalence of General Relativity and Shape Dynamics to readers interested in dualities of the type of AdS/CFT; and 2) to highlight that this mechanism can be used to explain certain results of holographic renormalization, providing an alternative to the AdS/CFT conjecture for these cases. To make contact with usual the semiclassical AdS/CFT correspondence, we provide, in addition, a heuristic argument that makes it plausible why the classical equivalence between General Relativity and Shape Dynamics turns into a duality between radial evolution in gravity and the renormalization group flow of a conformal field theory. We believe that Shape Dynamics provides a new perspective on gravity by giving conformal structure a primary role within the theory. It is hoped that this work provides the first steps towards understanding what this new perspective may be able to teach us about holographic dualities.


Introduction
A key insight of holography is that the behaviour of a conformal field theory under renormalization is governed by a diffeomorphism-invariant gravitational theory in one higher spacetime dimension. In this correspondence, the extra dimension plays the role of the renormalization scale. This gives the renormalization group a geometric setting and suggests deep connections between the dynamics of spacetime and the renormalization of quantum fields.
The holographic formulation of the renormalization group was elucidated in a beautiful series of papers [1,2,3,4,5,6,7,8] and underlies much of the very fertile applications of holography and the AdS/CFT correspondence [9,10,11] to condensed matter and fluid systems. In this paper, we offer new insights into the connection between gravitation and renormalization by giving a new explanation for why that connection exists. We give a novel derivation of a correspondence between a conformal field theory (CFT) defined on a d-dimensional manifold Σ with a metric h ab with finite volume V and a solution to a spacetime diffeomorphism invariant theory of a metric g µν defined on a manifold M of one higher dimension, such that Σ = ∂M and h ab is the pullback of g µν onto Σ. Through this correspondence, the behaviour of the conformal field theory under renormalization is coded into the dynamics of a spacetime of one higher dimension.
The new derivation we offer makes use of a recent reformulation of General Relativity in which spatial conformal invariance plays a central role. This formulation, called Shape Dynamics, is the result of trading the many fingered time aspect of spacetime diffeomorphism invariance -which allows one to arbitrarily redefine what is meant by surfaces of constant time -with local Weyl transformations on a fixed class of spatial surfaces. In a word, Shape Dynamics trades relativity of time for relativity of scale. Each is a gauge invariance that can be defined on the phase space of general relativity depending on one free function. They can be traded for each other because the constraint, S, that generates local scale transformations can be interpreted as a gauge fixing of General Relativity's Hamiltonian constraint H -or vice versa! That constraint S, as we show below, is the condition of constant mean extrinsic curvature (CMC) slicing, but it can also be read as a constraint generating local Weyl transformations that leave the total volume of space invariant. 1 When General Relativity is gauge-fixed to CMC gauge both H and S are imposed. Shape Dynamics arises by interpreting S as the generator of a gauge symmetry which is gauge fixed by H. The physics is the same -even if we choose a different gauge fixing for S.
The key idea is that Shape Dynamics provides insight into the relationship between gravity and CFT in one less dimension, because it is itself a conformal theory in one less dimension. This makes the existence of a correspondence between a gravity theory on M and a CFT on its boundary Σ = ∂M completely transparent because the former theory already enjoys invariance under local scale transformations on Σ. Furthermore, the possible anomalies for the Shape Dynamics conformal symmetry are the usual ones for the intrinsic geometry of Σ. This guarantees that the correspondence between gravity and a CFT in one less dimension provided through Shape Dynamics extends into the quantum domain.
In a previous paper, this correspondence was worked out in detail for the case of positive cosmological constant [12] in the context of Shape Dynamics. 2 To claim an explanation for an aspect of the AdS/CFT correspondence, it is necessary to redo the calculations for negative cosmological constant. This introduces no new difficulties and is carried out in Section 3 below.
