Seeing a c-theorem with holography

Robert C. Myers and Aninda Sinha

Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2J 2Y5, Canada(Received 12 June 2010; published 23 August 2010)

Using the anti-de Sitter conformal theory correspondence, we examine holographic renormalization

group flows in a framework where the bulk gravity contains higher curvature interactions. This holo-

graphic model allows us to distinguish the flow of the different central charges in dual theory. For

example, in four dimensions, one finds that the flow of the central charge a is naturally monotonic but that

of c is not. Our results agree with Cardy’s proposal to extend Zamolodchikov’s c-theorem to higher

dimensions. We are also led to formulate a novel c-theorem for a universal coefficient appearing in the

entanglement entropy of the fixed point conformal theories in any (including an odd) number of

dimensions.

DOI: 10.1103/PhysRevD.82.046006 PACS numbers: 11.25.Tq, 11.25.Hf

I. INTRODUCTION

Zamolodchikov’s c-theorem [1] is a remarkable resultfor quantum field theories in d ¼ 2. A direct outcome ofthe c-theorem is that in any renormalization group (RG)flow connecting two fixed points,

ðcÞUV � ðcÞIR: (1)

That is, the central charge of conformal field theory (CFT)describing the ultraviolet fixed point is larger than (or equalto) that at the infrared fixed point. The proof relies only onthe Euclidian group of symmetries, the existence of aconserved stress-energy tensor and unitarity in the fieldtheory.

There have been various suggestions on how such aresult might extend to quantum field theories in higher d.Cardy [2] conjectured a monotonic flow for the coefficientof the A-type trace anomaly. For even d, the trace anomalyfor a CFT is in a curved background is given by [3]

hTaai ¼

XBiIi � 2ð�Þd=2AEd (2)

where Ed is the Euler density in d dimensions and Ii are theindependent Weyl invariants of weight �d [4]. Cardy’sproposal is then that A satisfies a relation like Eq. (1) alongRG flows between two fixed points in any even d. Ofcourse, this coincides with Zamolodchikov’s result in d ¼2 where A ¼ c=12.

Efforts made towards proving Cardy’s conjectured c-theorem have focused d ¼ 4 in which case Eq. (2) containstwo terms, i.e., there is a single Weyl invariant I1 ¼CabcdC

abcd. The common nomenclature denotes the twocentral charges as c ¼ 16�2B1 and a ¼ A. Numerous non-trivial examples have been found supporting Cardy’s con-jecture, including perturbative fixed points [5] andsupersymmetric gauge theories [6]. The latter investiga-tions also demonstrated that the d ¼ 4 central charge cwillnot generically satisfy Eq. (1). Further, as we review below,support for a c-theorem in higher dimensions was found [7]

with the anti-de Sitter conformal theory (AdS/CFT) corre-spondence [8].However, a general proof is lacking and, in fact, a

counterexample to Cardy’s c-theorem in d ¼ 4 was pro-posed in [9]. If this counterexample survives further scru-tiny, there are two obvious possibilities: A ‘‘c-theorem’’exists in higher d but the quantity satisfying Eq. (1) is notthe central charge A. Alternatively, the central charge Asatisfies Eq. (1) under RG flows in higher d but only whensome more stringent requirements are imposed than in d ¼2. In the latter case, the challenge would be to identify theprecise conditions for which the analog of Eq. (1) issatisfied in higher (even) d.In the following, we find evidence for the second alter-

native within the framework of the AdS/CFT correspon-dence. We examine the c-theorem using a holographicmodel with a higher curvature gravity theory [10,11] whichallows one to distinguish the central charges a and c in thedual CFT. Hence, we are able to discriminate between thebehavior of these two central charges in RG flows, and wefind that only a has a natural monotonic flow. We are alsoable to extend our analysis to holographic CFT’s in arbi-trary higher d. Our results there suggest the followinggeneral conjecture:Placing a d-dimensional CFT on Sd�1 � R and calcu-

lating the entanglement entropy of the ground state be-tween two halves of the sphere, one finds a universalcontribution: Suniv / a�d [as detailed in Eq. (22)]. Further

in RG flows between fixed points, ða�dÞUV � ða�dÞIR.This conjecture then gives us a framework in which to

consider the c-theorem for d even or odd. As describedbelow, this conjecture actually coincides to Cardy’s pro-posal for even d. We now discuss the holographic origin ofthis conjecture.

