Universality in the shape dependence of

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Published for SISSA by Springer

Received: August 11, 2016 Revised: November 30, 2016

Accepted: December 1, 2016 Published: December 12, 2016

Universality in the shape dependence of holographic Rnyi entropy for general higher derivative gravity

Chong-Sun Chua,b and Rong-Xin Miaob

aPhysics Division, National Center for Theoretical Sciences, National Tsing-Hua University, 101 Section 2 Kuang Fu Road, Hsinchu 30013, Taiwan

bDepartment of Physics, National Tsing-Hua University, 101 Section 2 Kuang Fu Road, Hsinchu 30013, Taiwan

E-mail: mailto:[email protected]

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

Web End [email protected]

Abstract: We consider higher derivative gravity and obtain universal relations for the shape coe cients (fa, fb, fc) of the shape dependent universal part of the Rnyi entropy for four dimensional CFTs in terms of the parameters (c, t2, t4) of two-point and three-point functions of stress tensors. As a consistency check, these shape coe cients fa and fc satisfy the di erential relation as derived previously for the Rnyi entropy. Interestingly, these holographic relations also apply to weakly coupled conformal eld theories such as theories of free fermions and vectors but are violated by theories of free scalars. The mismatch of fa for scalars has been observed in the literature and is due to certain delicate boundary contributions to the modular Hamiltonian. Interestingly, we nd a combination of our holo-graphic relations which are satised by all free CFTs including scalars. We conjecture that this combined relation is universal for general CFTs in four dimensional spacetime. Finally, we nd there are similar universal laws for holographic Rnyi entropy in general dimensions.

Keywords: AdS-CFT Correspondence, Classical Theories of Gravity, Conformal Field Theory

ArXiv ePrint: 1608.00328

Open Access, c

The Authors.

Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP12(2016)036

Web End =10.1007/JHEP12(2016)036

JHEP12(2016)036

Contents

1 Introduction 1

2 Holographic Rnyi entropy for higher derivative gravity 52.1 Gauss-Bonnet gravity 52.1.1 fa(n) 6

2.1.2 fc(n) 8

2.1.3 fb(n) 9

2.2 General higher curvature gravity 112.2.1 fa(n) 14

2.2.2 fc(n) 16

2.2.3 fb(n) 16

3 The story of free CFTs 17

4 Universality of HRE in general dimensions 184.1 CFTs in three dimensions 214.2 CFTs in higher dimensions 22

5 Conclusions 26

A Equivalence between two stress tensors 26

B Solutions in general higher curvature gravity 28

C Universal laws in general dimensions 30

1 Introduction

One of the most mysterious features of quantum mechanics is the phenomena of entanglement. For system described by a density matrix , entanglement can be conveniently measured in terms of the entanglement entropy and the Rnyi entropy

SEE = Tr( log ), (1.1)Sn = 1

1 n

log Tr(n). (1.2)

For any integer n > 1, the Rnyi entropy Sn may be obtained from

Sn = log Zn n log Z1

1 n

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, (1.3)

where Zn is the partition function of the eld theory on a certain n-fold branched cover manifold. The Rnyi entropy provides a one parameter family of entanglememt measurement labeled by an integer n, from which entanglement entropy SEE can be obtained as a limit

SEE = lim

if Sn is continued to real n.

The study of entanglement entropy and the nature of quantum nonlocality has brought new insights into our understandings of gravity. It is found that entanglement plays an important role in the emergence of space-time and gravitational dynamics [15]. In addition to entanglement entropy, Rnyi entropy has drawn much attention recently, including the holographic formula of Rnyi entropy [6, 7], the shape dependence of Rnyi entropy [810], the holographic dual of boundary cones [11] and Rnyi twist displacement operator [12, 13].

Generally, for a spatial region A in a d-dimensional spacetime, the Rnyi entropy for A is UV divergent. If one organizes in terms of the short distance cuto , one nds it contain a universal term in the sense that it is independent on the UV regularization scheme one choose. In odd spacetime dimensions, the universal term is independent. In even space-time dimensions, the universal term is proportional to log and its coe cient can be written in terms of geometric invariant of the entangling surface = A. In four dimensions, the universal term of the Rnyi entropy has the following geometric expansion [14, 15],

Sunivn = log

Here the conformal invariants are

It was conjectured in [17] that

fb(n) = fc(n) (1.8)

holds for general 4d CFTs. This conjecture has passed numerical test for free scalar and free fermion [17]. According to [12], it seems that the relation (1.8) holds only for free CFTs. Evidence includes an analytic proof for free scalar. However, it is found to be violated by strongly coupled CFTs with Einstein gravity duals [9].

Sn (1.4)

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fa(n)2 R +fb(n)2 K fc(n) 2 C

. (1.5)

Z d2yCabab, K [integraldisplay] d2ytr( K2), (1.6)

where , R , Kij, Cabab are, respectively, the induced metric, intrinsic Ricci scalar, trace-less part of extrinsic curvature and the contraction of the Weyl tensor projected to directions orthogonal to the entangling surface . The shape dependence of the Rnyi entropy is described by the coe cients fa, fb, fc, which depend on n and the details of CFTs in general. The coe cient fa can be obtained by studying the thermal free energy of CFTs on a hyperboloid [6]. The coe cients fc and fb are determined by the stress tensor one-point function and two-point function on the hyperboloid background [12, 16]. Remarkably, it is found in [16] that fc is completely determined by fa:

fc(n) = nn 1

fa(1) fa(n) (n 1)fa(n)

Z d2yR , C

. (1.7)

