A practical approach to the Hamilton-Jacobi formulation of holographic renormalization

Abstract

Abstract

We revisit the subject of holographic renormalization for asymptotically AdS spacetimes. For many applications of holography, one has to handle the divergences associated with the on-shell gravitational action. The brute force approach uses the Fefferman- Graham (FG) expansion near the AdS boundary to identify the divergences, but subsequent reversal of the expansion is needed to construct the infinite counterterms. While in principle straightforward, the method is cumbersome and application/reversal of FG is formally unsatisfactory. Various authors have proposed an alternative method based on the Hamilton-Jacobi equation. However, this approach may appear to be abstract, difficult to implement, and in some cases limited in applicability. In this paper, we clarify the Hamilton-Jacobi formulation of holographic renormalization and present a simple algorithm for its implementation to extract cleanly the infinite counterterms. While the derivation of the method relies on the Hamiltonian formulation of general relativity, the actual application of our algorithm does not. The work applies to any D-dimensional holographic dual with asymptotic AdS boundary, Euclidean or Lorentzian, and arbitrary slicing. We illustrate the method in several examples, including the FGPW model, a holographic model of 3d ABJM theory, and cases with marginal scalars such as a dilaton-axion system.

Published for SISSA by Springer

Received: April 22, 2016

Accepted: May 22, 2016

Published: June 8, 2016

Henriette Elvang and Marios Hadjiantonis

Department of Physics and Michigan Center for Theoretical Physics, University of Michigan, 450 Church Str., Ann Arbor MI 48109, U.S.A.

E-mail: mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected]

Abstract: We revisit the subject of holographic renormalization for asymptotically AdS spacetimes. For many applications of holography, one has to handle the divergences associated with the on-shell gravitational action. The brute force approach uses the Fe erman-Graham (FG) expansion near the AdS boundary to identify the divergences, but subsequent reversal of the expansion is needed to construct the innite counterterms. While in principle straightforward, the method is cumbersome and application/reversal of FG is formally unsatisfactory. Various authors have proposed an alternative method based on the Hamilton-Jacobi equation. However, this approach may appear to be abstract, di -cult to implement, and in some cases limited in applicability. In this paper, we clarify the Hamilton-Jacobi formulation of holographic renormalization and present a simple algorithm for its implementation to extract cleanly the innite counterterms. While the derivation of the method relies on the Hamiltonian formulation of general relativity, the actual application of our algorithm does not. The work applies to any D-dimensional holographic dual with asymptotic AdS boundary, Euclidean or Lorentzian, and arbitrary slicing. We illustrate the method in several examples, including the FGPW model, a holographic model of 3d ABJM theory, and cases with marginal scalars such as a dilaton-axion system.

Keywords: AdS-CFT Correspondence, Gauge-gravity correspondence

ArXiv ePrint: 1603.04485

Open Access, c

The Authors.

Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP06(2016)046

Web End =10.1007/JHEP06(2016)046

A practical approach to the Hamilton-Jacobi formulation of holographic renormalization

JHEP06(2016)046

Contents

1 Introduction 1

2 Hamiltonian approach to holographic renormalization 42.1 Hamiltonian formalism of gravity 42.2 Hamilton-Jacobi formulation 52.3 Algorithm to determine the divergent part of the on-shell action 7

3 Pure gravity 9

4 Renormalization for the ABJM model 12

5 Renormalization for the FGPW model 14

6 Renormalization of a dilaton-axion model 18

7 Discussion 21

A Some useful formulas 22

B Six derivative counterterms for pure gravity 22

C One-point functions 23

1 Introduction

In many applications of gauge-gravity duality, there is a need to regulate divergences that appear near the boundary of the bulk theory; these are simply associated with UV divergences in the dual quantum eld theory. The divergences appear, for example, in calculations of conformal anomalies, correlation functions, and the free energy. The prescription for regulating divergences is to include suitable local counterterms. The resulting process of holographic renormalization is an old subject: it was discussed in the early days of AdS/CFT [1] and implemented in the classic calculations of conformal anomalies [2], the trace of the stress-tensor [3], and since then in countless other examples.

We focus on bulk spacetimes that are asymptotically AdS or Euclidean AdS. This includes duals of conformal theories (CFTs) as well as holographic renormalization group ows with a UV CFT. For a given gravity dual, the local counterterms are universal and one can calculate them once and for all in any given gravitational model. We distinguish between innite counterterms and nite counterterms. The former are unambiguous and can be determined using the bulk equations of motion. The nite counterterms, however,

1

JHEP06(2016)046

can typically only be xed using further constraints, such as supersymmetry. In this paper, we are concerned only with the innite counterterms.

There is a standard brute force procedure for determining the innite counterterms [2 5]. One expands the metric and elds near the AdS boundary using the Fe erman-Graham (FG) expansion [6]. Solving the equations of motion relates various coe cients in the FG expansion, but leaves unxed the coe cients that correspond to the source and vev rates for each eld. Using a suitable cuto , the on-shell action is evaluated near the AdS boundary by plugging in the FG expansion, subject to the equations of motion. This identies the divergences, however, they will be expressed in terms of the free coe cients in the FG expansion. This is not su cient, as local counterterms must be expressed directly in terms of the elds on the cuto surface. So starting with the most divergent terms, one works systematically backwards to convert each divergence to a local eld expression, thus basically reversing the FG expansion. This process identies the eld polynomials that are responsible for the divergences in the on-shell action. The counterterm action is then taken to be exactly minus those eld expressions; this ensures that the renormalized action Sbulk +Sct is nite. (This still leaves the possibility of ambiguities from nite counterterms;

we will discuss this briey in the Discussion section.)

While straightforward for many simple models with just one or two scalar elds, the brute force approach outlined above becomes increasingly tedious for models with multiple elds. Moreover, it is fundamentally unsatisfying that one rst abandons the eld expressions in favor of Fe erman-Graham only to reverse back to elds after identifying the innite terms. For this reason, another approach, based on the Hamiltonian formalism for gravity and the Hamilton-Jacobi equation, has been proposed for holographic renormalization.

Early in the studies of holographic renormalization group ows, de Boer, Verlinde, and Verlinde [7] proposed to use the Hamilton-Jacobi equation to derive rst-order equations for the supergravity model and they related it to the Callan-Symanzik equation. (See also [8, 9] and the lectures [10].) The specic application of the Hamilton-Jacobi equation to determine innite counterterms was studied by Kalkkinen, Martelli, and Mueck in [11, 12] and subsequently by Papadimitriou and Skenderis in [13] (see also [1416]).

One limitation of the method as formulated in [13] is that the dilatation operator is used to organize the calculation. This requires that the elds are eigenfunctions of the dilatation operator, but that makes it more challenging to handle scalars dual to operators with scaling dimension = d/2, because of their leading log-fallo .1 This is not an exotic case, but a very common one; for example, in a d = 4 eld theory, a scalar mass term is a relevant operator of dimension = 2. Another challenge is that, as presented in [13], the Hamilton-Jacobi method looks rather di cult to carry out in practice.

The goal of this paper is to straighten out and simplify the Hamilton-Jacobi approach for holographic renormalization. We will show that the application of the Hamilton-Jacobi equation

Son-shell

r + H = 0 (1.1)

1One can work around this, see for example [14]. The issue is also addressed in [16].

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(with the radial coordinate r playing the role of the usual time-coordinate), can be implemented via an algorithm that signicantly simplies the process of computing the innite counterterms. To avoid the issue of the dilatation operator and have an approach that applies more generally, we organize the calculation in terms of a derivative expansion (or inverse metric expansion), as also suggested in for example [7, 12, 16].