Of course, the correspondence we demonstrate is already known, as it is a consequence of the stronger AdS/CFT conjecture. What is new is the generality of the explanation for it. In principle, an explanation has already been conjectured (i.e., the Maldacena's conjecture that there is an exact isomorphism between certain specific supersymmetric conformal field theories and String Theories on certain asymptotically AdS space times). There are however three reasons to seek a more direct explanation for the connection between renormalization of CFT's and gravitation in one higher dimension: 1. Many of the existing results are weaker than the AdS/CFT conjecture requires, as they refer only to a connection between vacuum expectation values of certain operators in the CFT and the dynamics of classical General Relativity one dimension up. It would be interesting to have an explanation for these weaker results that does not require the existence of an isomorphism between the Hilbert spaces and observable algebras of two independently defined quantum theories -especially as we lack non-perturbative definitions of at least one side of the correspondence. This would also remove the potential for circularity which arises when the Maldacena conjecture is taken to be a statement of how String Theory is to be defined by the correspondence.
2. Generality. The existing results suggest the connection between renormalization and gravitation in one higher dimension is very general, whereas the Maldacena conjecture rests on properties of specific theories in specific dimensions and requires structures that do not seem to play a necessary role in the correspondence -at least at leading order -such as supersymmetry and compact extra dimensions. We will show that there exists a weaker correspondence that is indeed more general.
3. The Maldacena conjecture remains a conjecture so it would be interesting to demonstrate a weaker correspondence that, nonetheless, can explain some of the results in the literature. It should be emphasized that only those results in the literature that are not explained by the correspondence we demonstrate here can be considered evidence for the stronger Maldacena conjecture.
Stated informally, we show the following (see Sec. 2 below for the precise statement): Consider a CFT on a d-dimensional manifold Σ which has the topology of a sphere and a metric h ab (not necessarily spherical). The volume V = Σ √ h is fixed to a finite value, breaking conformal invariance. Consequently there is a trace anomaly proportional to a power of V −1 . More precisely, the renormalized 0|T ab |0 satisfies an equation 3 where c is the central charge, c r are numerical coefficients, r 0 and r are dimensions, which depend on d and C ab (r) [h, ∂h, . . . ; x) are symmetric tensors that are functionals of the metric on Σ and its derivatives. Formulas for these coefficients for several theories are known and are reviewed in [14]. Note that the operatorT ab transforms as a tensor density on Σ as a consequence of its definitionT where W [h ab ], the effective action, is a diffeo-invariant functional of the metric on Σ. The correspondence to General Relativity, known from [1, 2, 3] -and for which we will provide a transparent explanation -has the following form. We consider General Relativity, expressed in Hamiltonian form defined, as we recall, on a d + 1 spacetime manifold M whose boundary, ∂M = Σ, is an initial Cauchy surface. The metric induced on Σ is h ab while π ab is the tensor density which represents its conjugate momenta. We consider a solution to Einstein's equations (with possible matter fields and with appropriate signature to be specified below) determined by initial data (h ab , π ab ) on Σ, specified by S, a solution to the Hamilton-Jacobi equation on M. We have We express the solutions to the Hamilton-Jacobi equations in a large volume expansion. As we show in Section 2 below, the boundary momenta can be expanded in terms of the same functionals, C We note that the coefficients of the expansionc r will in general depend on the choice of matter fields. The correspondence then goes as follows. On Σ choose the canonical momenta for the gravitational field π ab to be equal to the vacuum expectation value of the energy momentum tensor, Then, with specific choices of matter fields, the coefficientsc r that govern the evolution of π ab under the Einstein equations exactly match their counterparts c r that govern the renormalization of 0|T ab |0 under changes in the renormalization scale V . Additionally, the cosmological constant must be taken to be related to the central charge. Furthermore, only a part of the local conformal symmetry on Σ acts as a gauge invariance in Shape Dynamics: the subgroup that stretches distances locally but leaves V , the total volume of Σ, invariant. Thus, in Shape Dynamics, V is "a physical observable" , that is a gauge invariant dynamical quantity, and it can be put in correspondence with the renormalization scale of the CFT.
This explanation directly answers several questions as to why the renormalization of a conformal field theory should be governed by a gravitational theory in one higher dimension.
1. It explains why there is a dual theory invariant under diffeomorphisms of Σ.
2. It explains why that theory is many-fingered time invariant, or invariant under spacetime diffeomorphisms in one higher dimension. This otherwise mysterious property is due to the isomorphism between Shape Dynamics, a theory with spatial conformal invariance, and General Relativity, which instead enjoys spacetime diffeo invariance.