In this paper, we apply the holographic approach developed in [9, 10, 13] to study the universal terms of the Rnyi entropy for CFTs in general spacetime dimensions that admit general higher derivative gravity duals. For 4d CFTs, expanding the coe cients (fa, fb, fc)

in powers of (n 1), we nd the leading and sub-leading terms are related to parameters

(c, t2, t4) of two point and three point functions of stress tensors [18, 19]:

fa(n) = a

c2(n 1) + c

3554 +7324t2 +1 81t4

(n 1)2 + O(n 1)3 (1.9)

fb(n) = c c

1112 +118t2 +1 45t4

(n 1) + O(n 1)2 (1.10)

fc(n) = c c

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1718 +7108t2 +1 27t4

(n 1) + O(n 1)2. (1.11)

It should be mentioned that the expansion (1.9) of fa has been obtained in [20] by using two-point and three-point function of the modular Hamiltonian. Here we provide a holographic proof of it. We note that (1.9) and (1.11) satisfy the relation (1.7). This can be regarded as a check of our holographic calculations. We also note that t2 = t4 = 0 for Einstein gravity and the eqs. (1.10), (1.11) reduce to the results obtained in [9] in this case. To the best of our knowledge, the universal dependence of fb on the coe cients t2, t4 as obtained in the relation (1.10) is new. This is one of the main results of this paper.

We remark that our holographic relations eqs. (1.9)(1.11) are also satised by free fermions and vectors.1 However, mismatch appears for free scalars. Actually, the discrepancy of fa in scalars has been observed in [20], which is due to the boundary contributions to the modular Hamiltonian. It was found that the boundary terms in the stress tensor of scalars are important at weak coupling and are suppressed in the strong coupling limit [20]. Although eqs. (1.9), (1.10), (1.11) are not satised by theories of free scalars, we nd that the following combinations

2fb(1) 3fc(1) = c [parenleftbigg]

1 + 1

12t2 +

1 15t4

, (1.12)

2fb(1) + 92fa(1) = c [parenleftbigg]

4 + 1

12t2 +

1 15t4

, (1.13)

are satised by all CFTs with holographic dual and all free CFTs including free scalars. We conjecture they are universal relations for all CFTs in four dimensions. Note that we have fc(1) + 32fa(1) = c from eq. (1.7), therefore eq. (1.12) and eq. (1.13) are not independent.

Without loss of generality, we focus on the conjecture eq. (1.12) in the rest of this paper.

In the notation of [12], our conjecture (1.12) for 4d CFTs can be written in the form

CD(1) 36hn(1) =

5 CT

1 + 112t2 +1 15t4

, (1.14)

where CT = 40

4 c, hn(n) and CD(n) are CFT data associated with the presence of the entangling surface. In general, for a d-dimensional CFT and an entangling surface

1We have assume fb(1) = fc(1) for free fermions and vectors. Numerical calculations support this assumption for free fermions [17].

(codimension 2), one denotes the coordinates orthogonal and parallel to the entangling surface by xa and y. The breaking of translational invariance in the directions transverse to can be characterized by the displacement operators Da(y). As a result, one has the following correlation functions [12]:

hTijin =

hn(n)

|xa|d

, (1.15)

hDa(y)Db(0)in = CD(n)

|y|2(d1)

. (1.16)

Here hn(n) is the coe cient xing the normalization of the one-point function for the stress tensor in the presence of the twisted operator for the n-fold replicated QFT, and CD(n)

is the normalization coe cient for the two-points correlation function of the displacement operators. In 4-dimensions, CD(n) and hn(n) related to the dependence of Rnyi and entanglement entropy on smooth or shape deformations [8, 2123]. The specic relation can be found in eqs. (2.12), (3.15), (3.19) of [12].

It should be mentioned that unlike fc and fb which are dened only in 4 dimensions, hn and CD have a natural denition in all dimensions. Therefore it is natural to ask if by using them one can generalize the results (1.12) and (1.13) to other dimensional spacetime. The holographic dual of hn and CD for Einstein gravity and Gauss-Bonnet Gravity in general dimensions are studied in recent works [10, 13]. Applying their results, one can express hn(1) and CD(1) in terms of CT and t2. Recall that we have t4 = 0 for Einstein gravity and Gauss-Bonnet Gravity. To get the information of t4, one has to study at least one cubic curvature term such as K7 and K8 in the action (2.43). Following the approach

of [10, 13], we obtain the holographic formulae of hn and CD for a d-dimensional CFT admiting a general higher curvature gravity dual:hn

CT = 2n

Mefd , (1.17)

CT =

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d2n

d + 1

(d 2)(n 1)

Me 2

, (1.18)

where Me is the e ective mass dened in eq. (4.14) and n is the coe cient in the function k(r) in eq.(4.15) which describes a deformation in the extrinsic curvature of the entangling surface. It is remarkable that these relations take simple and universal form for all the higher curvature gravity.

By using the holographic formula of hn and CD, we nd there are similar universal laws in general dimensions, which involves linear combinations of the terms CD(1), hn(1), CT , CT t2 and CT t4.2 In general, we have for a d-dimensional CFT,

hn(1)CT =

2 +1 d

(d 1)3d(d + 1) (d + 3)

2d5 9d3 + 2d2 + 7d 2 [parenrightbig]

(1.19)

+(d 2)(d 3)(d + 1)(d + 2)(2d 1)t2 + (d 2) 7d3 19d2 8d + 8

[parenrightbig]

D(1)

CT =

42 d + 1

1 d2 + dd2 d (d 2)(d 3)(d 1)2dt2 (d 2) 3d2 7d 8 (d 1)2d(d + 1)(d + 2)t4

. (1.20)

2In three dimensions, we have t2=0. And we have t2 = t4 = 0 in two dimensions.

