Holographic c-theorems in arbitrary dimensions

Holographic c-theorems in arbitrary dimensions

Robert C. Myers and Aninda Sinha
Perimeter Institute for Theoretical Physics
 Waterloo, Ontario N2L 2Y5, Canada
Centre for High Energy Physics, Indian Institute of Science,
 C.V.Raman Avenue, Bangalore 560012, India
Abstract:

We re-examine holographic versions of the c-theorem and entanglement entropy in the context of higher curvature gravity and the AdS/CFT correspondence. We select the gravity theories by tuning the gravitational couplings to eliminate non-unitary operators in the boundary theory and demonstrate that all of these theories obey a holographic c-theorem. In cases where the dual CFT is even-dimensional, we show that the quantity that flows is the central charge associated with the A-type trace anomaly. Here, unlike in conventional holographic constructions with Einstein gravity, we are able to distinguish this quantity from other central charges or the leading coefficient in the entropy density of a thermal bath. In general, we are also able to identify this quantity with the coefficient of a universal contribution to the entanglement entropy in a particular construction. Our results suggest that these coefficients appearing in entanglement entropy play the role of central charges in odd-dimensional CFT’s. We conjecture a new c-theorem on the space of odd-dimensional field theories, which extends Cardy’s proposal for even dimensions. Beyond holography, we were able to show that for any even-dimensional CFT, the universal coefficient appearing the entanglement entropy which we calculate is precisely the A-type central charge.

preprint: arXiv:1011.5819 [hep-th]

1 Introduction

Zamolodchikov’s c-theorem [1] states that for quantum field theories (QFT’s) in two dimensions, there exists a positive definite real function on the space of couplings , which satisfies the following three properties: i) is monotonically decreasing along renormalization group (RG) flows. ii) is stationary at RG fixed points , i.e., . iii) At RG fixed points, equals the central charge of the corresponding conformal field theory (CFT). This remarkable result requires only very simple conditions of the QFT’s with the proof relying only on the Euclidean group of symmetries, the existence of a conserved stress-energy tensor and unitarity in the field theory. A direct consequence of the c-theorem is that in any renormalization group (RG) flow connecting two fixed points,

(1)

That is, the central charge of the CFT describing the ultraviolet fixed point is larger than (or equal to) that at the infrared fixed point. An intuitive understanding of this result comes from the interpretation that the central charge provides a measure of the number of degrees of freedom of the underlying CFT. Its decrease along the RG flow can then be seen as a consequence of integrating out high-energy degrees of freedom in the Wilsonian approach to the renormalization group.

There have been various suggestions on how such a result might extend to quantum field theories in higher . One approach refers to the trace anomaly which is fixed by the central charge for two-dimensional CFT’s [2], i.e.,

(2)

Turning to , this expression generalizes to [2]

(3)

where is the square of the four-dimensional Weyl tensor and the expression in the second term is proportional to the four-dimensional Euler density. Hence, the question naturally arises: do any of , or satisfy a c-theorem under RG flows? Since is scheme dependent [3] and cannot be defined globally [4], it is not a useful charge to consider. With regards to the four-dimensional central charge , many counter-examples are now known where eq. (1) is not satisfied [5, 6]. Further, the latter investigations [6] also demonstrated that any linear combination of and will not generically satisfy eq. (1). This leaves us only to consider the central charge .

It was Cardy [7] who originally conjectured that should decrease monotonically along RG flows of four-dimensional QFT’s. Numerous nontrivial examples have been found supporting this conjecture, including perturbative fixed points [8] and supersymmetric gauge theories [6, 9]. While a counter-example to Cardy’s conjecture in was proposed in [10], recently, a certain flaw in this analysis was identified and so this possible counter-example is removed [11]. Further, as we review below, support for such a c-theorem in was found with the AdS/CFT correspondence [12, 13] – however, in the class of holographic theories studied there, and so both and satisfy eq. (1). While Cardy’s conjecture is supported by numerous nontrivial examples, a general proof is still lacking. A primary purpose of this paper is to report further evidence for the conjecture coming from a broader class of holographic models, where in particular the central charges and are distinct. An preliminary report of these results was given in [14].