However, our method so far does not yield explanations for specific correspondences found in the literature, such as those involving supersymmetry. The method by which we establish the relationship between conformal field theory and a gravitational theory is new and rests on the observation that both 0|T ab |0 and π ab satisfy five properties that determine their volume expansions in Sec. 2 and 3 identically up to the specification of the constants c r andc r . Four of these properties are taken from a set of axioms Wald proposed for 0|T ab |0 . Wald had also a fifth axiom, but this is not satisfied by standard examples so we replace it by a new axiom which specifies that the QFT is a CFT.
Both coefficients c r andc r depend on the matter contents of the two theories, but differently. We recover the known fact that in d = 4 the large N limit of N = 4 super Yang-Mills theory matches General Relativity in 5 dimensions. We argue also for a conjecture that given the coefficients c r determined by a CFT, a modified but still spacetime diffeomorphism invariant gravity theory can be constructed whose volume expansion has the same coefficients.
The main argument and results of the paper are stated in the next Section. They depend on results about the volume expansion in Shape Dynamics for negative Λ, which are computed in Section 3. Afterwards, in Section 4, we comment on our results.

The main argument 2.1 Conformal field theories and the modified Wald's axioms
In 1977, Wald [15] proposed five axioms for the renormalized 0|T ab |0 and proved the uniqueness of the coefficients in (1) for theories that satisfy them. However, the fifth axiom turns out not to be satisfied by known examples of renormalized quantum field theories. The remaining four axioms are satisfied and turn out to be a useful starting point for our argument, even though they are insufficient to prove uniqueness.
The first three axioms are the same as the axioms Wald motivated by coupling the expectation value of the energy momentum tensor to General Relativity. Wald's fourth axiom implements causality, which requires Lorentzian signature. The consequence of the causality axiom, that Wald uses in his uniqueness proof is that the expectation values ofT ab depends only on the germ of h ab (x) for fixed h. This implies, in particular, that if h ab and h ′ ab are two metrics which coincide in an open neighborhood of Our axiom 4 is slightly stronger than Wald's original axiom, however it applies also to Euclidean signature, which is necessary for our application.
To the first four axioms of Wald, we add a fifth axiom, which we distinguish with a prime.
where ρ is a test function and · is the average over Σ (not be confused with the expectation value). 5 The new fifth axiom says that the effective action W is invariant under VPCT's. To see this, first note that the effective action depends upon the spatial geometry through h ab so that the infinitesimal action of the spatial conformal transformations is given by: for some smearing function ρ. To find the infinitesimal action of the VPCT's, it suffices to subtract the mean of the above expression (see, for example, [16] for a demonstration of this or consult (13)) But, requiring this to be zero is precisely (7) since This implies that the scale invariance is broken only at the global level, which can be verified explicitly in known conformal field theories. At this stage, we can provide a simple argument to motivate this axiom. If we assume that the AdS/CFT correspondence holds, then the asymptotically AdS solutions of classical General Relativity are dual to a certain (known) CFT. Thus, by the equivalence of Shape Dynamics with General Relativity, Shape Dynamics must also be dual to a CFT. Of course, since our aim is to provide an independent explanation of a weak form of the AdS/CFT correspondence, this argument should only be taken as motivation. The deeper significance of this axiom will become clear shortly.
Axioms 1-5' can be satisfied if W defined by (2) has an expansion 5 We define the average over Σ of a scalar density ϕ of weight w as ϕ = 1 where the F (r) [h ab , ∂h ab ] are diffeomorphism and VPCT invariant functionals of the metric. Axioms 2-5' are satisfied by inspection of the form of (11), while the first axiom is satisfied trivially as only the vacuum expectation value is constrained by that form. The possible functionals, F (r) [h ab , ∂h ab ] that can appear at each order are constrained by their scaling behaviour as well as invariance under spatial diffeomorphisms and VPCT. The list of possibilities is tabulated in [17]. Important examples are where C abcd is the Weyl tensor, χ is the Euler invariant, and ∇ 2 = h ab ∇ a ∇ b . The coefficients c r then depend on the field content of the CFT and are known (see [14] for d = 2 and 4). Keeping these axioms in mind we now turn to Shape Dynamics, where we will show that they are also satisfied once an identification is made between bulk fields and boundary operators.