2 )d1hn(1) is obeyed by free fermions and conformal tensor elds3 but are violated by free scalars. However similar to the 4 dimensional case, there exist universal laws that include free scalar elds. For example, in three dimensions, we nd

CD(1) 16hn(1) =

3 CT

Note that the relation CD(1) = d (d+12)(

1 + t4 30

, (1.21)

works well for free fermions, free scalars and CFTs with gravity dual. As for the universal laws in higher dimensions, please refer to eq. (4.48). It is interesting to study whether these universal laws are obeyed by more general CFTs.

The paper is organized as follows. In section2, we study 4d CFTs which are dual to general higher curvature gravity and derive the relations between the coe cients (fa(1), fb(1), fc(1)) in the universal terms of Rnyi entropy and the parameters (c, t2, t4)

of two point and three point functions of the stress tensors in the conformal eld theory. In section 3, we compare these holographic relations with those of free CFTs and nd a combined relation which agrees with all the known results of the free CFTs. We conjecture this combined relation is a universal law for all the CFTs in four dimensions. In section 4, we consider three and higher general spacetime dimensions and derive the holographic dual of hn and CD for general higher curvature gravity and discuss the universal behaves of hn(1) and CD(1). Finally, we conclude in section5.

Notations: we use x (yi) and g (ij) to denote the coordinates and metric in the bulk (on the boundary). xa and y are the orthogonal and parallel coordinates on the entangling surface. ij is the induced metric on the entangling surface. For simplicity, we focus on Euclidean signature in this paper.

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2 Holographic Rnyi entropy for higher derivative gravity

In this section, we investigate the universal terms of Rnyi entropy for 4d CFTs that are dual to general higher derivative gravity. We rstly take Gauss-Bonnet gravity as an example and then generalize the results to general higher curvature gravity. Some interesting relations between the universal terms of holographic Rnyi entropy (HRE) and the parameters of two point and three point functions of stress tensors are found.

2.1 Gauss-Bonnet gravity

For simplicity, we consider the following Gauss-Bonnet Gravity which is slightly di erent from the standard form

I = 1

16GN

R + 12l2 + (R R 4R R + R2)

[bracketrightbigg]

+ IB, (2.1)

3The conformal tensor elds appear only in even dimensions.

where

1l2 (gg gg ), (2.2)

R = R + 4l2 g , (2.3)

R = R + 20

l2 (2.4)

and IB denotes the Gibbons-Hawking-York terms which make a well-dened variational principle and the counter terms which make the total action nite.

An advantage of the above action is that, similar to Einstein gravity, the radius of AdS is exactly l. While in the standard GB and higher derivative gravity, the e ective radius of AdS is a complicated function of l, which makes the calculations complicated. Below we set l = 1 for simplicity.

2.1.1 fa(n)

Let us briey review the method to derive fa(n) [6]. We focus on the spherical entangling surface, where tr K2 and Cabab vanish. Thus only fa appears in the universal terms of Rnyi entropy eq. (1.5). The main idea is to map the vacuum state of the CFTs in a spherical entangling region to the thermal state of CFTs on a hyperboloid. The later has a natural holographic dual in the bulk, the black hole that asymptotes to the hyperboloid. Using the free energy of black hole, we can derive Rnyi entropy as

Sn = n 1 n

T0 [F (T0) F (T0/n)] (2.5)

where T0 is the temperature of hyperbolic black hole for n = 1. Further using the thermodynamic identity, S = F/T , we can rewrite the above expression as

Sn = nn 1

[integraldisplay]

p8M + (1 + 8)2r44 1. (2.9)

RM dd+1xg (d = 4 here), the quantities R are given by

R = R

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T0/n SBH(T )dT (2.6)

where SBH(T ) is the black hole entropy. For our revised GB gravity (2.1), it takes the form

SBH = 1

4GN

ZH dy3h[1 + 2(RH + 6)] (2.7)

where H denotes horizon and RH is the intrinsic Ricci scalar on horizon.

The key point in this approach is nding the black hole solution that asymptotes to the hyperboloid on the boundary. We get

ds2bulk = dr2

f(r) + f(r)d2 + r2d 23 (2.8)

where d 23 is the line element for hyperbolic plane H3 with unit curvature, and f(r) is given by

f(r) = (1 + 12)r2

Here

M = r2H 1

[parenrightbig][parenleftBigg][parenleftBigg](1

+ 10)r2H 2

[parenrightbig]

, (2.10)

and rH denotes the position of horizon, f(rH) = 0. Note that f(r) has the correct limit: it becomes that of hyperbolic black hole (black hole in Einstein gravity) when M 0

( 0). In the large r limit, the boundary metric is conformal equivalent tods2boun = d2 + d 23, (2.11)

which is the expected metric on manifold S1 H3.To determine rH, we note that the Hawking temperature on horizon is given by

T = 1

4 rf(r)|r=rH =

2n. (2.12)

From the above equation, one can easily get T0 = 1

2 for the hyperbolic black hole with f(r) = r2 1 and rH = 1. Now let us solve eq. (2.12) and express rH in terms of (n 1)

rH = 1

n 1

2 49 + 912 + 42242

243(1 + 8)2 (n1)3 +O(n1)4. (2.13)

Substituting eq. (2.13) together with T = 1

2n , T0 =

2 and RH = 6/r2H into

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3 +

(10 + 96)27(1 + 8)(n1)2

eqs. (2.6), (2.7), we obtain

Sn = V 4GN

"1 1 + 82 (n 1) +754(5 + 48)(n 1)2 [parenleftBigg]118562 + 2514 + 133

162(8 + 1) (n 1)3[bracketrightBigg]

