# Holographic -theorems and higher derivative gravity

###### Abstract

In AdS/CFT, the holographic Weyl anomaly computation relates the -anomaly coefficient to the properties of the bulk action at the UV fixed point. This universal behavior suggests the possibility of a holographic -theorem for the -anomaly under flows to the IR. We prove such a -theorem for higher curvature Lovelock gravity, where the bulk equations of motion remain second order. We also explore gravity as a toy model where higher derivatives cannot be avoided. In this case, monoticity of the flow requires an additional condition related to the higher derivative nature of the theory. This is in contrast to the case of black hole entropy, where the second law follows from application of the full Einstein equations and the null energy condition.

^{†}

^{†}preprint: MCTP-10-57

## I Introduction

Central charges in conformal field theories can often be thought of as a proxy for the number of degrees of freedom exhibited by the theory. In this context, the Zamolodchikov -theorem Zamolodchikov:1986gt is a powerful result for two-dimensional conformal field theories. It states that there exists a -function which is monotonically decreasing along flows from the UV to IR, and which is equal to the central charge at the fixed points of the flow. This is a direct indication that UV degrees of freedom of the CFT are removed as the theory flows to the IR.

While two dimensional conformal field theories are rather special, there have been numerous attempts to generalize the -theorem to higher dimensions. However, one obstacle that needs to be surmounted in doing so is the realization that there may be multiple candidates for a satisfactory -function. For example, in four dimensions, the Weyl anomaly has the well known form

(1) |

where is the four-dimensional Euler density. In this context, Cardy demonstrated that, while the anomaly coefficient may not be the proper object to investigate, the coefficient appears to have the desired monotonicity property along flows Cardy:1988cwa . This has subsequently been confirmed in various situations Osborn:1989td ; Jack:1990eb ; Anselmi:1997am ; Anselmi:1997ys , and further investigated in the context of -maximization Intriligator:2003jj ; Kutasov:2003iy ; Kutasov:2003ux ; Barnes:2004jj .

While the above investigations have been carried out in a field theory context, AdS/CFT allows the possibility of a holographic version of the -theorem. At large , the and anomalies are equal, and can be computed holographically; for super-Yang-Mills, the result is simply Henningson:1998gx . This result may be extended to renormalization group flows, which correspond to radial flow in the bulk dual. A -theorem can then be proven at leading order in large by examining the flow equations for domain wall solutions interpolating between the UV and IR Alvarez:1998wr ; Girardello:1998pd ; Freedman:1999gp ; Sahakian:1999bd .

Recently, the leading order holographic -theorem has been extended to the case where the bulk theory may contain higher order curvature terms, corresponding to moving away from the leading behavior in AdS/CFT Sinha:2010ai ; Oliva:2010eb ; Myers:2010ru ; Oliva:2010zd ; Myers:2010xs ; Sinha:2010pm ; Myers:2010tj . In particular, the theories of interest include the addition of Gauss-Bonnet and ‘quasi-topological’ curvature-cubed terms to the bulk action. By generalizing the holographic -anomaly, computed for a general bulk action in Imbimbo:1999bj , Myers and Sinha constructed an appropriate -function that is monotonic along radial flow Myers:2010xs ; Sinha:2010pm ; Myers:2010tj . A key element of this holographic -theorem rests on the fact that the equations of motion contain no higher than second derivatives of the metric when expanded on the AdS background. Gauss-Bonnet gravity manifestly satisfies this requirement, as does the quasi-topological theory, which was explicitly constructed to have this property Oliva:2010eb ; Myers:2010ru .

Here we extend the investigation of Myers and Sinha in two directions. Firstly, we prove a holographic -theorem for the case of Lovelock bulk theories. These theories generalize Gauss-Bonnet gravity in dimensions by the addition of dimensional Euler densities where . While this result is mostly formal, in the sense that higher Lovelock terms are only present in theories in dimensions too large for practical applications, it nevertheless suggests that the holographic -theorem extends to arbitrary orders in the bulk curvature, so long as higher derivative terms in the equations of motion are able to be controlled.