Cardy’s conjecture [7] actually referred to any even number of spacetime dimensions. For even , the trace anomaly for CFT’s in a curved background can be written as [2]

(4)

where is the Euler density in dimensions and are the independent Weyl invariants of weight .111A note on our conventions: The stress tensor is defined by . The Euler density is normalized so that on a -dimensional sphere: . The Weyl invariants are constructed from contractions of curvatures or curvatures and covariant derivatives. There is some ambiguity in the construction of the which we (partially) fix by demanding that these invariants vanish when evaluated on a round -dimensional sphere, i.e., . Finally the last term is a conformally invariant but also scheme-dependent total derivative. That is, this last contribution can be changed or even eliminated by adding a (finite) covariant counter-term to the action. Cardy’s proposal is then that, in any even , for RG flows connecting two fixed points

(5)

Of course, this coincides with Zamolodchikov’s result in where and it matches the above discussion for where with our present choice of normalization.222Further, in comparing eqs. (3) and (4), we see for there is a single invariant corresponding to the Weyl tensor squared and for which .

One of the advantages of the investigating RG flows in a holographic framework is that the results are readily extended to arbitrary dimensions [12, 14]. In [14] and in the following, we examine holographic models with higher curvature gravity in the (+1)-dimensional bulk, which allows us to distinguish the central charges appearing in the trace anomaly (4) of the -dimensional boundary CFT’s for even . Hence we are able to discriminate between the behaviour of the various central charges in RG flows and we find that only has a natural monotonic flow, giving further support for Cardy’s conjecture (5). Our analysis of holographic RG flows applies in arbitrary higher and in fact, we find a certain quantity, denoted , satisfying an inequality such as that given in eq. (1) or (5) for any , that is, for both even and odd . There is no trace anomaly for odd and so some new interpretation for must be found in this case. Following [14], we identify this quantity with the coefficient of a universal contribution to the entanglement entropy for a particular construction for both odd and even . Our results suggest that these coefficients appearing in entanglement entropy play the role of central charges for odd-dimensional CFT’s and allow us to conjecture a c-theorem on the space of odd-dimensional field theories. However, we must emphasize that our higher curvature gravity actions are not derived from string theory. These theories should be regarded as toy models which allow us to explore of the role of higher curvature terms in holography. Ultimately, one would like to develop a better understanding of string theory in order study interesting holographic backgrounds with high curvatures and study the possibility of a holographic c-theorem in this framework.

An overview of the paper is as follows: We begin with a review of holographic c-theorem for Einstein gravity [12, 13] in section 2. Further, we describe a similar c-theorem for a particular higher curvature theory, known as quasi-topological gravity [15, 16]. In section 3, we extend our discussion to gravity theories with more general higher curvature interactions. We propose that the couplings of these new interactions should be tuned to remove any non-unitary operators from the dual boundary theory. With this constraint, we find that the resulting theories automatically satisfy a holographic c-theorem. In both of these sections, we are able to show that for even , the quantity which obeys the holographic c-theorem is precisely the central charge in eq. (4). In section 4, we demonstrate that appears in a certain calculation of entanglement entropy for odd or even . In particular, we place the -dimensional CFT on and calculating the entanglement entropy of the ground state between two halves of the sphere. We then find a universal contribution: . In this section, the entanglement entropy is calculated by relating it to the thermal entropy of the CFT on the hyperbolic plane, i.e., , with a particular temperature. In section 5, we describe a more conventional calculation of the entanglement entropy in the boundary CFT using the replica trick. The latter is translated to a holographic calculation in the bulk gravity theory and we reproduce the same results as in the previous section. Here we note that these calculations are distinct from the standard holographic calculations of entanglement entropy [17, 18], which are only applicable with the bulk theory is Einstein gravity. In section 5.4, we also extend our calculation using the replica trick to any CFT in even dimensions and show (without holography) that in general the universal coefficient is precisely the central charge . Returning to the holographic framework in section 5.5, we show that can be thought of as counting the degrees of freedom in the boundary CFT. In section 6, we compare our results with two other proposals for charges which may satisfy a c-theorem in higher dimensions. We explicitly show that does not correspond to the coefficient governing the leading singularity of the two-point function of the stress tensor or the leading coefficient in the entropy density of a thermal bath. Further, we show that the latter coefficient need not satisfy a holographic c-theorem within the class of boundary theories described by quasi-topological gravity. We conclude with a discussion of our results and possible future directions in section 7. We also have some appendices containing related calculations. In appendix A, we examine the holographic RG flows more generally for the theories considered in section 2. Appendix B makes some preliminary comments on establishing a holographic c-theorem when the matter fields couple to the higher curvature interactions. Finally in appendix C, we make some brief comments about possible c-theorems for holographic models with the non-relativistic Schrödinger symmetry.