II. HOLOGRAPHIC C-THEOREM

The AdS/CFT correspondence has emerged as a power-ful tool to study the behavior of strongly coupled CFT’s in

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diverse dimensions [8]. Within this framework, [7] consid-ered the c-theorem where one begins with (dþ 1)-dimensional Einstein gravity coupled to various matterfields:

I ¼ 1

2‘d�1P

Zddþ1x

ffiffiffiffiffiffiffi�gp ðRþLmatterÞ: (3)

The matter theory is assumed to have various stationarypoints where Lmatter ¼ dðd� 1Þ�2

i =L2 with some canoni-

cal scale L. The fixed points are distinguished by differentvalues of � as indicated by the subscript and at thesepoints, the gravity vacuum is simply AdSdþ1 with thecurvature scale given by ~L2 ¼ L2=�2

i . RG flows betweencritical points can be described with a metric of the form

ds2 ¼ e2AðrÞð�dt2 þ d~x2d�1Þ þ dr2: (4)

This metric becomes that for AdSdþ1 with AðrÞ ¼ r= ~L atthe stationary points. Now define [7]:

aðrÞ � �d=2

�ðd=2Þð‘PA0ðrÞÞd�1; (5)

where ‘‘prime’’ denotes a derivative with respect to r. Thenfor general solutions of the form (4), one finds

a0ðrÞ ¼ � ðd� 1Þ�d=2

�ðd=2Þ‘d�1P A0ðrÞd A

00ðrÞ

¼ � �d=2

�ðd=2Þ‘d�1P A0ðrÞd ðT

tt � Tr

rÞ � 0: (6)

Above in the second equality, the Einstein equations areused to eliminate A00ðrÞ in favor of components of the stresstensor. The final inequality assumes that the matter fieldsobey the null energy condition [12]. Given this monotonicevolution of aðrÞ with r and the standard connection be-tween r and energy scale in the CFT, aðrÞ always decreasesin flowing from the UV to the IR.

To make better contact with the dual CFT, it is simplestto focus the discussion on d ¼ 4 at this point. Then withthe holographic trace anomaly [13] for the AdS5 stationarypoints, one finds

aðrÞjAdS ¼ �2 ~L3=‘3P ¼ a: (7)

That is, the value of the flow function (5) matches preciselythat of the central charge a in the dual CFT at each of thefixed points. Hence with the assumption of the null energycondition, the holographic CFT’s dual to the gravity theory(3) satisfy Cardy’s c-theorem. Of course, one must add thatthe two central charges are precisely equal [13], i.e., a ¼ c,for the class of d ¼ 4 CFT’s dual to Einstein gravity.Hence these holographic models do not distinguish be-tween the flow of a and c.

It has long been known that to construct a holographicmodel where a � c, the gravity action must include highercurvature interactions [14]. In part, this motivated the

construction of quasitopological gravity [10]

I ¼ 1

2‘3P

Zd5x

ffiffiffiffiffiffiffi�gp �

12�2

L2þ Rþ �

2L2X4 þ 7�

4L4Z5

�;