Secondly, we address what happens when the bulk action is not restricted to second order equations of motion by examining gravity as a toy model. In this case, we find that the constructed -function may deviate from monotonicity by a term that is explicitly of higher derivative order. We suggest that this term, which may lead to a violation of the holographic -theorem, is related to the additional ghost modes of the theory. From an AdS/CFT point of view, this would correspond to a breakdown of unitarity in the dual field theory.

While this work was being prepared, we became aware of Myers:2010tj , which overlaps with some of our results.

## Ii A holographic -theorem for Lovelock gravity

While higher-curvature gravitational actions generically lead to higher derivative equations of motion and ensuing pathologies such as ghosts, a special class of higher-curvature actions may be constructed that nevertheless give rise to second order equations of motion for the metric. These are the Lovelock actions, which are constructed out of the -dimensional continuation of lower dimensional Euler densities Lovelock:1971yv

(2) |

Here the -th Lovelock term is the Euler invariant in dimensions

(3) |

In particular, is a cosmological constant, is the ordinary Einstein-Hilbert term and is the Gauss-Bonnet invariant. The equation of motion following from (2) is simply , where the generalized Einstein tensor is given by , with

(4) |

With a suitable choice of the cosmological constant, we take the bulk Lovelock action (2) to be dual to a -dimensional CFT. As demonstrated in Imbimbo:1999bj , the -dimensional type A trace anomaly (i.e. the term proportional to the Euler characteristic) is universal in holographic renormalization, and its coefficient may be expressed as

(5) |

where

(6) |

is the on-shell Lagrangian evaluated on the asymptotic AdS background with radius .

In order to construct a suitable -function, we need to promote the central charge (5) at the UV fixed point into a function of the radial flow. Here, for simplicity, we consider a radial slicing of the bulk space into flat slices of the form

(7) |

with curvature components

(8) |

This allows us to evaluate the individual Lovelock terms

(9) |

The second term is unimportant at AdS fixed points since we have

(10) |

As a result, the on-shell value of takes the form

(11) |

so that the central charge (5) may be expressed as

(12) |

The extension of into the bulk is by no means unique. As a first attempt to do so, we make the substitution

(13) |

so that

(14) |

This satisfies the requirement that reproduces the central charge at fixed points of the flow. In addition, it incorporates the metric function at the first derivative level, so that

(15) |

is linear in . Following Freedman:1999gp ; Myers:2010xs , we aim to demonstrate that is monotonic by appealing to the bulk equations of motion. To do so, we first compute the generalized Einstein tensor components (4) using the curvature components (8) for the bulk metric. The resulting two independent components are

(16) |

so that

(17) |

Comparison with (15) shows that, while the form of is suggestive, it nevertheless does not match with the difference . However, as indicated above, is not necessarily unique, and using the Einstein equation as a guide, we now construct a modified -function which is monotonic.

To proceed, we first note that, at AdS fixed points where the bulk matter sector contributes vanishing vacuum energy, the background satisfies the vacuum Einstein equation

(18) |

(Recall that the cosmological constant term is included in the gravitational sector through .) This allows us to add a vanishing on-shell contribution to (6), so that

(19) |

In this case, we are led to the definition

(20) |

Note that the shift removes the cosmological constant term from the definition of , and matches what is done in constructing a suitable -function in the leading two-derivative gravity. This -function can now be seen to satisfy

(21) |

where the inequality corresponds to the null energy condition. Therefore we have found an appropriate extension of the central charge which is indeed monotonic along flows from the UV to the IR. This result extends the observation of Myers:2010xs ; Myers:2010tj that a general holographic -theorem may be obtained in the presence of higher order corrections, provided the (linearized) equations of motion remain second order.