+O(n 1)4, (2.14) where V is the hyperbolic volume, which contributes a logarithmic term V univ =

2 log [6]. Now we can extract fa from eq. (2.14) as

fa(n) = a

c2(n 1) +

754(6c a)(n 1)2 + O(n 1)3

= a

c2(n 1) + c

3554 +7 324t2

(n 1)2 + O(n 1)3, (2.15)

where a =

8GN and c =

8GN (1 + 8) [24]. In the above derivation we have used

c a

c =

16t2 +

445t4, for 4d CFTs (2.16)

and t4 = 0 for GB gravity. Clearly, eq. (2.15) agrees with eq. (1.9) when t4 = 0. To get the information of t4, one must consider more general higher derivative gravity. We leave this problem to next section. Notice that the O(n 1)3 terms of fa eq. (2.14) is a complicated

function of a and c, which implies that there is no universal relations at this and higher orders. From the viewpoint of CFTs, terms of fa at order O(n1)3 are determined by four-

point functions of stress tensor [20]. Unlike two-point and three-point functions, four-point functions of CFTs are no longer universal. Thus, it is expected that there is no universal relation at order O(n 1)3 for fa. It depends on the details of CFTs at this and higher

orders. Similarly, one expects there is no universal relation at order O(n1)2 for fb and fc.

2.1.2 fc(n)

Now let us continue to derive fc(n). We take the approach developed in [9]. In general with a deformation of the eld theory metric, the change in the partition function is govern by one-point function of the eld theory stress tensor

log Zn = 12 [integraldisplay]@M

dx4 hT ijiij. (2.17)

The main idea of [9] is to consider specic deformation of the metric so that, on using (1.3), one may isolate the required shape dependent term in the universal part of the Rnyi entropy. For example, fc can be isolated with a deformation that a ects C but not K :

Sn = log

Z d2yfc(n)2 Cabab + , (2.18)

where are non-universal terms of the Rnyi entropy. This can be achieved by considering

on the entangling surface the following metric

ds2boun = d2 + 1

d2 + (ij + Qabijxaxb + O(3))dydyj

[bracketrightbig]

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(2.19)

where Qabij describes a deformation of the metric and give rises to an amount of Cabab as

Cabab = 1

3Qaii. (2.20)

Here in (2.19), we have adopted a local coordinate system (, , y) near , where for each point on , we introduce a one-parameter family of geodesics orthogonal to parametrized by , and denotes the radial distance to along such a geodesic. (x1, x2) ( cos , sin ) and {y, i = 1, , d 2} denotes an arbitrary coordinates sys

tem on . We note that in this computation of fc, the boundary metric (2.19) is conformal equivalent to a deformed conical metric.

To proceed with the calculation of fc(n), we consider the bulk metric that asymptotes to the deformed hyperboloid background (2.19):

ds2bulk = dr2

f(r) + f(r)d2 +

r2 2

d2 + (ij + q(r)Qabijxaxb + O(3))dydyj[bracketrightbig](2.21)

where q(r) is determined by the E.O.M in the bulk and approach 1 in the limit r .

Actually, to derive fc(n), we do not need to solve the E.O.M. That is because we already have ij = r2

2 Qabijxaxb Cabab, so we only need zero order of T ij in eq. (2.17) in order to extract the terms proportional to Cabab. In other words, we only need to calculate T ij on undeformed hyperboloid background.

We note that in the context of AdS/CFT, the stress tensor that appears in (2.17) can be taken either as the regularized Brown-York boundary stress tensor [25] or the holographic stress tensor [26]. The two are equivalent as we demonstrate in the appendix. In this section, we will consider the rst approach. The key point is to nd the regularized boundary stress tensor for our non-standard GB gravity (2.1). Notice that our non-standard GB

gravity (2.1) can be rewritten into the standard form, with only the coe cients of L0 = 1 and L2 = R di erent from the standard GB:

I = 1

16GN

R + d2 dl2 + L4( R)

[bracketrightbigg]

, (2.22)

= 1

16GN

1 + 2(d 1)(d 2)

R + d2 dl2 1 + (d + 1)(d 2)

+ L4(R)

[bracketrightbigg]

[bracketleftbigg][parenleftBigg]

where L4(R) denotes the standard GB term. The holographic regularization for GB gravity

is studied in [27]. Reparameterizing their formulas, we get for the Brown-York boundary stress tensor:

8GNT ij@M = 1 + 2 (d 1)(d 2)[parenrightbig] [bracketleftbigg]

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Kij@M K@M ij (d 1) ij +

(d 3)

d 2 [parenleftbigg]

Rij@M

2 R@M ij[parenrightbigg][bracketrightbigg]

Qij 13 Q ij[parenrightbigg] , (2.23)

where (x) is the step-function with (x) = 1 provided x 0, and zero otherwise. Kij@M is the extrinsic curvature on the AdS boundary and Qij is given by

Qij = 2K@MK@MikK k

@M j

2K@MikKkl@MK@Mlj + K@Mij(K@MklKkl@M K2@M) (2.24) +2K@MR@Mij + R@MK@Mij 2Kkl@MR@Mkilj 4R@MikK

k @M j.

Here R@M denotes the intrinsic curvature on the boundary.