We will be working with bulk actions of the form

S =

1

22

ZM dd+1xg R[g] gGIJ I J V ( ) , (1.2)

where we allow for a general metric GIJ = GIJ( ) on the scalar manifold. We consider domain wall solutions with arbitrary slicing and assume that the asymptotic UV structure of the metric is AdS (or Euclidean AdS). For such a system, we formulate the Hamilton-Jacobi problem for the on-shell action Son-shell; (1.1) is basically a partial di erential equation for

Son-shell and once derived, one no longer has to think about the Hamiltonian formulation of

general relativity. Instead, one systematically solves the Hamilton-Jacobi di erential equation for Son-shell by writing a suitable Ansatz for its divergent terms and then solving for

the coe cients in this Ansatz. The key point here is that scalars dual to relevant operators in the eld theory go to zero at the boundary. Therefore there can only be limited powers of each eld in the innite counterterms, and that makes the Ansatz nite.

Our method departs from previous approaches as follows.2 We consider Son-shell as

the action on the cut-o boundary; this breaks the general di eomorphism invariance in the radial direction and therefore we must take seriously the explicit dependence on the radial coordinate in Son-shell. Thus, the r partial-derivative in (1.1) plays a central role in our method. In fact, the coe cients in our Ansatz will be allowed to have explicit r-dependence, and the Hamilton-Jacobi equation then yields di erential equations for these coe cients that we can solve unambiguously in the near boundary limit.

We illustrate the use of the method in several contexts. To start out, we reproduce the purely gravitational counterterms [4] in d-dimensions. To show how the method works for a case with d odd, we reproduce the innite counterterms of the d = 3 ABJM dual model of [18]. We then turn to the example of the d = 4 FGPW model [19] whose two scalars have = 2 and = 3.

In the presence of a marginal scalar, more care must be taken. A marginal scalar generically goes to a nite value at the boundary and therefore the associated counterterms do not enjoy the same suppression as the scalars dual to relevant operators. We handle this by allowing the coe cients of our Ansatz for Son-shell to be functions of the marginal scalar.

We have applied this method successfully to calculate the counterterms for a ten-scalar model dual to (a limit of) N = 1 theory on S4 [20]; this indeed served as a motivation

for us to revisit the subject of holographic renormalization. However, for the purpose of presentation here, we restrict ourselves to simply show how our method reproduce the innite counterterms for the dilaton-axion system in [16].

The paper is organized as follows. In section 2, we present the Hamilton-Jacobi equation for the bulk and describe our algorithm for determining the innite counterterms.

2However, see [17] for a similar approach in dS space.

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Section 3 implements the method for pure gravity in d dimensions. The examples of the ABJM model and FGPW can be found in sections 4 and 5; these give very concrete illustrations of how we implement the algorithm. The more advanced case of marginal scalars is treated in section 6. The three appendices contain various technical details. Appendix A is a short list of useful identities for the metric variations of gravitational curvatures. Appendix B gives details of the calculation of the gravitational six-derivative terms needed for counterterms in d = 6. Finally, appendix C o ers explicit calculation of the one-point functions in FGPW to illustrate that the one-point functions determined from the renormalized action with our innite counterterms are indeed all nite.

2 Hamiltonian approach to holographic renormalization

We start with a brief description of the essential parts of the Hamiltonian formulation needed for holographic renormalization. We then formulate the problem of determining the on-shell action in terms of the Hamilton-Jacobi equation and we present our algorithm for calculating the divergent part of the on-shell action.

2.1 Hamiltonian formalism of gravity

We consider a general form of the bulk gravitational action:

S =

1

22

JHEP06(2016)046

ZM dd+1x g R[g] gGIJ I J V ( )

1 2

ZM ddx K . (2.1)

The last term in (2.1) is the Gibbons-Hawking boundary term which ensures that the variational problem is well-dened. In this term, ij is the induced metric on the boundary and K is its extrinsic curvature.

We choose a gauge for the bulk metric g such that the line element takes the form

ds2 = dr2 + ij(r, x)dxidxj , (2.2)

where latin indices i, j, . . . are in the range i, j = 1, 2, . . . , d and will denote boundary coordinates.

This allows us to decompose the Ricci scalar in the action to get

S =

1

22

ZM ddx dr

R[]+K2KijKijGIJ I J ijGIJi Ij J V ( ) , (2.3)

where the extrinsic curvatures are

Kij = 1

2ik

kj and K = 12ij

ij . (2.4)

The dots denote derivatives with respect to r. The boundary Gibbons-Hawking term does not appear in the expression (2.3), since it has been canceled by boundary terms that occur from partial integration of second derivative terms in the expansion of R[g].

4

In the Hamiltonian formulation of holographic renormalization, the radial coordinate r plays the role of the time coordinate. Therefore, the conjugate momenta to the elds are given by

ij = S

ij =

1

22 Kij Kij

and I = S

I =

12 GIJ

J , (2.5)

and the Hamiltonian is

H =ZM ddx ij ij + I I L

= 1

22

ZM ddx R[]K2+KijKij +GIJpIpJ ijGIJi Ij J V ( )

(2.6)

,

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where, for simplicity, we have introduced pI

2 I.

2.2 Hamilton-Jacobi formulation

The Hamilton-Jacobi formulation is well-known in classical mechanics [21]. With the radial coordinate r playing the role of time, the Hamilton-Jacobi equation takes the form

H + Son-shell

r = 0 . (2.7)

Just as in classical mechanics, it is key to emphasize that in the Hamilton-Jacobian formalism, the Hamiltonian is a functional of canonical momenta dened by

ij = Son-shell

ij and pI =

2

I =

2

Son-shell

I , (2.8)

as opposed to the canonical denitions (2.5). When the momenta are dened via equation (2.5) with the extrinsic curvature given by (2.4), the Hamiltonian constraint of Einsteins equation is simply H = 0. If this were used with the Hamilton-Jacobi equation (2.7), it would imply that the action has no explicit r-dependence; this is of course true for the di eomorphism-invariant gravitational bulk action whose metric equations-of-motion imply the Hamiltonian constraint. However, it is not true for the on-shell action, which is an action on the cut-o boundary. It has explicit r-dependence, as we shall see, and to determine it via the Hamilton-Jacobi equation we must use the denitions (2.8). With (2.8), the Hamilton-Jacobi equation (2.7) should be thought of as a rst-order partial di erential equation for Son-shell with respect to the elds, the metric, and r.

A practical approach is to use an Ansatz for the on-shell action: below we will be more explicit about how we choose an appropriate Ansatz, but for now we will develop the general formalism further. Let us write the Ansatz as

Son-shell = 1

2

ZM ddx U(, , r) . (2.9)

The function U is a function of the induced (inverse) metric ij on the boundary and the scalar elds I, and it has also explicit dependence on r. The cuto surface M becomes the boundary of the spacetime when 0.

5

Using the above Ansatz, the Hamilton-Jacobi equation takes the form

R[] + KijKij K2 + GIJpIpJ ijGIJi Ij J V ( ) + 2

U

r = 0 . (2.10)

We emphasize that this equation is to be understood as an integral equation, i.e. it holds up to total derivatives and we can manipulate it using partial integration in the boundary coordinates.

As discussed above, the conjugate momenta in (2.10) will be given by derivatives of U. For the scalar eld conjugates, this straightforwardly gives

pI = 2

Son-shell

I pI =

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U I . (2.11)

The conjugate momentum of the metric enters (2.10) via the extrinsic curvatures, since Kij = 22

ij

1d1 ijklkl

, as follows from (2.5). Now in the context of the Hamilton-Jacobi formalism, the extrinsic curvatures Kij in (2.10) must then be expressed in terms of ij as given by (2.8). This gives

Kij = 2ik

U kj

1d 1

U 2mnU mn

ij , (2.12)

where we have used ijjk = ki = (ij)jk = ij(jk) to express Kij in terms of derivatives with respect to the inverse metric rather than the metric; this will be useful later.