## Iii gravity

In proving the holographic -theorem for Lovelock gravity, we have constructed the function (20) entirely out of the first derivative of the metric function, . This ensures that reproduces the central charge at fixed points of the flow where . However, this construction also guarantees that is linear in the second derivative so that it may be connected to the equations of motion. This connection suggests that having second order equations is an essential aspect of obtaining the -theorem. On the other hand, bulk AdS duals are often considered to be effective theories where higher derivative corrections naturally arise (e.g. in the string expansion). Thus it is important to address whether any holographic -theorem could hold in such higher derivative theories as well.

Here we take one step towards a fully general investigation by considering the case of gravity. Such theories have been considered from a cosmological point of view, and are closely related to Brans-Dicke theory (see e.g. Sotiriou:2008rp ; DeFelice:2010aj ; Nojiri:2010wj ). However, here we are mainly interested in gravity as a toy model exhibiting higher derivative equations of motion. The action is given by

(22) |

where is a fixed but arbitrary function of the scalar curvature . The resulting equation of motion is

(23) |

where

(24) |

Since is second order in derivatives, the equation of motion is in general fourth order.

In order to construct an appropriate -function for gravity, we follow Freedman:1999gp ; Myers:2010xs and explore the difference in the Einstein equation components . To proceed, we take the same metric (7) as used above, and compute the Ricci components

(25) |

In this case, the Einstein equation (23) splits into

(26) |

Taking the difference of and gives

(27) |

Note that is a function of , which is in turn a function of according to (25). Thus the higher derivatives of are encoded in the terms in this equation.

Our aim is to construct a suitable which reproduces the -anomaly at the UV boundary and which is subject to a flow governed by (27). We start with the anomaly coefficient itself, given by (5)

(28) |

where this time is simply the on-shell value of at the asymptotic AdS fixed point

(29) |

As we saw above, the extension of to the interior is not unique. A straightforward choice would be to replace by , so that

(30) |

This is similar to our choice of in (14), although here we allow to contain through the dependence on the curvature scalar. However, differentiation of with respect to does not give any obvious correspondence with the difference (27). Thus we seek an improvement to , just as we did for the Lovelock case.

Since we do not want to destroy the matching of the -function with the actual -anomaly at AdS critical points, we can adjust by at most functions which vanish at such points. A natural possibility for such a function is to take the equations of motion at a critical point. In this case, the stress tensor vanishes, and furthermore the functions and become constant. The equation of motion (26) then simplifies to

(31) |

where . This suggests that we shift in (30) by , just as we did in the Lovelock case. The resulting -function then takes the form

(32) |

where is given in (25). Note that in the numerator of this expression is essentially the derivative of the Lagrangian with respect to (or equivalently with respect to ). This suggests a natural connection with an entropy function, as pointed out in Myers:2010xs ; Myers:2010tj . In fact, the appearance of in the entropy function for black holes was initially observed in Cognola:2005de .

With the definition (32), we now see that

(33) | |||||

where the second line follows from the equation of motion (27). If it were not for the term, we would then use the null energy condition, , to demonstrate that . This suggests that the higher derivative nature of gravity directly impacts the fate of the holographic -theorem. In particular, a non-trivial contribution is a direct sign that the gravitational background incorporates up to four derivatives of .

Taking a step back, it is perhaps not surprising that in this case monoticity of requires not just the weak energy condition on the matter sector, but also a further condition on the gravity sector. While we do not see a direct connection with unitarity, it is certainly plausible that this condition would be related to the absence of ghost modes in the background of the flow. One way to investigate this would be to map gravity onto Brans-Dicke theory. In this case, plays the role of the Brans-Dicke scalar. However, it is not clear to us how may be related to any obvious pathologies of the theory.