Substituting eq. (2.23) and ij = r22 Qabijxaxb into eqs. (2.17), (2.18), we obtain

fc(n) =

8GN

1 + 8 +

1718 32 3

(n 1) +217 + 4512 + 232322162(1 + 8) (n 1)2[bracketrightbigg]+ O(n 1)3

= c +

[parenleftbigg]

7 18a

4 3c

(n 1) + O(n 1)2

= c + c

(n 1) + O(n 1)2 (2.25)

Similar to fa(n), we have used ca

c = 16t2 +

1718 7 108t2

445 t4 for 4d CFTs and t4 = 0 for GB gravity.

Eq. (2.25) agrees with eq. (1.11) when t4 = 0. Note that eq. (2.25) and eq. (2.15) are consistent with the identity (1.7). This can be regarded as a check of our holographic calculations.

2.1.3 fb(n)

Now let us go on to calculate fb(n). The method is similar to that of fc(n): we consider the rst order variation (2.17) of the partition on the hyperboloid background deformed by an extrinsic curvature [9] and then extract fb(n) from

Sn = log

Z d2yfb(n)2 tr(K2) + . (2.26)

The main di erence from fc(n) is that now we need to calculate T ij on the deformed hyperboloid4n. This is because we have ij K, thus to extract K2 terms, we must get

T ij of order K.

To proceed, we deform the boundary hyperboloid by a traceless extrinsic curvature

ds2boun = d2 + 1

d2 + (ij + Kaijxa + O(2))dydyj

. (2.27)

Then the bulk metric becomes

ds2bulk = dr2

f(r) + f(r)d2 +

r2 2

d2 + (ij + k(r)Kaijxa + O(2))dydyj[bracketrightbig](2.28)

To get boundary stress tensor T ij of order O(K), we need to solve the E.O.M up to O(K). For traceless Kaij, we nd one independent equation

2 ff + rf 6

f + r2[parenrightbig][parenrightbig]+ f + r2[bracketrightbig] k(r)

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hrf 36 + 2f 3 [parenrightbig]

f (12 + 1)r2 4f

[parenrightbig]

+ 2r f

[bracketrightBig]

2 k(r)

k(r) = 0 (2.29)

Near the horizon, the solutions behave like k(r) (r rH)n/2. The solution is uniquely

determined by this IR boundary boundary condition and the UV boundary condition limr k(r) = 1. However, the IR boundary boundary condition k(r) (r rH)n/2

is not easy to deal with. Thus, we dene a new function

h(r) = k(r) exp[bracketleftbigg][integraldisplay]

rf2 2f + 12r + r

[parenrightbig]

dr f(r)

[bracketrightbigg]

(2.30)

and the E.O.M becomes

2(r 1)f + 4f + (12 + 1)((3r 1))

[bracketrightbig] h(r)

36 + 2f 3[parenrightbig]+ 4f

[parenrightbig] r

f + 2[parenrightbig][parenleftBigg][parenleftBigg]2f + 12r + r[parenrightbig][bracketrightbig] h(r)

2f + 12r + r[parenrightbig][bracketrightbig]h(r) = 0 (2.31)

Now the regularity condition at horizon simply requires h(rH) to be nite. Solving the above equation perturbatively, we get

h(r) = r + 1

r + h1(r)(n 1) + h2(r)(n 1)2 + (2.32)

with

h1(r) = r + 1r log

r + 1 r

+ fr

[parenrightbigg]

6r2 + 3r 1

6r3 , (2.33)

h2(r) = r + 1

2r log2 [parenleftbigg]

r + 1 r

[parenrightbigg]

6r2 + 3r 1

6r3 log

r + 1 r

[parenrightbigg]

+5 216r3 85r + 27

[parenrightbig]

r2 + 24 r r 360r3 155r + 69

[parenrightbig]

+ 4

[parenrightbig]

+ 20

[parenrightbig]

2160r7(1 + 8) , (2.34)

,where we have obtained solutions up to h5(r). For simplicity we do not list them here.

From eqs. (2.30), (2.32), (2.33), we can derive k(r). Expanding k(r) in large r, we nd

k(r) = 1

2r2 +

nr4 + O

[parenleftbigg]

1 r6

[parenrightbigg]

(2.35)

where

151104 2 + 34320 + 1945

[parenrightbig]

n =

18 +

n 1

(67 + 600 )432(1 + 8 ) (n 1)2 +[notdef][notdef]

7776(1 + 8 )2 (n 1)3

[notdef][notdef]

1362415104 3 + 471579456 2 + 54244296 + 2074355

[parenrightbig]

5598720(8 + 1)3 (n 1)4

+ 19865723572224 4 + 9304662564864 3 + 1627900276608 2 + 126143146752 + 3654194425[parenrightbig]

7054387200(1 + 8 )4 (n 1)5

+O(n 1)6. (2.36) Recall that log Z T ijij and (2.17) is calculated on the boundary with r . Thus, k(r) in the large r expansion is good enough for our purpose. Substituting eqs. (2.35), (2.28), (2.9), (2.13) into eqs. (2.23), (2.17), (2.26), we obtain

8GN fb(n) = (1 + 8 ) [parenleftbigg]

1112 + 10 [parenrightbigg] (n 1) +[notdef][notdef]

27840 2 + 5552 + 275

[parenrightbig]

JHEP12(2016)036

216(1 + 8 ) (n 1)2

[parenrightbig]

[notdef][notdef]

237436416 3 + 74097984 2 + 7667464 + 263115

155520(1 + 8 )2 (n 1)3

+ 3323533971456 4 + 1425617289216 3 + 228089069952 2 + 16137500288 + 426115725[parenrightbig]

195955200(1 + 8 )3 (n 1)4

+O(n 1)5. (2.37) Similar to fc(n), we can rewrite fb(n) in terms of a and c or c and t2. We have

fb(n) = c +

13a 5 4c

(n 1) + O(n 1)2

= c + c

(n 1) + O(n 1)2. (2.38)

To end this section, we notice an interesting property of solutions to GB gravity (2.1). Expanding in (n 1), we nd the solutions such as f(r) and h(r) are exactly the same

as those of Einstein gravity at the rst order (n 1). Di erences appear only at higher

orders. As we will prove in the next section, this is a universal property for general higher curvature gravity as long as we rescale the coe cient of R as 1.