It is convenient to dene

Yij = U

ij and Y = ijYij . (2.13)

One then nds from (2.12) that the dependence on extrinsic curvatures in the Hamilton-Jacobi equation (2.10) is given in terms of U as

K KijKij K2 = 4YijY ij

1d 1

(U 2Y )2 U2 . (2.14)

To summarize, our strategy for computing the on-shell action Son-shell is to use the

Ansatz (2.9) and solve the Hamilton-Jacobi equation

R[] + K + GIJpIpJ ijGIJi Ij J V ( ) + 2

U

r = 0 . (2.15)

with conjugate momenta given by (2.11) and K dened in (2.14). We remind the reader

that equation (2.15) has to hold only as an integral equation, so we are free to manipulate it using partial integration. While this was derived using the Hamiltonian formalism of gravity, we no longer need to think of the problem that way. Rather, we now have differential equation (2.15) for the on-shell action Son-shell. Next, we explain how to solve it

systematically.

6

2.3 Algorithm to determine the divergent part of the on-shell action

Let us next address how we propose to use the Hamilton-Jacobi formulation to determine the divergent part of the on-shell action and thereby the counterterms needed for a nite result. We outline here the general approach, however the method is much better illustrated by concrete examples; these follow in the next sections.

We assume that asymptotically the bulk metric approaches AdS space: in terms of the choice of coordinates (2.2), ds2 = dr2 + ij(r, x)dxidxj, this means that

ij e2r/L (0)ij + . . . as r , (2.16)

where L is the AdS radius. The boundary metric (0)ij can be Lorentzian or Euclidean, it can be at or curved. For example, recent applications of holography considered the dual eld theory on d-dimensional compact Euclidean spaces, such as spheres. In the following, (0)ij will be general.

The asymptotic behavior (2.16), gives edr(0). We are focusing only on the

divergent parts of the on-shell action, so we need terms in U only up to orders edr (possibly including also terms polynomial in r). Since the inverse metric ij scales as e2r, we can ignore any terms with more than

d 2

inverse metrics. Any (boundary) derivatives that appear in terms in U must necessarily be contracted pairwise by inverse metrics ij, so we

do not consider terms with more than d-derivatives. All in all, this makes it natural to organize the Ansatz for U in a derivative expansion:

U = U(0) + U(2) + . . . + U(2

d

2

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) , (2.17)

where the subscript represents the number of derivatives in each term. Curvature terms such as the boundary Ricci scalar, Ricci tensor, and Riemann tensor are each order 2 (i.e. they have two derivatives). Previous work, for example [7] and [16], have also organized the on-shell action as a derivative expansion.

For the 0th order in the derivative expansion, we have Y(0)ij =

U(0)

ij = 0, so (2.14)

simply gives

U2(0) . (2.18)

Thus at 0th order, the Hamilton-Jacobian equation (2.10) becomes

V ( ) = GIJ U(0)

I

U(0)

J

K(0) =

dd 1

dd 1

U2(0) + 2

U(0)

r . (2.19)

Without the last r-derivative term, we see that U(0) is essentially like a (fake) superpotential for the scalar potential V ; this was also noted [7] (see also [16, 22]). In general, it is not easy to solve for a superpotential for a given V ; however, we will not need to since our focus is on the generic asymptotically divergent terms only. As noted in the discussion below (2.8) the presence of the explicit r-derivative term in the Hamilton-Jacobi equation, and hence in (2.19), is crucial this point does not seem to have been appreciated in previous discussions of the method.

7

Let us for later convenience also record the results for K at two- and four-derivative

order:

K(2) =

K(4) = 4Y(2)ijY ij(2)

where Y(k)ij = U(k)ij .

We propose the following algorithm to determine the innite terms in the on-shell action:

Step 1: ansatz for U(2n). For each U(2n), we write a systematic Ansatz that includes all potentially divergent terms of this order with undetermined coe cients,3 for example

U(0) = A0 + A1 + A32 + . . . and U(2) = B0R + B1R + B2 + . . . (2.21)

where the coe cients Ai and Bi can have explicit dependence on r. The Hamilton-Jacobi equations will therefore give us di erential equations of these coe cients which we solve asymptotically, keeping only terms that give divergent contributions to the on-shell action.

Recall that the asymptotic behavior of a scalar with bulk mass m2I is I

I(0)e(d I)r/L, where m2IL2 = I( I d). The two solutions for I correspond to

the source and vev-rate fallo s. When a scalar approaches zero at the boundary, as is the case in many applications, we can immediately read o how many powers of the scalar can possibly appear in U(2n); the number of possible terms is nite and limited by the fact that we are only interested in the divergent terms.4 For example, if is a scalar with dimension = 3 in d = 4, then er, and we have to include powers up to 4 in U(0)

and can appear in U(2). (Note: such terms with edr fallo will be nite unless the r-dependence in the coe cient makes it divergent.) On the other hand, if in (2.21) is a = 2 scalar in d = 4, there can at most be quadratic powers of in U(0) and the term is not divergent, so it is not included in the Ansatz for U(2).

One can impose symmetries of the theory in order to further simplify the Ansatz for U(2n). If, for example, the bulk action has a symmetry , we can drop any terms

odd under this symmetry in the Ansatz.

Step 2: conjugate momenta. Next, using the leading asymptotic behaviors of the elds, we determine the leading asymptotics of the conjugate momenta. Using this together with pI = U

I xes some of the coe cients in U(0) quite easily.

Step 3: solving the Hamilton-Jacobi equation. We plug the Ansatz for U(2n) into the Hamilton-Jacobi equation and we solve it order by order by demanding that the coefcients of the di erent eld monomials vanish independently. When necessary, use partial integration to eliminate potentially non-independent terms that appeared by varying U. We start with U(0), then use those results to determine U(2), then U(4) etc.

3Terms are considered equivalent if related by partial integration.

4We will also discuss cases with a marginal scalar m2I = 0, for which there is no suppression near the boundary and generically the scalar goes to a non-zero constant. For such cases, we allow the coe cients

Ai in our Ansatz to be functions of the marginal scalar. An example is presented in section 6.

8

2d 1

U(0)

U(2) 2Y(2) 2U(0)U(2) , (2.20)

1d 1

U(2) 2Y(2)

22d 1U(0)

U(4) 2Y(4) U2(2) 2U(0)U(4) ,

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Step 4: counterterm action. Once the divergent terms in Son-shell have been deter

mined, the counterterm action is simply

Sct = Son-shell

div . (2.22)

This is added to the bulk action to get the regularized action Sreg = Sbulk +SGH +Sct from

which correlation functions can be computed and by construction are guaranteed to be nite. In many cases, counterterm actions are presented in term of the Fe erman-Graham radial coordinate related to r via = e2r/L, so that the line element is

ds2 = L2 d242 + ij dxidxj . (2.23)

We determine the divergent terms in the on-shell action using the r-coordinate, but convert to -coordinates for the nal presentation of our counterterm actions. In terms of the -coordinate, the cuto surface M, introduced in (2.9), is then located at = .

In the following sections, we demonstrate the procedure explicitly in a set of representative explicit examples. We start with pure gravity in d-dimensions with d = 2, 3, 4, 5, 6, then move on to a d = 3 ABJM dual model and the d = 4 two-scalar model known as FGPW. Finally, we illustrate how our method works with marginal scalars (dilaton + axion in d = 4).

3 Pure gravity

The simplest model one can consider is pure AdS gravity with no matter content in D = d + 1 dimensions. Counterterms obtained by renormalizing this model will be present in every other model and it is therefore useful to deal with them once and for all. The action we consider is given by (2.1) with no scalar elds and constant scalar potential

V =

U

r = 0 , (3.2) with K given by (2.14). Let us now apply the algorithmic procedure described in the

previous section in order to determine the necessary counterterms for this class of theories.