In searching for a holographic -theorem, we may also need to make a distinction between perturbative versus non-perturbative expansions in the higher derivative terms. For example, at the level, we may take , so that . The AdS vacuum with radius is given by the solution to (31), and in this case admits two branches

(34) |

This may be expanded for small

(35) |

We thus see that only the positive branch is smoothly connected to the finite radius background in the perturbative limit . In the context of AdS/CFT, it is natural to view as an expansion parameter in an effective theory with higher curvature interactions. In this case, we ought to restrict to only the positive branch. Perhaps the sign of can somehow be attributed to this choice. In particular, we have readily found numerical solutions for this example model coupled to a massless scalar where changes sign (from positive to negative) along the flow to the IR. However, all such resulting solutions involve a domain wall interpolating between the positive and negative branches of (34), and have the form

(36) |

This ‘kink up’ domain wall solution has the characteristic of a negative tension wall interpolating between two regions that both open up into AdS boundaries. Nevertheless, we have checked that the scalar matter is not responsible for this negative tension. Hence, it must have its origin in the higher derivative gravitational sector. This at least suggests that violation of the -theorem is closely related to pathologies in the gravity sector of the theory that would not arise when treated in a proper perturbative expansion where the equations of motion can be perturbatively arranged to use no higher than second order derivatives in the expansion.

## Iv Discussion

In both the Lovelock and the case, we have defined the -function based on a shifted form of the Lagrangian

(37) |

where is the Lagrangian density in the gravity sector (not including matter). Since we take the matter energy density to vanish at AdS fixed points, the addition of does not affect the identification of with the anomaly coefficient. However, this improvement allows to be related to the difference of the Einstein equations along the flow.

As noted in Myers:2010xs , the -function has a second interpretation in terms of entanglement entropy. This can be seen to arise in a natural manner, provided we perform the shift. In particular, consider a general higher curvature action of the form

(38) |

The corresponding Einstein equation may be written as

(39) |

where

(40) |

This generalizes the equation of motion given in (23). The higher derivative terms are manifestly present in the Einstein tensor. However, they vanish on the AdS background where is covariantly constant (as it is constructed out of the maximally symmetric Riemann tensor). Further taking to vanish in the asymptotic AdS region, we end up with

(41) |

The general -function (37) then takes the form

(42) |

If we were to consider black hole entropy in the presence of higher curvature corrections, it would be natural to use the Wald entropy formula Wald:1993nt ; Iyer:1994ys ; Iyer:1995kg

(43) |

where the integral is over the area of the horizon with unit binormal along and . This reduces to the familiar one-quarter of the horizon area (in units) in the absence of higher curvatures, where . Although this expression is intended to be evaluated at the black hole horizon, it can nevertheless be generalized into an entropy function Goldstein:2005rr ; Cremades:2006ke

(44) |

where we have used the explicit form of the metric (7).

Flows of have been investigated in the context of the second law of black hole thermodynamics in higher curvature gravity, including both Lovelock and gravity Jacobson:1995uq ; Cremades:2006ke ; Anber:2008js . In the case of gravity, a -theorem can be proven which generalizes the Hawking area theorem Hawking:1971vc by use of the Raychaudhuri equation Jacobson:1995uq ; Cremades:2006ke . In particular, we consider an affinely parameterized null congruence given by the tangent vector and define

(45) |

where is the expansion of the null congruence. The Raychaudhuri equation then gives

(46) |

and further application of the Einstein equation (23) reduces this to

(47) |

Provided the congruence is twist-free, and assuming the null energy condition along with positivity of , the terms on the right-hand-side are all non-positive, and as a result we may conclude that , which is the statement of the second law in gravity Jacobson:1995uq ; Cremades:2006ke .

For gravity with the metric written in the explicit form (7), we define , in which case

(48) |

so that

(49) | |||||

We see here that, even with a higher order equation of motion, the terms involving higher derivatives arrange themselves in just the proper manner to match with the Einstein equation combination (27). This is in contrast with the holographic -function, where given in (33) picks up an additional contribution proportional to . Nevertheless, it would be interesting to see if a connection can be made between the black hole entropy function (44) and the holographic -function (42).