2.2 General higher curvature gravity

In this section, by applying the methods illustrated in section2.1, we discuss the universal terms of Rnyi entropy for CFTs dual to general higher curvature gravity. In general, it is di cult to nd the exact black hole solutions in higher derivative gravity. Instead, we focus on perturbative solutions up to (n1)2. This is su cient to derive fa of order (n1)2 and

fb, fc of order (n 1). As we have argued above, it is expected that there is no universal

behavior at higher orders, due to the fact that the higher orders are determined by four and higher point functions of stress tensor, which depend on the details of CFTs.

1112 1 18t2

Let us consider the general higher curvature gravity I(R ). We use the trick in

troduced in [24] to rewrite it into the form similar as eq. (2.1). This method together with [28, 29] is found to be useful to study the holographic Weyl anomaly and universal terms of entanglement entropy [24, 3032].4 Firstly, we dene a background-curvature (we set the AdS radius l = 1 below)

~R = gg gg (2.39)

and denote the di erence between the curvature and the background-curvature by

R = R

~R . (2.40)

Then we expand the action around this background-curvature and get

I = 1

16GN

[integraldisplay]

dd+1xgL(R ) (2.41)

= 1

16GN

JHEP12(2016)036

[integraldisplay]

dd+1xg

L0+c(1)1 R+(c(2)1L4(R)+c(2)2 R R +c(2)3 R2)+8

Xi=1c(3)iKi(R)+O( R4)

[bracketrightbigg]

where L0 = L( ~R ) = L(R )|AdS is a constant dened by the Lagrangian for AdS

solution, and c(n)i are constants which parametrize the higher derivatives correction to the Einstein action up to third orders in the curvature with n denoting the order. Here L4(

denotes the GB term

L4( R) = R R 4

R R + R2, (2.42)

and Ki(

R) denotes the basis of third order curvature terms

Ki( R) = {

R3, R R R , R R R , R R R, R R R , R R R , R R ~ R ~, R R ~ R

}. (2.43)

We require that the higher derivative gravity has an asymptotic AdS solution. This would impose a condition c(1)1 = L0/2d [24]. Using this condition, we can rewrite the action (2.41) as

I = 1

16GN

L02d (R+d2d)+(c(2)1L4(R)+c(2)2 R R +c(2)3 R2)+8

Xi=1c(3)iKi(R)+O( R4) .

(2.44)

Rescaling GN

N = 2dL0 GN, c(n)i (n)i = 2dL0 c(n)i, we have

I = 1

16N

ZM (R + d2 d) + ((2)1L4(R) +(2)2 R R +(2)3 R2) +8

Xi=1(3)iKi(R) + O( R4).

(2.45)

Now it takes the form as eq. (2.1). For simplicity, we ignore the notation ~ below. The E.O.M of the above gravity is

R 2

2Lg = 0, (2.46)

4For recent discussions on entanglement entropy and the scale invariance, please see [33].

with P = L/R .

A couple of remarks on action (2.45) are in order.

Firstly, it is clear the hyperbolic black hole which is locally AdS is a solution to action (2.45). That is because R = 0 in AdS. We are interested of two kinds of perturbations: the rst one is g O(n 1) related to fa, fc, and the second one is

g O (n 1), K [parenrightbig]

related to fb. Remarkably, we have R O(n 1, K2)5 for the

deformed metric (2.28).Secondly, we are interested of the solutions up to O(n1)2 and O(K), or equivalently,

the action up to O(n 1)3 and O (n 1)2K2

. As a result, we can drop the O( R)4 terms in action (2.45) due to O( R)4 O (n 1)4, (n 1)3K2,

. Recall that the terms of

order O (n 1)aKb

[parenrightbig]

in the action contributes to terms at least of order O (n 1)a1Kb

and O (n 1)aKb1

in the E.O.M.

Thirdly, at the linear order in O(n 1, K), solutions to Einstein gravity are also

solutions to higher curvature gravity (2.45). In other words, the parameters(n)i do not appear in the solutions of order O(n1, K). Let us give a simple proof. Since Ki(

O(n1)3 and O (n1)2K2

, obviously they do not contribute to the solution at order O(n 1, K). Now we are left with three curvature-squared terms. Notice that R = 0 and R = 0

for all solutions to Einstein gravity with negative cosmological constant. Thus we only need to consider the GB term L4(

R), which contributes the following terms to the E.O.M

2L4(

JHEP12(2016)036

R)g , (2.47)

where P = L4(

R)/R . At leading order we have P

R 2

R R

4 R O (n 1)2, K2

and L4(

R) O (n 1)2, K2

. Thus, it is clear that the GB term

L4( R) does not a ect the E.O.M of order O(n 1, K). This is indeed the case as we have

seen in section 2.1. Now we nish the proof.