Shape Dynamics and Wald's axioms
Shape Dynamics is a gravitational theory whose solutions agree with a large set of solutions of the Einstein equations: those that have constant mean curvature slices. It is defined on a d + 1-dimensional manifold, M, with a preferred foliation by d-dimensional slices, Σ, labeled by an evolution parameter. 6 This evolution parameter can label either temporal or spatial "evolution", depending on the relative sign of the kinetic and potential terms in the Hamiltonian. We will be primarily interested in making contact with the best understood context of gauge-gravity duality, namely the case of Euclidean spacetime with AdS asymptotics. Thus, in this section we will use a radial Hamiltonian, with a negative cosmological constant, evolving Euclidean signature hypersurfaces Σ in a radial direction away from a (homogeneous) AdS boundary. In Section 3, we give a detailed derivation of the results used below in a more general context.
In the Hamiltonian language, the degrees of freedom of Shape Dynamics consist of a d-dimensional metric h ab on Σ and its conjugate momenta, π ab . The equations of Shape Dynamics are invariant under diffeomorphisms of Σ and reparameterizations of the evolution parameter, but not under general diffeomorphisms of M. Instead, the many-fingered time invariance of General Relativity is replaced by a local Weyl invariance of the fields on Σ, 6 SD is not refoliation-invariant, but it keeps a one parameter reparametrization invariance, which is nothing else than global reparametrization invariance. Therefore the time parameter t used at this stage has no physical meaning and can be replaced by any invertible function of t without changing the equations. Notice that Shape Dynamics swaps all but the global part of refoliation invariance with all but the global part of Weyl invariance. This fact has motivated [18,19], where the symmetry trading has been completed, swapping reparametrization invariance for global scale invariance, and obtaining a physical notion of time.
which are, however, restricted to be volume-preserving in that V , the volume of each slice, is gauge invariant.
The local constraints of the theory are where π = h ab π ab and £ ξ h ab = ∇ a ξ b + ∇ b ξ a is the Lie derivative of the metric. These constraints generate d-dimensional diffeomorphisms and VPCTs. The dynamics of the theory is governed by a single, global reparametrization constraint H SD defined by whereΩ and Ω(x) is the solution of and where σ a b = (δ a c h bd − 1 3 δ a b h cd )π cd is the traceless part of π ab . The negative cosmological constant has been written for convenience in terms of the AdS radius ℓ such that Explicit calculations show that the initial value problem, the equations of motion and, in particular, the action of the time-reparametrization constraints of Shape Dynamics, in a gauge determined by Ω(x) = 0 are identical with the initial value problem and equations of motion of General Relativity in a gauge determined by the condition that π/ √ h is a spatial constant (CMC gauge).
We thus see that Shape Dynamics and General Relativity are dynamically equivalent. The correspondence with conformal field theory is found by expressing the theory in terms of a solution to the Hamilton-Jacobi equations, S[h ab ], which is a function of the three metric and possibly matter fields on the boundary. We will consider the case where the topology of Σ is S d and M has a boundary ∂M = Σ| V →∞ = Σ.
We consider a solution to the constraint equations specified by data (h ab , π ab ) on the boundary Σ, given by the Hamilton-Jacobi functional S[h ab ]. The boundary momenta are We then make the observation that, for solutions of the equations of Shape Dynamics, these π ab satisfy the five modified Wald's axioms on the d-dimensional boundary. That is, we replace the expectation values 0|T ab |0 in Walds's axioms by and check that they are satisfied.
• Axiom 2: is satisfied by inspection of the large volume expansion (see Section 3.3), since lim V →∞ π ab = 0.
• Axiom 4: Dependence. That π ab [h; x) depends only on the germ of h ab (x) for fixed h follows by inspection for the leading orders of the large volume expansion. In this paper we will only be concerned with these.
• Axiom 5': Invariance under volume-preserving conformal transformations is now one of the constraints that defines the theory.