2 Holographic c-theorems – Take One

The c-theorem was first considered in the context of the AdS/CFT correspondence by [12, 13]. There one begins with (+1)-dimensional Einstein gravity coupled to various matter fields:

(6)

The matter theory is assumed to have various stationary points where with some canonical scale . While the latter is phrased in a general way, it is useful to keep in mind a simple example in the following. Namely, the matter sector here could naturally be a scalar field theory where the potential has a number of extrema with . The vacuum energy or cosmological constant is negative at all of the relevant stationary points and it is a convenient notation to introduce to distinguish the different values. At these points, the vacuum solution for the Einstein theory is simply AdS with the curvature scale given by .

Next one considers solutions of the above theory (6) where the scalar sits at one fixed point in the asymptotic (UV) region and makes a smooth transition to another fixed point in the interior (IR) region. Such a solution can be interpreted as a holographic representation of a renormalization group flow, in which the boundary CFT flows from one fixed point in the UV to another in the IR. The spacetime geometry for these holographic RG flows is conveniently described with a metric of the form

(7)

This metric becomes that for AdS with at the stationary points. Now one defines [12]:

(8)

where ‘prime’ denotes a derivative with respect to . Then for general solutions with the above metric (7), one finds

In the second equality above, the Einstein equations are used to eliminate in favour of components of the stress tensor. The final inequality assumes that the matter fields obey the null energy condition [19]. Combining this monotonic evolution of with with the standard connection between and energy scale in the CFT, always decreases in flowing from the UV (large ) to the IR (small ).

To make a more precise interpretation of the bulk solutions in terms of the boundary CFT, it is simplest to focus the discussion on at this point. In this case, the holographic trace anomaly [20] allows one to calculate the two central charges, and , of the four-dimensional CFT – see eq. (3). For any of the AdS vacua with a curvature scale , one finds

(10)

As we emphasize below, the equality of the two central charges results because the bulk theory is Einstein gravity [20]. However, the important observation is that the value of the flow function (8) will precisely match that of the central charges in the dual CFT at each of the fixed points. Hence with the assumption of the null energy condition, the holographic CFT’s dual to Einstein gravity (22) satisfy Cardy’s proposed c-theorem. That is, for these holographic RG flows, is always larger at the UV fixed point than at the IR fixed point. Of course, these holographic models do not distinguish between the flow of and , and so the central charge obeys the same inequality as well.

It has long been known that to construct a holographic model where , the gravity action must include higher curvature interactions [21]. In part, this motivated the recent construction of the higher curvature theory, known as quasi-topological gravity [15]. This gravitational theory should be regarded as a toy model which allows us to explore the behaviour of a broader class of holographic CFT’s. It was demonstrated in [14] that this bulk theory also naturally exhibits a holographic c-theorem, as follows: The action for quasi-topological gravity can be written as [15]

(11)

where is the four-dimensional Euler density, as used in Gauss-Bonnet gravity [22],

(12)

and is the new curvature-cubed interaction [15, 23]

This action is written for any (boundary) dimension , although we set for to avoid the singular behaviour of the pre-factor of in eq. (11). By introducing interactions quadratic and cubic in the curvature, this holographic model allows one to explore the full three-parameter space of coefficients controlling the two- and three-point functions of the stress tensor in a general -dimensional CFT [24]. The reader should keep in mind that this action (11) was not derived from string theory. Rather, as noted above, it was constructed as a toy model to allow us to explore a broader class of holographic CFT’s while maintaining control within the gravity calculations. However, we should also note that the gravitational couplings, and , in eq. (11) are constrained to be not very large, otherwise one finds that the dual CFT is inconsistent – for a more precise discussion, see [16, 25, 26, 27]. We emphasize that the discussions here and in [15, 16] should only be regarded as an initial exploration of the role of higher curvature terms in holography. Ultimately, one would like to develop our understanding of string theory to the point where we can study interesting holographic backgrounds with high curvatures.