Finally, let us discuss the regularized boundary stress tensor of action (2.45). Let us rstly discuss the curvature-squared terms in action (2.45). Such terms are studied in [34] at the rst order of c(2)2 and c(2)3. Reparameterizing their formulas, we nd for the

Brown-York boundary stress tensor

8NT ij@M = (1 + 2c(2)1(d 1)(d 2))[bracketleftbigg]Kij@M K@M ij (d 1) ij +

(d 3)

d 2 [parenleftbigg]

Rij@M

2 R@M ij[parenrightbigg] [bracketrightbigg]

Qij 13 Q ij[parenrightbigg] , (2.48)

where d = 4 and Qij is given by eq. (2.24). Remarkably, the terms c(2)2 R R and c(2)3 R2 do not contribute to the regularized boundary stress tensor. This is actually expected since for an asymptotically AdS spacetime, we can rewrite the metric in Fe erman-Graham gauge

ds2 = g dxdx = 1

2 d

2 + 1

ijdxidxj, (2.49)

5Note that we have R proportional to O(K2) instead of O(K). The reason is as follows: K depends on the orientation, while R is orientation independent. Thus R must be proportional to even powers of K. Substituting f(r) = r2 1 and k(r) = r2 1/r into the metric (2.28), one can check that indeed

R O(K2).

+2c(2)1

where ij = (0)ij +

(1)ij + and the boundary is at

0. Near the boundary, we

have [24]

gR g O[parenleftBigg][parenleftBigg]

!, (2.50)

gL4(R) O [parenleftbigg]

1 , (2.51)

gK7(R) gK8(R) O(1), (2.52)

g R R gR R gKi6=7,8(R) gO(R4) O(). (2.53)

Clearly, only terms (2.50), (2.51) in action (2.45) are divergent and need to be regularized near the boundary. No counter terms are needed for the other terms for d = 4. In addition to the counter terms which make the action nite, one may worry about the Gibbons-Hawking-York (GHY) boundary terms which make a well-dened variational principle. For general higher curvature gravity, the GHY-like term is proposed in [35]. For Ki(

R), we have

IGHY

Z@M d4x

2 P ijK

j @M i

), (2.54)

where P = Ki(

R)/R . So the GHY-like terms for Ki(

R) are harmless. The GHY-like terms and counter terms for curvature-squared are discussed in [34], which yield eq. (2.48).

In conclusion, the regularized boundary stress tensor for higher curvature gravity (2.45) is given by eq. (2.48) in dimensions less than ve (d = 4). It should be stressed that the GHY-like terms and counter terms for K7(

R) and K8(

R) are necessary when d 6.2.2.1 fa(n)

Applying the methods of section 2.1.1, let us calculate fa(n) in general higher curvature gravity (2.45). Recall that Rnyi entropy on spherical entangling surface is given by

Sn = n

1 n

1 T0

JHEP12(2016)036

[integraldisplay]

T0/n SBH(T )dT (2.55)

with SBH(T ) the black hole entropy

SBH = 1

8GN

ZH dy3h LR . (2.56)

To suppress the massive modes and ghost modes with M 1/c(n)i, we work in pertur

bative framework with c(n)i 1. After some calculations, we nd the black hole solution as

ds2bulk = dr2

f(r) + f(r)F (r)d2 + r2d 23 (2.57)

where d 23 is the line element for hyperbolic plane H3 with unit curvature, and f(r), F (r) are given by

f(r) = r2 1 +

2(n 1)

3r2

(n 1)2,

+O(n 1)3 (2.58) F (r) = 1

8(52c(3)7 + 3c(3)8)

3(1 + 8c(2)1)r8

r6(336c(2)1 + 192c(3)7 96c(3)8 + 35) 24r2(c(2)1 + 228c(3)7 3c(3)8) + 4608c(3)7[parenrightBig]

27(1 + 8c(2)1)r8

(n 1)2 + O(n 1)3. (2.59)

From the conditions

f(rH) = 0, (2.60)

T = 1

pf(r)r[f(r)F (r)]|r=rH =12n, (2.61)

we nd a consistent solution

rH = 1

(n 1)

3 +

27(n 1)2 +

4(4c(2)1 84c(3)7 3c(3)8) 27(1 + 8c(2)1)

(n 1)2 + O(n 1)3. (2.62)

Substituting the above equations into eqs. (2.55), (2.56), we obtain

8GN

fa(n) = 1

1 + 8c(2)1

2 (n1)+

JHEP12(2016)036

154(336c(2)1 +192c(3)7 96c(3)8 +35)(n1)2 + . (2.63)

Using the following relations [32],

a =

8GN , c =

8GN (1 + 8c(2)1), (2.64)

t2 = 12

1 + 8c(2)1

(4c(2)1 192c(3)7 + 96c(3)8), (2.65)

t4 = 2160

1 + 8c(2)1

(2c(3)7 c(3)8), (2.66)

we can rewrite eq. (2.63) as

fa(n) = a

c2(n 1) + c

3554 +7324t2 +1 84t4

(n 1)2 + (2.67)

which is eq. (1.9) advertised in the Introduction.

We remark that although we work in linear order of c(n)i in the above derivation, our result eq. (2.67) applies to nite c(n)i. For the case c(2)2 = c(2)3 = 0, eqs. (2.58), (2.59), (2.62)

are exact in c(n)i. For small but non-zero c(2)2 and c(2)3, we have performed a fth order perturbation and nd that eq. (2.67) remains unchanged.