The π ab then satisfy the five modified axioms of Wald.
where again the functionals F (r) [h ab , ∂h ab ] are the same functionals in (11), invariant under spatial diffeomorphisms and VPCT's. In Appendix A, it is shown that this is indeed a valid ansatz and, furthermore, that it will continue to hold for the quantum effective action. The volume expansion for Shape Dynamics was developed for positive Λ in 3 + 1 dimensions in [12], and is derived for general Λ and d and for both Riemannian and Euclidean signature in Sec. 3. The calculations described in Appendix B show that the coefficients,c r depend on the choice of matter fields. It follows that both 0|T ab |0 and π ab satisfy the same axioms and have an expansion of the same form in terms of the same functionals, C We then find our main result: Given a CFT on a d-dimensional manifold Σ with metric h ab defined above, whose 0|T ab |0 satisfy the modified Wald axioms, and a choice of matter fields for Shape Dynamics such that the coefficients c r andc r match, there is a solution to the dynamical equations of Shape Dynamics on a d + 1-dimensional manifold M such that Σ = ∂M, gotten by taking boundary data and relating the cosmological constant Λ = − d(d−1) Notice that our main result is not that we can extract Cauchy data for Shape Dynamics from a CFT. The result is rather that the large volume expansion of Shape Dynamics is consistent with a set of reasonable axioms for the expectation value of the energy momentum tensor of a CFT. It is then very interesting that in d = 4 there is such a match relating the best understood CFT which is the large N limit of N = 4 super-Yang-Mills theory to pure Shape Dynamics, i.e. no matter fields.
The term at zeroth order in volume in the expansion of W for this conformal field theory is equal to [1] S d=4 This matches precisely to the same term in the expansion of S in Shape Dynamics with no matter fields, as computed below.

Shape Dynamics is equivalent to General Relativity
The main result of [16] is that the bulk dynamics of classical General Relativity in d+1 dimensions is locally 7 indistinguishable from a classical conformal field theory in d+1 dimensions. The simplest way of showing the physical equivalence between General Relativity is by constructing a linking gauge theory, which can be partially gauge-fixed to give General Relativity, or using a different partial gauge-condition to give Shape Dynamics. This linking theory is defined on an extension of the ADM phase space, which is obtained by adding a conformal factor and its canonically conjugate momentum density to the ADM phase space.
We will not present the entire construction here; the part of the construction that is relevant for this paper will be given in section 3.1. The physical equivalence of Shape Dynamics and General Relativity can be seen very explicitly by working with the linking theory: Suppose we could work out the observable algebra and the equations of motion in the linking theory. Then we would have a very simple way of obtaining the observable algebra and equations of motion of the two gauge-fixed theories: One simply needs to impose the two gauge-fixing conditions and work out the two phase space reductions. This procedure leads to a one to one correspondence between the observables of the two theories and ensures that the equations of motion for the observables coincide. This is why Shape Dynamics and General Relativity make indistinguishable physical predictions.

Correspondence between General Relativity and conformal field theory
Using the physical equivalence of General Relativity and Shape Dynamics, we can conclude Given a CFT on a d-dimensional manifold, Σ with metric h ab defined above, whose 0|T ab |0 satisfy the five modified Wald axioms, and a choice of matter fields for Shape Dynamics such that the coefficients c r matchc r , there is a solution to the dynamical equations of General Relativity on a d + 1-dimensional manifold M such that Σ = ∂M, gotten by taking boundary data Furthermore, given the match mentioned above Given the large N limit of N = 4 super=Yang-Mills theory on a 4 dimensional manifold, Σ with metric h ab defined above, whose 0|T ab |0 satisfy the five modified Wald axioms, there is a solution to the dynamical equations of pure General Relativity on a 5-dimensional manifold M such that Σ = ∂M, gotten by taking boundary data (h ab , π cd ) = (h ab , 0|T cd |0 ) and relating the cosmological constant Λ = − d(d−1)

Derivation of volume expansion in Shape Dynamics
In this technical section, we provide the detailed derivations of the key expressions used in Section 2. The results presented here generalize the results of [12] to arbitrary dimension, signature, and cosmological constant and allow us to work with either a radial or a temporal evolution parameter. This is necessary for the context provided in the last section.
In [12] the volume expansion for solutions to the Hamilton-Jacobi equation for Shape Dynamics was presented for the case of positive cosmological constant and Lorentzian signature. In this Section we repeat the calculation for arbitrary sign of Λ and both Euclidean and Lorentzian signatures. We refer the readers to [16] for motivation and details as our goal here is to obtain results that can be compared with existing calculations of the trace anomaly and its dual.