We have also introduced a factor of in the cosmological constant term above in anticipation of our consideration of holographic RG flows below. The idea is that as in eq. (11), the gravity theory is coupled to a standard matter theory, e.g., a scalar field, with various stationary points which yield different values for the parameter . At any of these stationary points, there is an AdS solution with a curvature scale where

(14)

In general, this equation yields three roots for . However, for any choice of the couplings and , at most one of these roots corresponds to a ghost-free AdS vacuum which supports nonsingular black hole solutions, as described in detail in [15]. Further, in the case that the couplings, and , are not large, this will be the root that is continuously connected to the single root (i.e., ) that remains in the limit . Implicitly, we will be working in this regime of the coupling space and with this particular root in the following.

Originally, this action (11) was constructed to give a theory for which AdS black hole solutions could be easily found analytically [15]. However, another remarkable property of quasi-topological gravity is that the linearized graviton equations in the AdS vacuum are only second order in derivatives [15]. In fact, up to an overall numerical factor, the linearized equations are precisely the same as those for Einstein gravity in the AdS background. If we focus on for a moment, the techniques of [20] can be applied to calculate the central charges [16]

(15)
(16)

These expressions make clear that as long as the higher curvature couplings are nonvanishing.

Now to examine holographic RG flows in quasi-topological gravity (for arbitrary ), we adopt the same metric ansatz (7). As above, at a stationary point of the matter sector with a fixed , the vacua again correspond to . In this case, we construct a new flow function as [14]

Now examining the radial evolution of , we find

Again, the gravitational equations of motion allow us to introduce the components of the stress tensor in going from the first to second line. Further, as above, we also assume the null energy condition for the final inequality to hold. We apply this constraint here in the spirit of constructing a toy model with reasonable physical properties. One might note that violations of the null energy condition have been argued to lead to instabilities quite generally [28]. In the context of a full string theory or theory of quantum gravity, we expect that this condition will be relaxed but to make progress here, we assume that the matter sector continues to obey the null energy condition as a pragmatic choice – see section 7 for further discussion. With the latter assumption then, evolves monotonically along the holographic RG flows and we can conclude that the corresponding ‘central charge’ is always larger in the UV than at the IR fixed point.

Note that there is a technical point which we must address for odd . In this case, it could be that the expression in the second line of eq. (2) is negative if . However, we can rule out this possibility as follows: By construction, our flow geometry will have an asymptotically AdS region at large where . Now imagine that in the interior, is negative over some region and positive from out to the asymptotic boundary. Hence at the radius , we must have had and . However, this leads to a contradiction. If we evaluate the equation of motion at and combine this result with the null energy condition, we find . Hence our assumption that there is some region where must be incorrect.333In the special case that , we can assume that it can be expressed in terms of a Taylor expansion around . To leading order, we would have with and is some positive and even integer. Then, as above, we again find a contradiction in the vicinity around and reach the same conclusion.

Let us denote the fixed point value of the flow function (2) as

(19)

Then with eq. (2), our holographic model satisfies a holographic c-theorem which specifies that

(20)

Having found that satisfies a c-theorem, one is left to determine what this quantity corresponds to in the dual CFT. Inserting into eq. (19) and comparing with eq. (16), we see that is precisely the central charge . Motivated by Cardy’s conjecture (5) for a c-theorem in QFT’s in even dimensional spacetimes, it is natural to compare to the coefficient in eq. (4). In fact, using the approach of [29],444See discussion around eq. (88) for more details of this calculation. one readily confirms that there is again a precise match

(21)

Hence again, we find support for Cardy’s conjecture with this broad class of holographic CFT’s. However, we must seek a broader definition of in order to understand our results for odd . We address this question in sections 4 and 5, where we show that emerges in a certain calculation of entanglement entropy.