Step 1: since there are no scalars, the general Ansatz for each order of the expansion of U is

U(0) = A(r) , U(2) = B(r)R , U(4) = C1(r)RijRij + C2(r)R2 , (3.3)

where the four-derivative terms are only needed for d 4.5 We are not including terms

like R since it is a total derivative and it will not contribute in the on-shell action. For d 6, we need

U(6) = D1R3 +D2RRijRij +D3R jiR kjR ik +D4RijRklRikjl +D5R R+D6Rij Rij . (3.4)

5In U(4), one could also have included a term with the square of the Riemann tensor. However, it is not hard to see that its coe cient will be set to zero in the HJ equation.

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JHEP06(2016)046

d(d 1)

L2 . (3.1) The Hamilton-Jacobi equation (2.15) simplies to

R[] + K +

d(d 1)

L2 + 2

This is not a complete list of independent six-derivative terms, but it turns out to be a su cient list.

It is important that all the coe cients in the above expressions for U depend on the radial coordinate r, as this will capture the explicit r-dependence of the on-shell action.

Step 2: this step is irrelevant for the pure gravity case since there are no matter elds.

Step 3: we now solve Hamilton-Jacobi equation (3.2) order by order to determine the unknown coe cients A, B, C1,2 and Di.

At zero-derivatives, (3.2) with K(0) given by (2.18) gives

2 A

dd 1

A2 + d(d 1)

L2 = 0 , (3.5)

where the dot denotes di erentiation with respect to r. For large r, the solution to the di erential equation is

A(r) =

d 1

L + O edr/L

JHEP06(2016)046

. (3.6)

The subleading terms in the large-r expansion of A give only nite contribution to the on-shell action and we can drop it to simply have

U(0) =

d 1

L . (3.7)

This captures the leading divergence associated with the cosmological constant.At two-derivative order, the HJ equation (3.2) with (2.20) gives

R

2d 1

U(0) U(2) 2Y(2)

2U(0)U(2) + 2

U(2)

r = 0 . (3.8)

ij = BRij, so Y(2) = BR.

With the solution for U(0) in (3.7), we obtain the following di erential equation for B:

2 B + 2d 2

L B + 1 = 0 . (3.9)

The di erential equation for B has solution

B(r) =

(r2 + O(1) for d = 2

The inverse-metric variation of U(2) simply gives Y(2)ij =

U(2)

for d > 2 (3.10)

In both cases, the subleading terms are not important since they give nite contributions to the on-shell action. The result is therefore

U(2) =

(r2R for d = 2

L2(d2) + O e(d2)r/L

(3.11)

The linear r behavior in the d = 2 case is our rst illustration of the explicit r-dependence in the on-shell action and the importance of keeping the Son-shellr -term in the Hamilton-Jacobi

equation.

10

L2(d2) R for d > 2

For the four-derivative terms, we calculate the inverse-metric variation of U4 using the formulas in appendix A. In particular, we nd Y(4) = 2C1RijRij + 2C2R2 (up to total derivatives that can be dropped). Using this together with the results for Y(2) above, we can calculate K(4) given in (2.20). At 4th order, the HJ equation (3.2) is simply K(4)+2U(4)r = 0

and collecting terms gives

"2 C1 + 2(d 4)

L C1 +

Ld 2

#

2 C2 + 2(d 4)

L C2 dL24(d 1)(d 2)2

R2 = 0 .

Demanding the coe cients of the RijRij and R2 terms to vanish independently results in two di erential equation for the coe cients C1 and C2, which have solutions

C1 =

(L2r8 + O(1) for d = 4

L3 2(d2)

2(d4) + O e(d4)r/L

2 RijRij +

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for d > 4 (3.12)

C2 =

(

L2r

24 + O(1) for d = 4

dL3 8(d1)(d2)

2(d4) + O e(d4)r/L

for d > 4 (3.13)

Again, the subleading terms can be dropped because they give only nite contributions to the on-shell action. Thus, the result for U(4) is

U(4) =

L2r8 RijRij 13R2

for d = 4

RijRij d4(d1) R2 for d > 4 (3.14)

Step 4: we now have all information needed to write the counterterm action.

Sct =

1 2

L3 2(d2)

2(d4)

ZM ddx U = 1 2

ZM ddx hU(0) + U(2) + . . . + U(2d 2

)

i

. (3.15)

Summarizing the above results, the purely gravitational counterterms are

d = 2: Sct = 1 2

ZM ddx 1L log

L4 R

,

d = 3: Sct = 1 2

ZM ddx 2L +L2 R

,

d = 4: Sct = 1

2

ZM ddx 3L +L4 R log

L3 16

RijRij 1 3R2 ,

d = 5: Sct = 1

2

ZM ddx 4L +L6 R +L3 18

RijRij 5 16R2 ,

(3.16)

d = 6: Sct = 1

2

ZM ddx 5L +L8 R +L3 64

RijRij 3 10R2

log L5 256

Rij Rij 120R R

+2RijRklRikjl + 1

5RRijRij

3 100R3

,

11

where we have used = e2r/L. The results for the six-derivative terms displayed for d = 6 are derived in appendix B.

These purely gravitational counterterms reproduce results well-known in the literature, see for example [4], but it is relevant to present them here in the context of our approach to holographic renormalization. In particular, they will appear in the following examples.

4 Renormalization for the ABJM model

ABJM theory [23] is the N = 6 superconformal Chern-Simons theory in d = 3 dimensions

with gauge group U(N) U(N) and Chern-Simons levels k and k. Its holographic dual

is M-theory on AdS4 S7/

Zk. In the limit of large tHooft coupling ( = N/k), M-theory reduces to eleven dimensional supergravity on AdS4 S7/

Zk. The recent paper [18] by Freedman and Pufu explores the gauge-gravity dual description of F -maximization for ABJM theory on a 3-sphere using a 4-dimensional holographic dual. We will use the model of [18] as a very simple example to illustrate our approach to holographic renormalization.

The ABJM holographic model [18] is described by the Euclidean bulk action

Sbulk =

1

22

ZM d3x drg R[g] Lm

JHEP06(2016)046

, (4.1)

where 2 = 8G4 and the matter Lagrangian is

Lm = 2

3

Xa=141 zaza . (4.2)

In the Euclidean theory, the scalars za and za are independent complex elds, not related by complex conjugation. However, since only products of za and za appear in this Lagrangian, it is useful to dene za

Xa=1zaza(1 zaza)2+ V (z, z) , V (z, z) = 1 L2

6 3

12 (a + ia) , za

12 (a ia), where a and a are elds

that can take complex values.Under this, the matter Lagrangian becomes

Lm =

3

Xa=1aa + aa

1 12(a)2 12(a)2

2 +V , V =

1 L2

63

Xa=1

41 12(a)2 12(a)2 . (4.3)

Expanding the potential for small elds, we nd

V = 1

L2

6 2(aa + aa) (aa + aa)2 + . . .

, (4.4)

so the six elds a and a all have mass 2/L2. By our general discussion, this means

that their asymptotic fallo is generically er/L.

For simplicity, let us start out with a model with just one pair of the elds and ; since the ABJM dual has the three pairs appear the same way and they do not mix, it is easy to generalize the result back to that case. Thus setting the elds with a = 2, 3 to zero, we will consider the model described by the potential

V = 1

L2

2

4

1 122 122

. (4.5)

12

In the notation (2.1), we have scalars I = (, ) and the metric on the scalar target space is the GIJ = 1 122 122

2 IJ with I, J = 1, 2. The Hamilton-Jacobi equation (2.15) for this model is then

R + K

1 122 122 2ij(ij + ij)

+ 1 122 122 2 p2 + p2

1L2 2

4

1 122 122 !