Ideally, it would be desirable to construct a general proof of a holographic -theorem for the -function of (42) using techniques such as the generalized Raychaudhuri equation (46). This would allow the -theorem to be separated from the particular bulk AdS flow parameterization of (7). However, unlike the Raychaudhuri equation itself, which is independent of dynamics, the incorporation of into the generalized expansion necessarily brings higher derivative dynamics into the picture. Thus it appears unlikely that the proof of a holographic -theorem for higher curvature gravity can be fully decoupled from the exact form of the higher order interactions. (In fact, we do not expect any -theorem to hold unless additional unitarity constraints are imposed.) Nevertheless, we anticipate that it would be possible to make a general statement in the restricted case where the linearized equations of motion remain second order in the AdS background.

###### Acknowledgements.

The idea of a holographic -theorem for gravity came out of discussions with R. Myers at the Mitchell Institute Application of AdS/CFT workshop. JTL also wishes to acknowledge earlier conversations with C. Herzog and D. Marolf on AdS -theorems and the Raychaudhuri equation. We acknowledge the hospitality of the Khuri lab at the Rockefeller University, where this work was initiated. The work of WS was supported in part by the National Science Foundation under grant number PHY-0903134. This work was supported in part by the US Department of Energy under grant DE-FG02-95ER40899.## References