The Shape Dynamics Hamiltonian
The ADM Hamiltonian constraint, in d dimensions and with arbitrary signature and cosmological constant, is where the inverse DeWitt supermetric, G abcd , is given by s is the signature of the direction in M orthogonal to Σ, and the general cosmological constant is given by where k = 0, ±1 specifies the character of Λ. In the context of Section 2, we have s = 1 (for a radial Hamiltonian) and k = −1 for a negative cosmological constant. The recipe for constructing Shape Dynamics (see [16]) is to apply a volume-preserving conformal transformation to the ADM Hamiltonian constraint (29), and equate the result to a spatial constant, H SD . Solving this equation with respect to the conformal factor of the VPCT, and plugging the result back in, gives the Shape Dynamics Hamiltonian constraint H SD .
A VPCT transforms h ab and π ab as whereΩ for any n, which will be chosen for convenience. The traceless part of of π ab transforms homogeneously with the opposite weight of the metric the Ricci scalar transforms as there is a preferred choice for n, which simplifies the transformation law of R we will use this choice of n from now on. Inserting (37) and the transformations (32) into (29) gives (we use here the local expression of the constraint) which reduces weakly to The Shape Dynamics Hamiltonian is given by simultaneously solving

Volume expansion
It is convenient to isolate the d-dimensional volume and its conjugate momentum: from the other degrees of freedom. For this, we define the fixed-volume metric and its conjugate momentum: where V 0 = Σ √h is some reference volume. The Poisson algebra of the new variables is This explicitly isolates the V -dependence of the theory. In terms of the new variables, equations (40) become where barred quantities and means · 0 are calculated usingh ab . We now fix a gauge for the conformal factor (Yamabe gauge) such that and indicate this choice of conformal section using a tilde, e.g.h ab . We will solve equations (44) and (45) by inserting the expansion ansatz and solving order by order in V −2/d . Using this expansion, (45) is trivially solved by We can now outline the procedure for finding the solution order-by-order: • For n = 0, we have trivially • For n = 1, we can take the mean and use the fact that the solution to the Yamabe problem is unique [20]. This leads to • For n = 2, we use the expansion and get Taking the mean and using integration by parts to drop boundary terms (Σ is compact without boundary) we get Inserting this into (52) gives This equation admits the solution For negative Yamabe class, this solution is not unique ifR happens to be in the (discrete) spectrum of∇ 2 . The issue of non-uniqueness is discussed in [21].
• For n = 3 . . . (d − 1), the same reasoning will apply. Using the result ω (n−2) = 0, we can now use the expansionΩ which leads to Taking the mean leads to: which, again, has the solution ω (n−1) = 0 .
• For n = d, the solution can still easily be worked using the previous expansions for n = d and including theσ a bσ b a term. The resulting equation is taking the mean, we get ω (d−1) can then be solved by inverting the following equation where ∆ is the d-dimensional conformal Laplacian Thus, all higher order terms will be non-local because they will involve inverting the conformal Laplacian.
Collecting the first three non-zero terms, we get

Hamilton-Jacobi equation
Using the substitutions where S = S(h ab , α ab ) is the HJ functional, depending on the metric h ab and on d(d + 1)/2 integration constants α ab , we can solve the Hamilton-Jacobi equation associated to Eq. (65) order by order in V using the ansatz for odd d or for even d (because the previous ansatz is not valid with even d).
We can get a recursion relation for the solution by taking the asymptotic boundary condition lim Using this we get: • For n = 0, the solution is trivial • For n = 1, the solution is equally straightforward. The result for d = 2 is In d = 2 this term gives the conformal anomaly. It is found to be • For n = 2, we use and find In d = 4 this we get the anomaly. It is (77) • For n > 2, If we ignore higher order terms in the V -expansion of H SD then we get a cute recursion relation for S (n) In general though, there will be contributions from higher order terms that depend on the inverse of ∆ but these are not important for d < 5.