+ 2U

r = 0 , (4.6)

where K is given by equation (2.14) and the conjugate momenta p and p are the

and derivatives of the on-shell action (2.11). We now proceed to determine the innite counterterms for this model.

Step 1: since we are working in d = 3 dimensions we need to include in our Ansatz only terms with up to two derivatives:

U = U(0) + U(2) . (4.7)

Terms with four or more derivatives give nite contributions to the on-shell action.

Keeping only potentially divergent contributions means that for U(0) we only need to consider terms up to cubic order in the scalar elds. However, we get strong constraints on the Ansatz from the symmetries of the model: it is invariant under the transformations , , and . With these symmetries imposed, the most general Ansatz

at zero-derivative order isU(0) =

2L + A(r)(2 + 2) . (4.8)

The constant term is xed from the purely gravitational calculation of section 3. At two-derivative order, the only potentially divergent term that preserves the symmetries of the theory is purely gravitational and it was calculated in section 3:

U(2) =

L

JHEP06(2016)046

2 R . (4.9)

We can skip Step 2 because the model is so simple.

Step 3: we are now able to solve equation (4.6). Keeping only zero-derivative terms and using that K(0) = 32U2(0) from (2.14) we nd that

32U2(0) +

1

1

22

1

22 2

(p2(0) + p2(0)) V (, ) + 2

U(0)

r = 0 , (4.10)

where,

= 2A . (4.11)

Putting everything together and collecting terms that are proportional to (2 + 2) gives

the following di erential equation for A(r):

A + 2A2 + 3

LA +

1L2 = 0 . (4.12)

p(0) =

U(0)

= 2A , p(0) =

U(0)

13

This has solution

A =

1

2L + O

er/L

. (4.13)

Since A was the only unknown coe cient in the Ansatz for U, this concludes the calculation of the innite contributions in the on-shell action. Specically, we have found that

U(0) =

1 L

2 + 122 +122 = 1L (2 + zz) . (4.14)

Step 4: the counterterm action for the ABJM model is obtained by generalizing our result to the three avors of za and za elds:

Sct = 1 2

ZM d3x"1 L

2 +3

Xa=1 zaza

JHEP06(2016)046

+ L

2 R

#

. (4.15)

This result is in perfect agreement with the counterterm action given in equations (6.4)(6.5) in [18]. For the applications in [18] one further needs to use supersymmetry to determine the nite counterterms; we do not discuss this here.

5 Renormalization for the FGPW model

The FGPW model [19] is the holographic dual of the single-mass limit of N = 1 gauge

theory in at space. This non-conformal eld theory is obtained from N = 4 SYM theory

by softly breaking the supersymmetry to N = 1 as follows. In N = 1 language, N = 4 SYM

consists of a vector multiplet and three chiral multiplets. The eld theory dual to FGPW is obtained by giving a mass to one of the chiral multiplets. In the UV, the conformal theory of N = 4 SYM is recovered, while in the infrared, the theory ows to a Leigh-Strassler

xed point. The holographic dual FGPW model captures the RG ow of this theory via a at-space sliced domain wall solution which approaches asymptotic AdS5 in the UV and another AdS5 in the IR. The ratio of the AdS radii in the UV and IR translates to the ratio of UV and IR central charges a in the eld theory. More generally, the authors of [19, 24] derived the rst version of a holographic version of the c-theorem.

The holographic FGPW model is described by a D = 4 + 1-dimensional bulk action

S =

1

22

ZM d4x drg R[g] Lm

, (5.1)

with matter Lagrangian given by6

Lm = + + V (, ) = 2 + 2 + ijij + ijij + V (, ) . (5.4)

6In the paper [19], the scalar potential V is given in terms of a superpotential W as

VFGPW = 1 L2

1 2

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

@W @1

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

2

2

+ 1

2

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

@W @3

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

43 W 2[parenrightbigg] , (5.2)

with

[bracketleftBig] p6 . (5.3)

Here, we have conformed to our normalization conventions by rescaling the scalars 1 = /2 and 3 = /2, and taken the potential to be V = 4VFGPW.

W = 1

42

cosh(21)(6 2) (36 + 2)[bracketrightBig]

and = e3/

14

The scalars and are dimension = 3 and = 2 elds dual to the fermion and scalar mass deformations of N = 4 SYM. They approach zero near the UV boundary as

0 er/L and (0r +

~

0) e2r/L , (5.5)

as r . For the purpose of holographic renormalization, we only need to keep the terms

in the potential that can give divergent terms in this limit, so we expand the potential in small elds to nd

V (, ) = 1

L2 12 42 32 + c4 + . . .

. (5.6)

The masses of the scalars, m2 = 3/L2 and m2 = 4/L2, are directly related to the

scaling dimensions = 3 and = 2 via m2IL2 = I( I 4).

The actual FGPW model has c = 1 in (5.6), but here we keep the coe cients general. This will serve to illustrate how the counterterms carry information that is specically dependent on coe cients in the scalar potential; i.e. one should in general expect model-dependent terms in the counterterm action.

The HJ equation (2.15) for the FGPW model takes the form

R[] + K + p2 + p2 ijij ijij V (, ) + 2

U

r = 0 . (5.7)

JHEP06(2016)046

with K dened in (2.14) and momenta

p = U

p =

U

. (5.8)

Since we are working in d = 4 dimensions we need to keep terms with up to four derivatives, so we write

U = U(0) + U(2) + U(4) . (5.9)

We now proceed with solving for the divergent terms of the on-shell action following the algorithmic procedure described in section 2.3:

Step 1: we begin by writing the most general Ansatz for each U(i). We only keep terms that can give divergent contributions. With the scalar fallo s (5.5) and each inverse metric giving e2r, the most general Ansatz at 0th order is

U(0) =

3L + A1 + A2 + A32 + A4 + A53 + A62 + A72 + A84 , (5.10)

where the constant term is xed by the purely gravitational analysis in section 3. Each of the coe cients Ai is considered a function of r.

At order 2 we use the Ansatz

U(2) =

L

4 R + B1R + B2R + B3R2 + B4

. (5.11)

We did not include ()2, since it is equivalent to after partial integration.

15

At order 4, the only option are the purely gravitational terms we have already solved, so we have

U(4) =

L2r

8

RijRij 13R2 . (5.12)

Since the full FGPW model (5.2)(5.3) is symmetric under , we can immediately

set the following coe cients in the Ansatz to zero:

A1 = A4 = A5 = B1 = 0 . (5.13)

Step 2: at the leading order, the conjugate momenta obtained from (2.5) must agree with those in (5.8). From (2.5), we have

p =

p =

, (5.14)

and via (5.5) this gives

p =

2 L

JHEP06(2016)046

1

L 2r

+ O e2r/L/r

, p =

1L + O e3r/L

. (5.15)

On the other hand (5.8) gives

p(0) =

U(0)

= A2 + 2A6 + A72 , p(0) =

U(0)

= 2A3 + 2A7 + 4A83 . (5.16)

Comparing (5.15) to terms in (5.16) at similar orders, we can directly infer that some of the coe cients Ai must vanish:

A2 = A7 = 0 . (5.17)

Furthermore, we learn that A3 = 12L and A6 = 1L 1 L2r

. However, let us leave A3 and A6 unxed for now for the purpose of illustrating how they are xed using the HJ equation.

Step 3: we proceed to solve the HJ equation (5.7). We start from the terms at 0th order. Keeping only terms without spatial derivatives and using K(0) = 43U2(0) from (2.18) we nd that

43U2(0) + p2(0) + p2(0) V (, ) + 2

U(0)

r = 0 . (5.18)

To solve this, we set the coe cient of each combination of elds to zero. For example, collecting the terms proportional to 2 gives

A3 + 4LA3 + 2A23 +

32L2 = 0 = A3 =

1

2L + O e2r/L

. (5.19)

This is the solution for A3 we anticipated from comparing (5.15) and (5.16).