(8)

where X4 is the ‘‘Gauss-Bonnet’’ interaction, a combina-tion of curvature-squared terms which corresponds to theEuler density in four dimensions, and Z5 is a particularinteraction involving curvature-cubed terms developed in[10,15]. This holographic model allows one to explore thefull three-parameter space of coefficients controlling thetwo- and three-point functions of the stress tensor in ageneral four-dimensional CFT [16]. However, one mustkeep in mind that this action (8) was not derived fromstring theory. Rather it was constructed to facilitate theexploration of a broader class of holographic CFT’s whilemaintaining control within the gravity calculations.Further, the gravitational couplings in Eq. (8) can be con-strained by various consistency requirements of the dualCFT [11]. A generalization of the model (8) to d ¼ 2CFT’s and c-theorems was considered in [17].In comparison to the action presented in [10], we have

replaced the cosmological constant term by 12�2=L2 withthe study of RG flows in mind. The idea is that as in Eq. (3)the gravity theory is coupled to a standard matter actionwith various stationary points which yield different valuesfor the parameter �2. The curvature scale of the AdS5vacua are related to the scale L appearing in Eq. (8) by~L2 ¼ L2=f1, where

�2 � f1 þ �f21 þ�f21 ¼ 0: (9)

Using standard techniques [13], one determines the centralcharges of the CFT’s dual to these AdS5 vacua [11]:

a ¼ �2 ~L3=‘3Pð1� 6�f1 þ 9�f21Þ; (10)

c ¼ �2 ~L3=‘3Pð1� 2�f1 � 3�f21Þ: (11)

Considering the metric ansatz (4), it is straightforward toconstruct two flow functions:

aðrÞ � �2

‘3PA0ðrÞ3 ð1� 6�L2A0ðrÞ2 þ 9�L4A0ðrÞ4Þ; (12)

cðrÞ � �2

‘3PA0ðrÞ3 ð1� 2�L2A0ðrÞ2 � 3�L4A0ðrÞ4Þ: (13)

These are chosen as the simplest extensions of Eq. (5) withd ¼ 4 which yield the two central charges at the fixedpoints, i.e., aðrÞjAdS ¼ a and cðrÞjAdS ¼ c. Now for ageneral RG flow solution, one finds

a0ðrÞ ¼ �3A00ðrÞA0ðrÞ cðrÞ ¼ � �2

‘3PA0ðrÞ4 ðT

tt � Tr

rÞ � 0;

(14)

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where as before we are using the gravitational equations ofmotion and assuming that the matter sector obeys the nullenergy condition. We might note that the null energycondition plays a central role in gravitational physics[12] and that violations of this condition are argued tolead to instabilities quite generally [18]. While it is pos-sible that our assumption is unduly conservative in thepresent context, we regard it as a reasonable restrictionon the matter sector in order to produce a holographicframework with a consistent dual CFT. With the latterassumption, aðrÞ evolves monotonically along the holo-graphic RG flows and we can conclude that the centralcharge (10) satisfies the analog of Eq. (1). One can alsoconsider the behavior of cðrÞ along RG flows but there is noclear way to establish that c0ðrÞ has a definite sign. Hence,this holographic model provides another broad class offour-dimensional theories which support Cardy’s proposalthat the central a (rather than any other central charge)evolves monotonically along RG flows.

Given this result and those above for Einstein gravity, itis straightforward to extend our holographic analysis ofquasitopological gravity to an arbitrary spacetime dimen-sion. Beginning with the equations of motion which areproportional to Tt

t � Trr, one can engineer the following

flow function [19]

adðrÞ � �d=2

�ðd=2Þð‘PA0ðrÞÞd�1

��1� 2ðd� 1Þ

d� 3�L2A0ðrÞ2 � 3ðd� 1Þ

d� 5�L4A0ðrÞ4

�:

(15)

By construction, adðrÞ satisfies the following:

a0dðrÞ ¼ � �d=2

�ðd=2Þ‘d�1P A0ðrÞd ðT

tt � Tr

rÞ � 0; (16)

where we again assume the null energy condition. Notethat here we must also ensure that A0ðrÞ> 0 for odd d—thedetails of this proof will be given in [20]. If we define thefixed point value as a�d � adðrÞjAdS, then

a�d ¼�d=2 ~Ld�1

�ðd=2Þ‘d�1P

�1� 2ðd� 1Þ

d� 3�f1 � 3ðd� 1Þ

d� 5�f21

�;