3 Holographic c-theorems – Take Two

In section 2, we have shown that quasi-topological gravity naturally gives rise to a holographic c-theorem. Further at the fixed points, we identified the quantity that decreases along the RG flows as the coefficient of the A-type trace anomaly of the dual conformal field theory, for even . Now we would like to test how robust this result is by expanding our considerations of holographic RG flows to a broader class of gravitational theories. In the following, we begin with a completely general curvature-cubed action and develop a series of constraints so that the resulting holographic model is physically reasonable. Again, our analysis here should be considered as an exploration of holography with certain toy models which display credible physical properties.

To begin, we will consider an action of the form

(22)

where and contain general interactions quadratic and cubic in the curvature

(23)

In constructing , we began with all possible six-derivative interactions and eliminated terms which are redundant because of index symmetries or the Bianchi identities or which are total derivatives [15]. In fact, this construction yields two additional independent terms, and . However, we have discarded these terms in eq. (3) for simplicity as we would find that the corresponding coefficients are always set to zero with the constraints introduced in the following discussion.

We again assume that there is a matter sector, which exhibits stationary points with different values of in the cosmological constant term in eq. (22). At any of these critical points, there is an AdS with a curvature scale where

(25)

with

Of course, we have arranged eq. (25) to take the same form as eq. (14) in the previous section. In general, this cubic equation again yields three roots for . However, as in the previous section, when the couplings, and , are not large, there will be one root that is continuously connected to the single root (i.e., ) that remains in the limit of Einstein gravity, i.e., . Implicitly, we will be working in this regime in the following and will refer to this particular root. We have not analyzed the general theory in great detail but we expect that, as for quasi-topological gravity, the vacua corresponding to any other (real) roots will be problematic [15].

While we could examine holographic RG flows with this action with general curvature-squared and -cubed interactions, it seems unreasonable to expect that any such arbitrary gravity theory should yield a holographic c-theorem, just as it is unreasonable to expect that any arbitrary quantum field theories should satisfy a c-theorem. In particular, we do not expect that non-unitary QFT’s will satisfy a c-theorem. Hence we must ask how should we constrain the new couplings in this gravitational action (22) in order to produce a physically credible model. Quasi-topological gravity has a number of interesting properties which make it a reasonable toy model for holographic studies. One striking feature of the theory was that although the general equations of motion are fourth order in derivatives, if the equations of motion for gravitons propagating in the AdS vacuum are only second order [15]. While this feature greatly facilitates holographic investigations of this theory, as explicitly seen in [16], there is a deeper significance to this property, as we now discuss.

Given the general gravitational action (22), the full equations of motion will be fourth order in derivatives, as explicitly shown in [30]. Further even if considering the equations of motion for graviton propagation in a general background solution or in the AdS vacuum, these linearized equations are still fourth order. To gain some intuition for such higher order equations, we establish an analogy with a higher-derivative scalar field equation (in flat space) – following [16]. To begin, we would think of a simple massless scalar (i.e., ) as providing the analog of the linearized Einstein equations. Then we modify this equation with the addition of a fourth order term to model the graviton equations produced in our generalized gravity theory (22)

(27)

Here we imagine is some high energy scale (the analog of ) and is the dimensionless coupling that controls the strength of the higher-derivative term (the analog of and ). The (flat space) propagator for this scalar can now be written as

(28)

Now the pole is associated with the regular modes which are easily excited at low energies. The second pole is associated with additional ‘physical’ modes that appear at the high energy scale. Depending on the sign of , these new modes may have a regular mass () or be tachyonic (). However, the key point is that these extra high energy modes are ghosts (for either sign of ) because the overall sign of their contribution to the propagator (28) is negative. This appearance of ghosts is a generic feature of higher derivative equations of motion and so one must worry that the fourth order graviton equations generically emerging from eq. (22) indicate that these gravitational theories contain ghosts. From a holographic perspective, this indicates that the graviton couples to more than the usual stress tensor in the boundary CFT. The massive ghost modes indicate that metric fluctuations also mixes with an additional tensor operator which is non-unitary. That is, the new operator produces states with negative norm in the CFT. Hence from either perspective, there is a fundamental pathology with such a theory.