Similarly, one nds

2-terms: A6 + 4

LA6 + 2A26 +

2L2 = 0 = A6 =

1L +

1

2r + O

L2 r2

,

(5.20)

4-terms: A8

16L2 (1 + 3c) = 0 = A8 =

16L2 (1 + 3c) r + O 1

.

16

Terms proportional to 2 vanish directly; had we had a term b2 in the expansion of the scalar potential, the HJ equation would have shown that b 6= 0 is not consistent with

the EOM.

Having calculated all the unknown coe cients in the U(0) Ansatz, let us write down the nal result (with r = L2 log ):

U(0) =

1 L

3 +

1 + 1 log

2 + 122 +112 1 + 3c

4 log

. (5.21)

We can identify each of the contributions. The rst one is related to the cosmological constant and it is xed for all models in D = 4 + 1 dimensions, as we saw in the pure gravity case in section 3. The terms that are quadratic in the elds are uniquely xed by the mass terms in the scalar potential and are as such universal for all models. Finally, the 4-terms are clearly model-dependent, as can be seen from the explicit dependence on c.

With the 0th order result in hand, we are now able to continue solving HJ equation for the two-derivative terms. Keeping only such terms from equation (5.7) gives

R

8 3U(0)

U(2) 1 2Y(2)

+2p(0)p(2) +2p(0)p(2) ijijijij +2U(2)r = 0 ,(5.22)

where we used K(2) from (2.20). U(0), p(0) and p(0) are known from (5.16) and (5.21),

while we calculate p(2) and p(2), and Y(2) from the Ansatz (5.11) for U(2):

p(2) =

U(2)

= B2R ,

p(2) =

U(2)

= 2B3R + 2B4

JHEP06(2016)046

,

L

4 Rij + B2Rij + B3Rij2 + B4ij ,

(5.23)

Y(2)ij =

U(2)

ij =

where we are dropping total derivatives. The result for Y(2)ij implies Y(2) = U(2). In the HJ equation (5.22), we organize the terms according to the eld monomials and set the coe cients of divergent terms to zero. The terms simply proportional to R directly vanish because we have already solved the purely gravitational part of the problem. The remaining terms allow us to solve for the coe cients B2,3,4:

R-terms: B2 + 1r B2 = 0 = B2 = O

1 r

,

R2-terms: B3

112 = 0 = B3 =

112r + O 1

(5.24)

As in the zero weight case the subleading terms related to integration constants are not important because they lead to nite contributions to the action. The nal expression for U(2) is then

U(2) = L

14R 1 4

,

-terms: B4 + 1

2 = 0 = B4 =

1

2r + O 1

.

1 6R

log . (5.25)

17

The rst term is purely gravitational. The second term is independent of details of the higher order terms in the potential and thus xed for all models that contain a scalar with m2L2 = 3. Finally, notice that the combination of the Laplace operator

and the Ricci

scalar R that appears in the last term is proportional, up to an overall constant to the conformal Laplacian.

Step 4: we have now fully determined the counterterm action necessary to cancel the divergences of the on-shell action. In particular we will have Sct = 12

R

d4x U and

therefore,

Sct = 1 2

ZM d4x(1 L

JHEP06(2016)046

3 +

1 + 1 log

2 + 122 +112 (1 + 3c) 4 log

+ L

14R 1 4

1 6R

log 116L3

RijRij 13R2 log

). (5.26)

This is our nal result for the FGPW model.

As a test, we have calculated the one-point functions of the QFT operators that are dual to the elds of the FGPW model. The one-point function of the operator dual to eld I will be given by7

hOI i = lim 0

I/2

Sren

I , (5.27) where the regularized action (ignoring possible nite counterterms) is

Sreg = Sbulk + SGH + Sct . (5.28)

In order to check that the expressions obtained are indeed nite, one must impose the equations of motion on the coe cients in the Fe erman-Graham expansion of the elds. We nd that with our innite counterterms, all three one-point functions in FGPW are indeed nite. Details are presented in appendix C.

6 Renormalization of a dilaton-axion model

In this section we present the procedure of renormalization of a dilaton-axion model. The purpose of this example is to illustrate how the procedure for holographic renormalization applies to theories that include marginal scalars. Specically, we examine the renormalization of the dilaton-axion model previously studied in [16]: the 5d bulk action is

Sbulk =

1

22

ZM d4x drg R[g] Lm

, (6.1)

with

12L2 . (6.2)The elds and are massless and therefore correspond to marginal QFT operators with scaling dimension = 4. Z denotes an arbitrary function of the dilaton eld . Near

7In the special case where I = d/2 the one-point function has an extra factor of log .

18

Lm = + Z()

the asymptotic boundary, these scalars generically do not vanish but instead approach a nite value. In particular, their asymptotic behavior is given by

(x, r) = (0)(x) + O

e2r/L

e2r/L . (6.3)

As a consequence, we cannot regard the e ective action as a power-expansion in these elds, as higher powers are not suppressed. Instead, we will take the Ansatz to involve general functions of and .

By dening the eld to be = (, ) and the Kahler metric to be G = 1 0

0 Z()

!,

, (x, r) = (0)(x) + O

we conclude that the HJ equation (2.15) now becomes

R[] + K + p2 +

1Z()p2 ijij Z()ijij +

JHEP06(2016)046

12L2 + 2

U

r = 0 . (6.4)

The momenta are dened, in the usual way (2.11), as derivatives of U.

Let us now examine step-by-step the procedure introduced in the previous sections and spot any important di erences.

Step 1: with d = 4, we need to keep terms with up to four derivatives:

U = U(0) + U(2) + U(4) . (6.5)

Taking into account that any possible function of the elds could give divergent contributions in the on-shell action we write the following Ansatz for the zero, two and four derivative parts of U respectively:

U(0) = A(, , r) , (6.6)

U(2) = B0R + B1() () + B2()2 + B3()2 , (6.7)

U(4) = C1R2 + C2RijRij + C3R + C4R + C5R()2 + C6R()2 + C7R() ()+C8Rijij + C9Rijij + C10Rijij + C11(

)2 + C12( )2

+C13 + C14ijij + +C15ijij + +C16ijij (6.8) +C17 ()2 + C18

()2 + C19

()2 + C20

() ()

+C21 ()2 + C22

2 +C25()2()2 + C26(() ())2 + C27()2() () + C28()2() () .

The coe cients A, Bi and Ci are all considered functions of the radial coordinate r as well as the elds and . We have omitted terms that up to total derivatives can be decomposed to the ones already included. For example, since B = i(Bi)B()2 B()

(), such a term can be absorbed in B1 and B2, so it is redundant to include it in the

Ansatz.

Step 2: we use equation (2.5) and the asymptotic behavior of the elds (6.3) to determine the leading behavior of p and p to be

p =

= O

e2r/L

() () + C23 ()2

2 + C24

()2

, p = Z()

= O

e2r/L

. (6.9)

19

On the other hand, our Ansatz for U(0) gives

p(0) =

U(0)

= A , p(0) =

U(0)

= A . (6.10)

By comparing the two sets of expressions for the momenta, we understand that the coe -cient A can neither depend on nor , and thus p(0) and p(0) vanish. This leaves U(0)

to be purely gravitational and thus we can use directly our result from section 3:

U(0) =

3L . (6.11)

Step 3: we now proceed to solve HJ equation and determine the unknown coe cients of our Ansatz. Since the zero-derivatives contribution has already been xed, we start our analysis with the two-derivative terms. At this order, the HJ equation simplies to

R

8 3U(0)

U(2) 12Y(2) ()2 Z()()2 + 2U(2)r = 0 , (6.12)

using p(0) = p(0) = 0. Here, Y(2) = ijY(2)ij is the trace of the tensor

Y(2)ij =

U(2)

ij = B0Rij ijB0 +

B0ij + 12B1ij

+ 12B1ij + B2ij + B3ij .