(17)

and the result in Eq. (16) guarantees that

ða�dÞUV � ða�dÞIR: (18)

Having found that a�d satisfies a c-theorem, one is left to

determine what this quantity corresponds to in the dualCFT. By construction for d ¼ 4, this is precisely the cen-tral charge a. Motivated by Cardy’s general conjecture foreven d, it is natural to compare a�d to the coefficient A in

Eq. (2). In fact, using the approach of [21], one readilyconfirms that there is a precise match:

a�d ¼ A for even d: (19)

Hence again, we find support for Cardy’s conjecture withthis broad class of holographic CFT’s. However, we mustseek a broader definition of a�d to also incorporate odd d.Examining the results for black hole entropy in quasito-

pological gravity [10], we observe the following: If thedual CFT is placed on a (d� 1)-dimensional hyperbolicplane (i.e., the d-dimensional space is Hd�1 � R), theenergy density �E of the ground state is negative. If atemperature is introduced and the system is heated up tothe point where �E ¼ 0, the entropy density becomes

s ¼ ð4�Þd=2�ðd=2Þa�dTd�1 ¼ 2�

�d=2�ðd=2Þ a�d

Ld�1; (20)

where L denotes the radius curvature of Hd�1. Now thisleads to an interesting question: The bulk geometry isprecisely AdSdþ1 in ‘‘unusual’’ coordinates, i.e.,

ds2 ¼ dr2

ðr2~L2 � 1Þ ��r2

~L2� 1

�dt2 þ r2d�d�1

2 ; (21)

where d�d�1 denotes the line element on Hd�1 with unitcurvature, and so why should there be any entropy at all?The answer is that the foliation of AdSdþ1 in Eq. (21)divides the boundary into two halves—see Fig. 1. Hence,the entropy is interpreted as the entanglement entropybetween these two halves—related ideas were discussedin [22]. Note that if we integrate over the hyperbolichorizon out to some maximum radius which is then giventhe conventional interpretation of a short-distance cutoff,i.e., � ¼ L2=�max, the leading contribution in Eq. (20)takes the form S / a�dðL=�Þd�2. Hence despite doing a

(d� 1)-dimensional integral, the leading divergence pro-duced by the hyperbolic geometry has precisely the power

FIG. 1 (color online). A slice of constant t through the AdSdþ1

metric in Eq. (21). This slice bears some similarity to theEinstein-Rosen bridge in a Schwarzschild black hole [12].Note that only half of the AdS boundary is reached in the limitr ! 1. The other half is reached from the second asymptoticregion ‘‘behind the horizon.’’

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expected for the ‘‘area law’’ contribution to the entangle-ment entropy in a d-dimensional CFT [23]. However, thisdivergent contribution is not universal. Rather a universalcontribution is extracted from the subleading terms. Theform of the universal contribution to the entanglemententropy depends on whether d is odd or even:

Suniv ¼� ð�Þðd=2Þ�14a�d logðL=�Þ for even d;ð�Þðd�1Þ=22�a�d for odd d:

(22)

With a Weyl transformation, the boundary metric can bebrought to that on Sd�1 � R and Eq. (22) can then beinterpreted as the universal contribution to the entangle-ment entropy between the two halves of the Sd�1 forground state of the CFT. Hence, this framework allowsus to interpret the coefficient a�d in terms of entanglement

entropy in the dual CFT for odd or even d. Further ourprevious analysis shows that this coefficient obeys a c-theorem in the holographic RG flows.

III. DISCUSSION

Although our holographic calculations here referred toquasitopological gravity for concreteness, our findingshold for a much wider class of higher curvature gravitytheories in arbitrary dimensions [20]. In all of these in-stances, we still assume that the null energy conditionholds and that the gravitational couplings are independentof the matter fields. It will be interesting to explore theextent to which these conditions can be relaxed while stillrequiring the existence of a holographic c-theorem. In anyevent, having identified this interesting behavior in the RGflows of a broad class of holographic CFT’s, it is natural toconjecture that the same c-theorem should apply morebroadly for RG flows outside of any holographic frame-work. Hence, we propose the general conjecture presentedabove in the introduction. Now one would like to deter-mine what evidence one might find for this general con-jecture, again, outside of a holographic framework.