However, as the analysis of quasi-topological gravity indicates [15], this problem can be evaded at least in the AdS vacuum. That is, we can tune the coupling constants in eq. (22) to special values, and , so that the linearized graviton equations in the AdS vacuum are only second order in derivatives. This tuning eliminates the appearance of ghosts in the gravity theory and of non-unitary operators in the boundary CFT. In the scalar field analogy above, specially tuned simply corresponds to . However, note that as we approach from finite values, the mass of the ghost modes diverges. Hence in the context of our holographic model, we can understand that the non-unitary operators are removed from the spectrum of the boundary CFT because, as we adjust the higher curvature coupling constants to approach the ghost-free model, the conformal dimension of these operators diverges. Further note, that after we have fixed the couplings to and , we are able to calculate arbitrary -point functions of the stress tensor in the vacuum of the boundary CFT, with the usual perturbative expansion in terms of Witten diagrams [31]. Hence to begin, we impose this requirement to constrain the gravitational action (22) as a tentative step towards producing a physically interesting holographic model. Afterwards, we will examine whether RG flows in these theories also obey a holographic c-theorem.

To identify the constraints leading to second order linearized equations of motion for fluctuations, we proceed as follows: First, we write the AdS metric as555We will assume . The case has been considered in [32, 33, 30, 34].

(29)

Next (using Mathematica), we consider the linearized equations of motion around this background, including all possible metric fluctuations where are allowed to depend on all coordinates. Isolating the coefficient of in these equations, we find that this coefficient can be set to zero with

(30)

Then we look at the coefficient of and set this to zero with

(31)

Remarkably, one finds that this two constraints alone are sufficient to eliminate all of the higher order contributions to the linearized equations of motion! In the context of RG flows, will change between the various fixed points. As a result, it is prudent to demand that the above constraints hold for any value of .666Note that this condition cannot be satisfied when , as explained in [30]. Thus we are led to the following constraints:

(32)
(33)

While eq. (33) follows directly from eq. (30), the constraint in eq. (3) comes from taking a linear combination of the expressions appearing in both eqs. (30) and (31). Note that the conditions (32) on the yield the Gauss-Bonnet combination of curvature-squared interactions [22], as expected, i.e., .

With the constraints (323) above, we have ensured that we have a reasonable (i.e., unitary) boundary theory for the AdS vacua. Hence we might examine if these theories satisfy a holographic c-theorem. So following the experience developed in the previous section, we substitute in the RG flow geometry (7) and examine the gravitational equation of motion proportional to . However, unfortunately, the resulting equation as terms proportional to the third and fourth derivative of the conformal factor, i.e.,  and . In order to get a simple c-theorem as in the previous section, we can eliminate these terms by fixing

Of course, with hindsight, the interpretation of this problem is obvious. We have ensured that the non-unitary operators corresponding to the ghost-like graviton modes have been removed from the CFT spectrum at any fixed points. However, when the boundary theory is perturbed away from the fixed points, the non-unitary operator come in from infinity to ‘pollute’ the RG flow. The solution is then also obvious. We should demand that the linearized equations of motion for any fluctuations around the RG flow geometry (7) are second order in derivatives. This will ensure that the boundary QFT does not contain any non-unitary operators along the RG flows, as well as at the fixed points.

As before we considering general fluctuations around the RG flow metric (7), we examine higher order contributions to the linearized equations of motion. If we have already imposed eqs. (323), the coefficients of three new terms proportional to , and can be set to zero with eq. (3) and

(36)
(37)

The seven constraints in eqs. (3237) are necessary and sufficient to produce to two-derivative equations for metric fluctuations around a general RG flow (7). This seems like the best approach to constructing a holographic model with a physically reasonable boundary theory in the present study of RG flows.

As noted above with eq. (3), the above constraints also ensure that, with the RG flow metric (7), the gravitational equation of proportional to takes the same simple form found in the previous section. Hence we construct the flow function

Here, with the constraints (3237), the couplings defined in eq. (3) reduce to

(39)

Now by construction the radial derivative of yields

(40)

As before, we again assume the null energy condition holds to produce the final inequality. If, as before, we denote the fixed point value of the flow function (3) as

(41)

then our higher curvature theories satisfy the holographic c-theorem

(42)

Hence we are led to conclude that demanding unitarity of the boundary theory along the RG flows also guarantees that the theory obeys a holographic c-theorem.