JHEP06(2016)046

(6.13)

After plugging everything into the HJ equation, one uses partial integration to eliminate terms that were not in our original Ansatz and therefore were not independent. Demanding that the coe cient of each independent term in the resulting HJ equation is zero, one nds that the two-derivative contribution to the on-shell action is

U(2) =

L R ()2 Z()()2 . (6.14)

For terms with four spatial derivatives equation (6.4) simplies to

8 3U(0)

U(4) 12Y(4) + 4Y(2)ijY ij(2) 4 3

U(2) 1 2Y(2)

2 Y 2(2)

+ p2(2) +

1 Z()p2(2) + 2

U(2)

r = 0 . (6.15)

The canonical momenta that appear in this equation are

p(2) =

U(2)

=

L + L

4 Z()()2

L

2 Z()() () .

(6.16)

p(2) =

U(2)

=

L

It is useful to notice that

Y(4) = ij

U(4)

ij = 2U(4) + total derivatives , (6.17)

20

and the complicated tensor Y(4)ij is not needed for the calculation. The total derivatives of Y(4) will not contribute to HJ equation since they are multiplied by U(0), which is a constant, and total derivatives can be dropped by the equation.

Demanding that the di erent kinds of terms that appear in the four-derivative equation vanish independently yields the following solution for U(4):

U(4) = L3

16

RijRij 13R2 2

Rij 13Rij (ij + Z()ij)

+ 12Z()()2 2+ Z()

+ Z()

Z() () ()

2

(() ())2 ()2()2 log . (6.18)

Step 4: this concludes the calculation of the counterterms that cancel the innities of the on-shell action for the dilaton-axion model. For completeness, let us write down the general result.

Sct = 1 2

ZM d4x 3L +L 4

R ()2 Z()()2

L3

16

RijRij 13R2 2

Rij 13Rij (ij + Z()ij)

JHEP06(2016)046

+23 ()2 + Z()()2

2 + 2Z()

+ 12Z()()2 2+ Z()

+ Z()

Z() () ()

2

+23 ()2 + Z()()2

2 + 2Z()

(() ())2 ()2()2 log

. (6.19)

This result for the counterterms agrees with the one found by a more complicated route in [16].

7 Discussion

We have presented a simple implementation of the Hamiltonian approach to holographic renormalization. The idea of using the Hamilton-Jacobi equation is not new, but we hope that our presentation and algorithm makes the method more accessible and useful for others to use. For our own purposes, it has shown great value in the application to the holographic renormalization of a 10 scalar model dual to N = 1 gauge theory on S4, an

analysis that will be presented elsewhere [20].

Determining the innite counterterms is typically only one part of holographic renormalization. One often needs the nite counterterms too, but just as in standard quantum eld theory, this typically amounts to being a scheme-dependent question. However, in the presence of supersymmetry, one can x the nite counterterms to be compatible with the supersymmetries in the problem. In the case of at-sliced domain walls, this can be done using the Bogomolnyi-trick of writing the bulk action in terms of sums of squares that each vanish on the BPS equations. This rewriting requires a partial integration that leaves a

21

boundary term that exactly becomes the counterterm action and encodes both innite and nite counterterms. In the case of non-at slicing, one can then argue that the universality of the counterterms allows one to pick the nite counterterms of the at-space Bogomolnyi boundary term and use them in conjunction with the more general innite counterterms discussed in this paper. This has worked successfully in several cases, for example [18] and [25]. The prescriptions does, however, have a bit of an ad hoc feel to it and it would be interesting to understand better the relationship between the BPS equations for curved domain walls and how/if they can be used to determine directly the innite and nite counterterms.

Acknowledgments

We are grateful to Nikolay Bobev, Dan Freedman, Finn Larsen, Ioannis Papadimitriou, and Silviu Pufu for useful discussions. Some calculations were done using Mathematica and the package xAct. HE and MH are supported in part by NSF CAREER Grant PHY-0953232. HE is a Cottrell Scholar of the Research Corporation for Science Advancement. MH also acknowledges a Fulbright Fellowship by the Institute of International Education.

A Some useful formulas

We present here a list of formulas that are useful to computing the metric variations of various contractions of curvature tensors:

[integraldisplay] ddx X

R ij(y) = [parenleftBig]

JHEP06(2016)046

RijX + ( X) ij ijX[parenrightBig]

, (A.1)

[integraldisplay] ddx X

(RklRkl)

ij(y) = [parenleftBig]

2RikRkjX + kl(XRkl) ij +

(XRij) 2ki(XRkj)[parenrightBig]

, (A.2)

[integraldisplay] ddx X

Rkmln

ij(y) = [parenleftbigg]

1

2 mlX in kj

1

2 nlX jm ki +

1

2 klX im jn[parenrightbigg] , (A.3)

[integraldisplay] ddx X

Y ij(y) = [parenleftbigg]XijY +i(XjY )

1

2 k(XkY ) ij[parenrightbigg]+[integraldisplay] ddx

X Y

ij(y) . (A.4)

The elds on the r.h.s. of these equations depend on y.

B Six derivative counterterms for pure gravity

In d = 6 dimensions one needs to consider counterterms with up to six derivatives. For the pure gravity case, the six-derivative Ansatz is given by equation (3.4). In this Ansatz, it is possible to include terms with contractions of two or three Riemann tensors, but it is easy to show that the coe cients of such terms will be zero.

The HJ equation at six-derivative order becomes

K(6) + 2

U(6)

r = 0 . (B.1)

22

The total derivatives of Y(4)ij that appear in K(6) are now important because they are

multiplied by the non-constant Y(2)ij = BRij. In particular, we have that

Y(4)ij = C1

2RklRikjl + 12 Rij + Rij ijR

+ C2

2RRij + 2 Rij 2ijR

.

(B.2)

The coe cients B and C1,2 are those calculated in section 3. Additionally, in the product Y(2)ijY

ij

(4) , terms proportional to RijijR can be changed to RijRij =

1

2 R R by

adding appropriate total derivatives and using the Bianchi identity. Finally, by using the variation rules of appendix A, one realizes that Y(6) = 3U(6) up to total derivative terms that can be ignored because Y(6) is only multiplied by the constant U(0). Putting everything together and demanding that the coe cient of each of the independent terms is zero gives di erential equations for the coe cients D1,2,3,4,5,6:

R3-terms: D1 + d 6

L D1

dL416(d 1)2(d 2)3

= 0 ,

RRijRij-terms: D2 + d 6

L D2 +

L44(d 1)(d 2)2(d 4)

= 0 ,

R jiR kjR ik-terms: D3 +

d 6

L D3 = 0 ,

JHEP06(2016)046

(B.3)

RijRklRikjl-terms: D4 + d 6

L D4 +

2L4(d 2)3(d 4)

= 0 ,

R R-terms: D5 + d 6

L D5

L44(d 1)(d 2)3(d 4)

= 0 ,

Rij Rij-terms: D6 + d 6

L D6 +

L4(d 2)3(d 4)

= 0 .

Keeping only divergent contributions from the solutions of these equations, we obtain the result (for d = 6)

U(6) =

L4r

128

Rij Rij 120R R + 2RijRklRikjl + 15RRijRij 3100R3 . (B.4)

C One-point functions

In this appendix we calculate the one-point functions for the quantum eld theory operators dual to the elds of the FGPW model and explicitly check that the counterterm contributions cancel the divergences that come from the bulk action. One may consider three di erent one-point functions, hOi, hOi and hTiji, where the QFT operators O/

are dual to the bulk elds / respectively and the QFT energy-momentum tensor Tij is

dual to the metric ij.