Comments and conclusions
We make several comments on these results • The reason why a conformal field theory on a d-dimensional manifold with a fixed metric is related to a gravitational theory in one higher dimension now becomes transparent. The correspondence maps the constraint generating d-dimensional spatial diffeomorphisms of the gravitational theory to the conservation law of the energy-momentum tensor of the CFT. The remaining constraint, the Hamiltonian constraint generating many fingered time in d + 1 dimensions is shown, through the correspondence with Shape Dynamics, to be equivalent to the generator of volume-preserving conformal transformations in that theory, plus a single global time-reparametrization constraint (14). The latter is mapped by the correspondence using Wald's axioms to the trace anomaly of the energy-momentum tensor of the CFT.
Thus, we can now understand why the geometry of the renormalization group of a CFT is given by General Relativity, possibly coupled to matter fields, a diffeomorphism invariant theory in one higher dimension.
• We can conjecture a more general correspondence. Given a CFT we get an expansion of the form (1) with coefficients, c r . We can define a theory with the gauge symmetries of Shape Dynamics by the corresponding expansion, Eq. (4), where we equate the values of the coefficients,c r = c r . This gives us some gravitational theory with a possibly different global Hamiltonian constraint replacing (14). We can reconstruct H SD by its volume expansion. This gives what may be called a generalized Shape Dynamics theory which by construction is matched to the original CFT.
We can then conjecture that by gauge symmetry trading of the VPCT for many fingered time, this can be matched to some spacetime diffeomorphism invariant theory in d + 1 dimensions. This may not be General Relativity, but by construction will have spacetime diffeomorphism invariance. If this matching can be achieved there is a general correspondence between any CFT and some gravitational theory in one higher dimension.
• We have studied only the 0|T cd |0 but it is possible that the renormalization group flow of other conserved currents can be explained by expanding Wald's axioms to them, and then by coupling Shape Dynamics to suitable matter fields.
• The correspondence we show is more general than the conjectured AdS/CFT correspondence as stated originally by Maldacena, in that it is not restricted to supersymmetric theories and String Theory and the properties of 10 dimensional supersymmetry algebra play no role in establishing the general correspondence at this level. However supersymmetry certainly plays a role in specific examples of correspondences, such as those involving supersymmetric Yang-Mills theory.
• The correspondence we demonstrate here is however weaker than the original Maldacena conjecture in that we claim only the classical bulk gravitational theory plays a role. Rather then a conjectured equivalence between the Hilbert spaces and observables algebras of a bulk and boundary theory, we demonstrate only an equivalence between the expectation values of operators in the CFT and solutions of the classical field equations of the bulk theory.
• However, some of the evidence used to support the stronger Maldacena conjecture can now be explained by the weaker and more general correspondence we give here. Hence, only results that are not explicable by our weaker correspondence can count as evidence for the stronger Maldacena conjecture.
• In particular, Shape Dynamics does not, so far, directly explain the correspondence between pure General Relativity and N = 4 super-Yang-Mills theory in the N → ∞ limit except as a coincidence of the coefficients in the two expansions. This coincidence is well represented by the work of Skenderis et al, and also from a purely gravitational perspective [3,7].
• We can also conjecture that the correspondence we have demonstrated here extends to a stronger correspondence between a quantization of Shape Dynamics and the quantum conformal field theory. The evidence we possess for an extension of the correspondence into the quantum regime is the apparent absence of anomalies (for odd dimensions) of our spatial conformal transformations (see appendix A). In spite of their classical correspondence, the quantization of Shape Dynamics is unlikely to coincide with a quantization of General Relativity due to the very different structure of their constraints.
A APPENDIX: The spatial Weyl (conformal) anomaly for Shape Dynamics In this section we show that the spatial Weyl anomaly candidates for Shape Dynamics can depend neither on the momenta nor on the time derivatives of the fields. The results obtained here will appear in fuller form in [22]. For the proof we use the results of Barnich [23], showing that in the Hamiltonian case, one can restrict the search for anomalies to the function space of polynomials of the canonical variables and their spatial derivatives (i.e. no time derivatives are necessary). After this is established, we can resort to cohomological methods (which don't require an explicit knowledge of the action functional) to look for anomaly candidates.