In fact, the unitarity constraints, (3237), are more than sufficient to produce a holographic c-theorem. If we only require that the equation is second order in derivatives and that a holographic c-theorem arises, the necessary constraints can be written as

(43)
(44)
(45)

Considering the curvature-squared couplings , it is easy to see that the solution of eq. (32) also satisfies eq. (43) above. However, eq. (43) admits a two parameter space of solutions and so the unitary solution is only a special case within this larger set. Clearly, analogous comments apply for the curvature-cubed couplings . However, the conclusion seems to be that some ‘unphysical’ models with nonunitary operators still satisfy a holographic c-theorem. It is perhaps not too surprising that such circumstances can arise since the RG flows only probe a small part of the full boundary theory. What is more important is that all of the holographic models with a unitary boundary theory are guaranteed to satisfy a holographic c-theorem.

In fact, the above discussion is incomplete for the constraints needed to ensure that the boundary theory is unitary. Having imposed the above constraints, the quadratic action for gravitons in the AdS vacua takes the form of a Fierz-Pauli action

(46)

where the constant pre-factor is given by . As we will see in section 6, this coefficient controls the strength of the leading singularity in the two-point function of the stress tensor in the boundary CFT. In order to avoid ghost-like gravitons and to have a unitary boundary theory, we should also impose . However, it is straightforward to see that this constraint is always satisfied because of our assumption about which root of eq. (25) we are considering – recall the discussion below eq. (3). We begin by denoting the right-hand side of eq. (25) as . Now we are choosing to be the smallest positive root of the equation . Note that since and , the function must have a positive slope at this root, i.e., . However, recognizing that our expression above is precisely , we have established that for this root.777One might be concerned that we could find but this only occurs outside of our chosen domain with and ‘not large’ – see a full discussion for quasi-topological gravity in [15].

There is a technical issue for and 5, which we now briefly address. Let us focus on the case to be specific. Analogous comments will apply for and we return to this case below. If we examine eq. (3), it seems that our construction is singular with . However, this is a spurious singularity as the factor of in the second term can be absorbed into our definition of in eq. (39). More importantly, this contribution to is proportional to and so is simply a constant in . This is related to an obvious ambiguity in constructing the flow function, namely, we can always add a constant to , which, of course, does not effect the radial evolution. While in general there is no motivation to add an extra constant, it turns out that there is a natural choice to make here. Quite generally, we will find that the flow function is proportional to a Wald-type formula [35]

(47)

For further discussion of this relation, we refer the reader to sections 4 and 5, as well as refs. [32, 30]. In any event, if we want to preserve this relation for , we should add a constant term to the flow function as follows

(48)

The appearance of this constant above is related to the fact that, in the gravity action (22) with the constraint (32), the curvature-squared terms are proportional to the four-dimensional Euler density (12). The latter does not effect the gravitational equations of motion since the bulk theory is four-dimensional with but it still contributes to black hole entropy [36].888The full story must be more involved since it seems that with a large coefficient, this term would lead to violations of the second law in black hole mergers. Hence it seems that if this topological term were to appear in the action of a complete theory of quantum gravity, the corresponding dimensionless coupling must be restricted to be relatively small. Now returning to , analogous comments apply but it is now the contribution proportional to which is a constant. We again fix this contribution through eq. (47) to produce

(49)

Let us examine the interactions that result in our toy model (22) after the unitarity constraints (3237) are imposed. First as already noted above, the constraints (32) on the couplings require that the curvature-squared interaction takes precisely the form of that appearing in Gauss-Bonnet gravity (12). Hence these interactions make two-derivative contributions to the equations of motion in a general background for [22]. As also discussed above, for , this term is a topological invariant in the four-dimensional bulk and so does not contribute to the equations of motion. However, it still plays a role in fixing our normalization for , as given in eq. 48. Note that for , this interaction simply vanishes because of a Schouten identity for curvatures in three or fewer dimensions. One can understand this heuristically as arising because if one attempts to evaluate

(50)

in less than four dimensions, the result must vanish because the indices do not run over enough values.