These one point functions can be calculated by variations of the renormalized action

Sren = lim

0

Sreg = lim

0

(Sbulk + SGH + Sct) , (C.1)

23

where the regularized action Sreg is the sum of the bulk action (5.1), the Gibbons-Hawking

boundary term, and the counterterm action (5.26). In particular, the three correlation functions are given by:

hOi = lim 0

log

Sreg

1 , hOi = lim

0

1 3/2

Sreg

2 ij . (C.2)

Sreg

1 , hTiji = lim

0

1

The variation of the bulk action gives only a boundary term since the rest of the contributions are set to zero by the equations of motion. Namely, one gets

Sbulk

=

12

2 L

,

JHEP06(2016)046

Sbulk

=

12

2 L

,

Sbulk

ij =

1

22

L ij mnmnij

. (C.3)

On the other hand, the variation of the counterterm action has been already calculated during the renormalization process and it is related to the conjugate momenta of the elds:

Sct

= =

12 p ,

Sct

= =

12 p ,

Sct

ij = ij =

12

Yij

1

2Uij

. (C.4)

After putting everything together, the following expressions are obtained:

hOi=

12 lim

0

log

2L +2 L

1 + 1 log

, (C.5)

hOi=

12 lim

0

1 3/2

2L+1 L+

13L (1+3c) 3

L

2

1 6R

log

, (C.6)

hTiji=

1 2

2

1

2L (ij ijmnmn) Yij +

1

2Uij

, (C.7)

with

Yij = L

4 Rij +

L24(Rij2 + 4ij 2ij ()2ij

ij)

+ L3

96 (4RRij 12RklRkilj +

Rij + 2ijR 6

Rij)

log , (C.8)

and U as calculated in section 5.

To determine whether the above expressions are nite, one has to use the Fe erman-Graham expansions for the metric and the scalar elds of the theory:

ij = 1

(0)ij + (2)ij + (2,1)ij log

+ (4)ij + (4,1)ij log + (4,2)ij log2

+ O(2) (C.9)

24

=1/2(0) + 3/2 (2) + (2,1) log

+ O(5/2) (C.10)

= (0) + (0,1) log

+ O(2) (C.11)

Notice that for the special case of the -eld there is a logarithmic term even in leading order in . (This is generally true for all elds with scaling dimension = d/2.) All the coe cients of the above expansions can be determined in terms of (0)ij, (4)ij, (0), (0,1), (0) and (2) using the equations of motion for the elds and the metric. These undetermined coe cients encode information about the boundary QFT. Namely, the leading order coe cients (0,1) and (0) are related to the source of the respective QFT operators, while coe cients (0) and (2) are related to their vev rate. Additionally, the leading coe cient (0)ij in the expansion of is the background metric of the boundary QFT. Finally, although (4)ij is not fully determined, its trace and covariant divergence can be related to the other expansion coe cients using Einsteins equation.

The substitution of the expansion (C.11) for into hOi directly leads to cancellation

of all of the divergences, without using the equations of motion, and the result is

hOi =

1 2

2L(0) . (C.12)

Plugging the expansion (C.10) for into hOi leads to direct cancellation of the

divergent terms in leading order, i.e. those proportional to 1/, however, a logarithmic divergence remains:

hOi =

1 2

JHEP06(2016)046

2 L(2) +

2 L(2,1)

+ 1

2 lim

0

2 L(2,1)

13L(1 + 3c)3(0) +

L (0) 1 6R(0)

(0)

log , (C.13)

where R(0) R[(0)] is the Ricci scalar obtained by the metric (0) and

(0)(0)

1

(0) i p

(0)ij(0)j(0)

. (C.14)

In order to see the desired cancellations, one has to calculate the expansion coe cient (2,1) via the equation of motion for the eld ,

L2 + 422 + 4 + 22 Tr(1) + 3 2c3 = 0 . (C.15) By the asymptotic expansions for and the metric, the terms proportional to 3/2 give

(2,1) =

1 4

L2 (0) + Tr(1(0)(2)) 2c2(0) (0) . (C.16)

Finally, (2) is determined using Einsteins equation:

R[g] = + + 13L2 V (, )g . (C.17)

The ij component of this equation is

L2Rij[] = 222ij + 2ij + 2 Tr(1)ij 22mnminj (C.18)

25

122 Tr(11)ij + 2 Tr(12)ij + Tr(1)ij + L2ij + L2ij + 22()2ij + 22()2ij + 2

L2 V (, )ij .

Expanding it and keeping terms up to O(1) one nds

(2)ij =

L2 2

R(0)ij 1 6R(0)(0)ij

162(0)(0)ij. (C.19)

Now using these results for (2,1) and (2) in hOi exactly cancels the logarithmic term and

gives the following nite result for the one-point function:

hOi =

1 2

(0) + 13L(1 + 3c)3(0) . (C.20)

A similar approach leads to the renormalized one-point function for the energy-momentum tensor. A direct substitution of the asymptotic expansions in equation (C.7) leads to the cancellation of the leading O(2) divergences. However, the remaining diver

gences can be canceled only after solving Einsteins equation for (4,1) and (4,2). Terms proportional to log give

(4,2)ij =

162(0,1)(0)ij , (C.21)

while terms proportional to give

(4,1)ij = L4

8

Rkl(0)R(0)ikjl 1 3R(0)R(0)ij

JHEP06(2016)046

2 L(2)

L (0) 1 6R(0)

L4

32

Rkl(0)R(0)kl 13R2(0) (0)ij

+ L4

16

(0)R(0)ij 13ijR(0) 16 (0)R(0)(0)ij

+ L2

4 (0)

13ij +16(0)ij (0) 1 6R(0)ij

(0)

(C.22)

L2 6

i(0)j(0) 1 4kl(0)k(0)l(0)(0)ij

1

24 (1 + 3c) 4(0)(0)ij

1 3(0)(0,1)(0)ij .

Then, the renormalized energy momentum tensor will be given by:

hTiji =

2 L(4)ij

1 L

132(0) (0)(0,1) +232(0,1) 172 (1 3c) 4(0) + (0)(2) (0)ij

+ L

8 kl(0)k(0)l(0) + (0) (0) 1 9R(0)

(0)

(0)ij

L

4 (0)

ij 1 2R(0)ij

(0) + L3 32

R(0)klRkl(0) +19R2(0) + (0)R(0)

(0)ij

+ L3

4

R k(0)i R(0)kj 32Rkl(0)R(0)ikjl +14ijR(0) 34 (0)R(0)ij

. (C.23)

26

The trace of the stress-tensor one-point function gives a much simpler expression, since the trace Tr(1(0)(4)) can be obtained from the component of Einsteins equation, which gives

2 Tr(11)22 Tr(12)2 Tr(1) = (2)2+(2)2+

Keeping only terms of order O(2) in this yields

Tr(1(0)(4)) =

13(22(0) + 2(0,1)) +

After plugging in the above result the trace anomaly becomes

hT ii i =

L3 8

L2

3 V (, ) .

(C.24)

L4

16

R(0)ijRij(0) 29R2(0)

L28 (0)

(0) 5 18R(0)

(0)

(C.25)

1 9

1 + 32c 4(0) (0)(2) .

1

L

4(0)(0,1) 22(0,1) 16 (1 + 3c) 4(0) 2(0)(2)

+ L

2

(0) (0)(0) + ij(0)i(0)j(0)

JHEP06(2016)046

R(0)ijRij(0) 13R2(0) . (C.26)

It must be mentioned that the above results for the one-point functions are true only up to contributions from nite counterterms in the action.

Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

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