Although cohomological methods are able to pinpoint which are the candidates for a given anomaly, they cannot yield their normalization, for which one must resort to explicit calculations using the effective action. We find that the requirements on a possible anomaly (in the cohomological language, the ghost number one element of the Stora-Zumino chain of descent) cannot be met if there there are elements containing spatial derivatives and powers of momenta. Furthermore, we find that the only possible contribution to the anomaly that involves the momenta is where σ is a ghost number one scalar. But this vanishes on-shell and thus there are no anomalous terms containing the momenta. If one wishes, one could now continue the calculation to find the remaining terms, which are the usual geometric terms for the Weyl anomaly, which now only depend on the intrinsic geometry of the each of the g ab .

A.1 The calculation
These considerations allow us to consider both our fundamental fields and the symmetry groups acting on them to belong to each given hypersurface Σ. Thus in the present case the Hamiltonian treatment of the d+1 theory will effectively reduce to the Lagrangian treatment of a d-dimensional Euclidean theory with two extra fields ξ a , p ab , with determined tensor indices and conformal weights. The symmetric 2-tensor p ab = g ac g bd π cd / √ g has conformal weight n(1 − d/2) (where n is the conformal weight of g ab ).
The assumption here is to take our space of cochains as being formed by integrated local polynomials of the fields g ab , g ab , p ab and covariant derivatives ∇ a , with the above restrictions. So for instance we do not allow the appearance of parity breaking terms such as the epsilon tensor.
Thus, for the first term in the Stora-Zumino chain of descent we must match the the following tensor indices and conformal weights: Multiplying the second line by 2 n and subtracting the first line we obtain d N p + 2N ∇ = d. So either N p = 1 and N ∇ = 0 or N p = 0 and N ∇ = d. The important part of this calculation is that we can see that the momenta appear only when there is no covariant derivative, and therefore no derivatives of the metric involved. It can thus appear only as the term π, which vanishes on-shell and is thus not a candidate for the anomaly. The other anomaly terms that can appear consist only of metric-dependent terms, and these have been classified in all dimensions.

B APPENDIX: Scalar Field (d ≤ 4)
The goal of this appendix is to illustrate how, in the presence of matter fields, the coefficients c r in the volume expansion (21) of Hamilton's principle function, S[h ab ], for Shape Dynamics will depend upon the matter fields. We will demonstrate this using the simple example of a single real scalar field, ϕ.
The matter Hamiltonian for this scalar field is: Adding this to the Hamiltonian constraint (29) and performing the phase space extension and canonical transformation described in Section 3.1, we obtain the Shape Dynamics Hamiltonian coupled to a scalar field In [24], it was shown that a consistent coupling of matter to Shape Dynamics requires that the matter fields be invariant under VPCT. They can, however, transform non-trivially under homogeneous conformal transformation, and the weight of this transformation represents the anomalous scaling, ∆, of the matter fields. For a real scalar field, we find that the dilatation operator has the following action in the extended theory The volume-dependence can then be extracted using the canonical transformation (ϕ, π ϕ ) → (φ,π ϕ ): The volume expansion of the Shape Dynamics Hamiltonian can now be performed. In general, consistency of the equations will limit the possibilities for U(ϕ) for a particular anomalous scaling, ∆. Because the volume expansion will, in general, depend on ϕ and π ϕ , the volume expansion of the Hamilton-Jacobi equation will be modified. This will, in turn, affect the expansion coefficientsc r .
For a simple illustration of this, consider the case where ∆ = d. In this case, ϕ has no conformal scaling. The zeroth-order equation is which is only consistent and non-trivial (i.e., ϕ = const) if is the scalar field is free so that U(ϕ) = 0. We can now pick a gauge wherē Then, the first order equations lead to Subsequent orders will, therefore, be unchanged from the results of Section 3.2 withR →R ϕ until n = d. At order n = d, we obtain: taking the mean, we get Using the substitution the volume expansion of the Hamilton-Jacobi equation becomes It is clear that we can still use the ansatz S (0) = const of homogeneous asymptotics to seed a recursion relation for the general volume expansion of S. However, because of the ϕ-dependence ofR ϕ and the δS δϕ term, the higher order expansion coefficientsc r will depend upon ϕ as we intended to show. The explicit solution of S for different matter fields is currently being investigated.