Turning to curvature-cubed interactions where the eight couplings are constrained by the five equations (3337). Hence in general, one expects a three-parameter family of unitary interactions, which we will describe in terms of a basis of three independent interactions. As the first of these, one can readily verify that the cubic Lovelock interaction satisfies these constraints. This interaction is proportional to the six-dimensional Euler density with

(51)

In analogy to the comments about the Gauss-Bonnet interactions, this particular curvature-cubed interaction will only contribute two-derivative terms to the equations of motion in a general background for [22]. For , it does not contribute to the equations of motion but still plays a role in determining , as given in eq. 49. For , this term simply vanishes and so should not be counted as one of the basis interactions.

As a second basis interaction, we can take the quasi-topological term given in eq. (2), which also satisfies the constraints (3337). While this term only makes two derivative contributions to the gravitational equations in any RG flow geometry, we should recall that fourth order terms can appear in other backgrounds [15, 16]. Further, we should note that this term was only constructed for and and so it cannot be counted amongst the basis interactions in or .

There are, in fact, two other candidates for the third basis interaction both of which are constructed from Weyl tensor:999Recall the definition of Weyl tensor in a (+1)-dimensional spacetime is (52) using the standard notation: .

(53)

In fact, these terms do not contribute to the linearized equations of motion around the RG flow geometry (7) at all, which can be deduced as follows. These backgrounds are conformally flat and since these terms are cubic in the Weyl tensor, the contribution to the quadratic action (46) for the graviton fluctuations must be proportional to at least one power of the Weyl tensor. Therefore there can be no contribution to the linearized equations of motion. Similarly, these terms do not contribute to the background equations which determine for a particular RG flow and so the corresponding coupling constant would not appear in in determining the curvature scale of the fixed points (25) or in the flow function (3). However, we emphasize that these interactions (53) would effect other properties of the boundary QFT. For example, they would contribute in a calculation of the three-point function of the stress tensor.

Now it may seem that we have an overabundance of basis interactions, since we have enumerated four possible unitary interactions above but our initial count of the constraints indicated that there should only be a three parameter family of such interactions. However, as noted in [23], the interactions listed above are not all independent for . In particular, in this case, we have the relation

(54)

Hence we can use any three of the above interactions as our basis for the curvature-cubed interactions in our holographic model with unitary RG flows when . For , is not defined and so our basis would be , and . For , Schouten identities reduce the number of number of possible interactions with and [30]. Hence for , we would have a two parameter family of interactions with and . For , is also not defined and so we are reduced to a one parameter family with only [30].

To close this section, we re-iterate that having found that satisfies a c-theorem (42) for our generalized holographic models, one must again ask what this quantity corresponds to in the boundary CFT. Motivated by Cardy’s conjecture (5) and our results in the previous section, we first compare to the central charge for even . With the approach of [29], we again find a precise match

(55)

Hence again, we have found evidence to support Cardy’s conjecture with this broad class of holographic CFT’s. However, we are again left without an interpretation of for odd .

4 and Entanglement Entropy – Take One

Above, we have identified a quantity in eq. (41) for a broad class of holographic models which varies monotonically in RG flows. For even , we have shown this quantity equals the coefficient of the A-type trace anomaly. However, a broader definition of is required to interpret our results when the boundary theory has a odd number of spacetime dimensions. We address this question here and in the next section, where we show that emerges in a certain calculation of entanglement entropy. Note that our discussion here and in the next section does not make use of the usual holographic calculation of entanglement entropy [17, 18]. The latter only applies for Einstein gravity while here our bulk theory involves higher curvature interactions. Of course, if we eliminate these additional terms, our results for the entanglement entropy reduce to those calculated with the standard approach [17, 18].

We begin with the following observation: Recall quasi-topological gravity theory (11) for which we considered holographic c-theorems in section 2. Black hole solutions and thermodynamics were studied in some detail for this theory in [15]. The horizon entropy for any of the (static) black hole solutions found there is given by the following:

(56)

where is the horizon radius. The final factor yields the ‘area’ of the horizon with being the induced metric on a spatial slice of the horizon. Also is an integer with values