- (1) A. B. Zamolodchikov, Irreversibility of the Flux of the Renormalization Group in a 2D Field Theory, JETP Lett. 43, 730 (1986) [Pisma Zh. Eksp. Teor. Fiz. 43, 565 (1986)].
- (2) J. L. Cardy, Is There a c Theorem in Four-Dimensions?, Phys. Lett. B 215, 749 (1988).
- (3) H. Osborn, Derivation of a four-dimensional theorem, Phys. Lett. B 222, 97 (1989).
- (4) I. Jack and H. Osborn, Analogs for the theorem for four-dimensional renormalizable field theories, Nucl. Phys. B 343, 647 (1990).
- (5) D. Anselmi, D. Z. Freedman, M. T. Grisaru and A. A. Johansen, Nonperturbative formulas for central functions of supersymmetric gauge theories, Nucl. Phys. B 526, 543 (1998) [arXiv:hep-th/9708042].
- (6) D. Anselmi, J. Erlich, D. Z. Freedman and A. A. Johansen, Positivity constraints on anomalies in supersymmetric gauge theories, Phys. Rev. D 57, 7570 (1998) [arXiv:hep-th/9711035].
- (7) K. A. Intriligator and B. Wecht, The exact superconformal R-symmetry maximizes , Nucl. Phys. B 667, 183 (2003) [arXiv:hep-th/0304128].
- (8) D. Kutasov, A. Parnachev and D. A. Sahakyan, Central charges and symmetries in super Yang-Mills, JHEP 0311, 013 (2003) [arXiv:hep-th/0308071].
- (9) D. Kutasov, New results on the ‘-theorem’ in four dimensional supersymmetric field theory, arXiv:hep-th/0312098.
- (10) E. Barnes, K. A. Intriligator, B. Wecht and J. Wright, Evidence for the strongest version of the 4d -theorem, via -maximization along RG flows, Nucl. Phys. B 702, 131 (2004) [arXiv:hep-th/0408156].
- (11) M. Henningson and K. Skenderis, The holographic Weyl anomaly, JHEP 9807, 023 (1998) [arXiv:hep-th/9806087].
- (12) E. Alvarez and C. Gomez, Geometric holography, the renormalization group and the -theorem, Nucl. Phys. B 541, 441 (1999) [arXiv:hep-th/9807226].
- (13) L. Girardello, M. Petrini, M. Porrati and A. Zaffaroni, Novel local CFT and exact results on perturbations of N = 4 super Yang-Mills from AdS dynamics, JHEP 9812, 022 (1998) [arXiv:hep-th/9810126].
- (14) D. Z. Freedman, S. S. Gubser, K. Pilch and N. P. Warner, Renormalization group flows from holography supersymmetry and a -theorem, Adv. Theor. Math. Phys. 3, 363 (1999) [arXiv:hep-th/9904017].
- (15) V. Sahakian, Holography, a covariant -function and the geometry of the renormalization group, Phys. Rev. D 62, 126011 (2000) [arXiv:hep-th/9910099].
- (16) A. Sinha, On the new massive gravity and AdS/CFT, JHEP 1006, 061 (2010) [arXiv:1003.0683 [hep-th]].
- (17) J. Oliva, S. Ray, A new cubic theory of gravity in five dimensions: Black hole, Birkhoff’s theorem and C-function, Class. Quant. Grav. 27, 225002 (2010) [arXiv:1003.4773 [gr-qc]].
- (18) R. C. Myers, B. Robinson, Black Holes in Quasi-topological Gravity, JHEP 1008, 067 (2010) [arXiv:1003.5357 [gr-qc]].
- (19) J. Oliva, S. Ray, Classification of Six Derivative Lagrangians of Gravity and Static Spherically Symmetric Solutions, Phys. Rev. D82, 124030 (2010) [arXiv:1004.0737 [gr-qc]].
- (20) R. C. Myers and A. Sinha, Seeing a -theorem with holography, Phys. Rev. D 82, 046006 (2010) [arXiv:1006.1263 [hep-th]].
- (21) A. Sinha, On higher derivative gravity, -theorems and cosmology, Class. Quant. Grav. 28, 085002 (2011) [arXiv:1008.4315 [hep-th]].
- (22) R. C. Myers and A. Sinha, Holographic -theorems in arbitrary dimensions, JHEP 1101, 125 (2011) [arXiv:1011.5819 [hep-th]].
- (23) C. Imbimbo, A. Schwimmer, S. Theisen and S. Yankielowicz, Diffeomorphisms and holographic anomalies, Class. Quant. Grav. 17, 1129 (2000) [arXiv:hep-th/9910267].
- (24) D. Lovelock, The Einstein tensor and its generalizations, J. Math. Phys. 12, 498 (1971).
- (25) T. P. Sotiriou, V. Faraoni, Theories Of Gravity, Rev. Mod. Phys. 82, 451-497 (2010) [arXiv:0805.1726 [gr-qc]].
- (26) A. De Felice and S. Tsujikawa, theories, Living Rev. Rel. 13, 3 (2010) [arXiv:1002.4928 [gr-qc]].
- (27) S. Nojiri, S. D. Odintsov, Unified cosmic history in modified gravity: from theory to Lorentz non-invariant models, Phys. Rept. 505, 59-144 (2011) [arXiv:1011.0544 [gr-qc]].
- (28) G. Cognola, E. Elizalde, S. Nojiri, S. D. Odintsov, S. Zerbini, One-loop gravity in de Sitter universe, JCAP 0502, 010 (2005) [hep-th/0501096].
- (29) R. M. Wald, Black hole entropy is the Noether charge, Phys. Rev. D 48, 3427 (1993) [gr-qc/9307038].
- (30) V. Iyer and R. M. Wald, Some properties of Noether charge and a proposal for dynamical black hole entropy, Phys. Rev. D 50, 846 (1994) [gr-qc/9403028].
- (31) V. Iyer and R. M. Wald, A Comparison of Noether charge and Euclidean methods for computing the entropy of stationary black holes, Phys. Rev. D 52, 4430 (1995) [gr-qc/9503052].
- (32) K. Goldstein, R. P. Jena, G. Mandal and S. P. Trivedi, A C-function for non-supersymmetric attractors, JHEP 0602, 053 (2006) [arXiv:hep-th/0512138].
- (33) D. Cremades and E. Lozano-Tellechea, Holography, the second law and a -function in higher curvature gravity, JHEP 0701, 045 (2007) [arXiv:hep-th/0608174].
- (34) T. Jacobson, G. Kang and R. C. Myers, Increase of black hole entropy in higher curvature gravity, Phys. Rev. D 52, 3518 (1995) [arXiv:gr-qc/9503020].
- (35) M. M. Anber and D. Kastor, C-Functions in Lovelock Gravity, JHEP 0805, 061 (2008) [arXiv:0802.1290 [hep-th]].
- (36) S. W. Hawking, Black holes in general relativity, Commun. Math. Phys. 25, 152 (1972).