For even dimensions, one can calculate the entangle-ment entropy for a CFT divided by a smooth boundarymaking use of the trace anomaly [23,24]. In general, theresult will depend on all of the coefficients appearing inEq. (2). However, our conjecture refers to a very specificbackground geometry and a specific boundary dividing thisspace. Applying the approach of [23,24] for this geometryin arbitrary even d, one finds that the coefficient of theuniversal entanglement entropy (22) is precisely a�d ¼ A[20]. While this matches Eq. (19) for our holographicmodel, our result here is a general statement about Sunivon Sd�1 � R with any CFT in even d. Hence our conjec-tured c-theorem coincides precisely with Cardy’s proposalfor arbitrary even d. Therefore any evidence for Cardy’sconjecture also supports the present conjecture. However,the present results also frame Cardy’s conjecture in thecontext of entanglement entropy. Taking into account both

of these perspectives should prove useful in better under-standing the precise conditions under which A satisfiesEq. (1) for higher (even) dimensions, e.g., in particular,for d ¼ 4.We emphasize that specifying the geometry in which

one calculates the entanglement entropy is an importantfeature of our conjecture. As noted above, our prescriptionfor the geometry was crucial to have Suniv / A in even d.We can infer that the entanglement entropy decreasesalong RG flows for several known examples in d ¼ 2and 3 [25–27]. However, we cannot say that these ex-amples provide direct support of our conjecture primarilybecause they do not concern the precise geometry specifiedthere. We also anticipate that with further study one canidentify other geometries for which the universal entangle-ment entropy is proportional to a�d [20]. For example,

Suniv / a for a d ¼ 4 CFT when calculated for a sphericalboundary in flat space [24].Of course, entanglement entropy has previously been

considered in the context of RG flows and c-theorems. Inparticular, [28] establishes an entropic c-theorem in d ¼ 2based purely on considerations of Lorentz symmetry andthe strong subadditivity. This construction is distinct fromour conjecture in d ¼ 2, where in fact the latter simplycoincides with Zamalodchikov’s c-theorem [1].The higher curvature terms play an important role in our

holographic analysis as they allow us to unambiguouslyidentify a�d with the A-type anomaly in any even d.

Similarly we are also able to distinguish a�d from C, the

coefficient appearing in the free energy density, i.e., f ¼CTd. This coefficient was also considered in efforts toextend the c-theorem to higher dimensions [29]. In ourholographic model, it does not appear that this coefficientalways varies monotonically in RG flows [20].In closing, we note that our calculation of the holo-

graphic entanglement entropy did not make reference tothe standard proposal conjectured by [23]. We would alsonote that this is also the first calculation of such a quantitythat includes the contributions of higher curvature inter-actions in the bulk gravity theory. However, in the Einsteingravity limit, i.e., � ¼ 0 ¼ �, our result (22) coincideswith that calculated using the standard proposal. Thepresent calculation may point the way to a more systematicderivation of holographic entanglement entropy.

ACKNOWLEDGMENTS

We thank I. Affleck, E. Fradkin, J. Gomis, S. Hartnoll,D. Kutasov, A. LeClair, M. Metlitski, C. Nayak, S.Sachdev, G. Vidal, and X.-G. Wen for discussions.Research at Perimeter Institute is supported by theGovernment of Canada through Industry Canada and bythe Province of Ontario through the Ministry of Research& Innovation. R. C.M. also acknowledges support fromNSERC, under Discovery Grant No. and funding from theCanadian Institute for Advanced Research.

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HSd d

dxffiffiffig

pEd ¼ 2. The Weyl invar-

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[19] For the reasons explained in [10], one should set � ¼ 0 ind ¼ 5 which avoids the singularity in last term of Eq. (15).

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