2.2.2 fc(n)

Now let us study fc(n) in higher curvature gravity (2.45). Similarly, we consider the rst order variation (2.17) of the partition function with Tij computed on the undeformed hyperboloid background. Here T ij is the regularized boundary stress tensor given by eq. (2.48). The bulk metric takes the form

ds2bulk = dr2

f(r) + f(r)F (r)d2 +

r2 2

, (2.68)

which approaches the deformed hyperboloid eq. (2.19) for q() = 1. Recalling f(r), F (r)

as given in eqs. (2.58), (2.59) and substituting all these equations together with ij =

r22 Qabijxaxb into eqs. (2.17), (2.18), we obtain

8GN

d2 + (ij + q(r)Qabijxaxb + O(3))dydyj

JHEP12(2016)036

fc(n) = (1 + 8c(2)1) + [parenleftbigg]

3 c(3)7 +

17 18

3 c(2)1

3 c(3)8[parenrightbigg]

(n 1) + . (2.69)

Applying eqs. (2.64), (2.65), (2.66), we can rewrite fc(n) as

fc(n) = c + c 1718 7108t2 1 27t4

(n 1) + . (2.70)

Notice that fa(n) (2.67) and fc(n) (2.70) are consistent with identity (1.7). This is a non-trivial check of our holographic approach, in particular, the regularized boundary stress tensor eq. (2.48).

2.2.3 fb(n)

Finally, let us discuss fb(n) in the higher derivative gravity. Similar to the case of the GB gravity, the key point is to nd deformed black hole solutions up to order O(K)

ds2bulk = dr2

f(r) + f(r)F (r)d2 +

r2 2

d2 + (ij + k(r)Kaijxa + O(2))dyidyj[bracketrightbig](2.71)

For traceless Kaij, there is one independent equation of k(r). We nd the solution at the linear order in (n1) is exactly the same as that of Einstein gravity, which agrees with the

arguments below eq. (2.46). Modications from the higher-curvature terms only appear at higher orders. Remarkably, at the next order O(n 1)2, only c(2)1, c(3)7 and c(3)8 contribute.

Following the approach of section 2.1.3, we obtain k(r) in large r expansion as

k(r) = 1

2r2 +

nr4 + O

[parenleftbigg]

1 r6

, (2.72)

where

n =

18 +

n 1

12 +

(600c(2)1 + 4224c(3)7 2112c(3)8 67)

432(1 + 8c(2)1)

(n 1)2 + O(n 1)3. (2.73)

Substituting eqs. (2.72), (2.58), (2.59) and ij =

r22 Kaijxa into (2.17) and (2.26), we

obtain

fb(n) =

8GN

(1 + 8c(2)1) +

1112 10c(2)1 + 32c(3)7 16c(3)8[parenrightbigg](n 1) +

[bracketrightbigg]

(2.74)

= c c

1112 +118t2 +1 45t4

(n 1) + (2.75)

as declared in the Introduction.

Now we have obtained fa(1), fb(1) and fc(1) by using holographic methods. Interestingly, they only depend on the parameters of stress tensor two-point and three-point functions. When c(2)2 = c(2)3 = 0, our derivations are nonperturbative in the coupling constants of higher curvature gravity. For small but non-zero c(2)2 and c(2)3, we have performed a fth order perturbation and nd that they remains unchanged. In conclusion, our obtained results eqs. (1.9), (1.10), (1.11) are universal laws for strongly coupled CFTs that are dual to general higher curvature gravity. It is expected that there are no such universal laws at the next order, since the next order terms would involve the stress energy four-point functions which no longer admit any universal form.

3 The story of free CFTs

In this section, we discuss the universal terms of Rnyi entropy for free CFTs. We nd the holographic relations found in section2 also apply to free fermions and free vectors but not to free scalars. We nd a combined relation which is obeyed by all free CFTs and strongly coupled CFTs with holographic dual. It seems that this combined relation is universal for all CFTs in four dimensions.

For the theory consisting of ns free real scalars, nf free Weyl fermions and nv free vectors, the functions fa(n) and fc(n) have been calculated explicitly in [8, 15, 3639]. We list the results as follows:

fa(n) = 1

360

JHEP12(2016)036

ns (1+n)(1+n2)4n3 +nf(1+n)(7+37n2)16n3 +nv (1+n+31n2+91n3) 2n3

, (3.1)

fc(n) = 1

120

ns (1+n)(1+n2)4n3 +nf(1+n)(7+17n2)16n3 +nv (1+n+11n2+11n3) 2n3

. (3.2)

One can check that the above fa(n) and fc(n) satisfy the identity eq. (1.7). Assuming fb(n) = fc(n), we have

fb(n) = 1

120

ns (1 + n)(1 + n2)4n3 +nf(1 + n)(7 + 17n2)16n3 +nv(1 + n + 11n2 + 11n3) 2n3

. (3.3)

This is at least the case for free scalars [12]. Numerical calculations also support fb(n) =

fc(n) for free fermions [17].

According to [40, 41], the stress tensor three-point functions for CFTs in general space-time dimensions are completely determined in terms of the three parameters A, B, C as,

CT = d/2d(d + 2) [d/2][(d 1)(d + 2)A 2B 4(d + 1)C], (3.4)

t2 = 2(d + 1)

(d 2)(d + 2)(d + 1)A + 3d2B 4d(2d + 1)C (d 1)(d + 2)A 2B 4(d + 1)C

, (3.5)

t4 =

(d + 1) d

(d + 2)(2d2 3d 3)A + 2d2(d + 2)B 4d(d + 1)(d + 2)C

(d 1)(d + 2)A 2B 4(d + 1)C

, (3.6)

JHEP12(2016)036

where for free 4d CFTs, we have

A = 8

276 (ns 54nv), (3.7) B =

2276 (8ns + 432nv + 27nf), (3.8)

C =

276 (2ns + 432nv + 27nf). (3.9)

and CT = 404 c.

Substituting eqs. (3.7)(3.6) into the holographic relations eqs. (1.9), (1.10), (1.11) for fa, fb, fc and comparing with those of free CFTs eqs. (3.1), (3.2), (3.3), we nd exact agreements for fermions and vectors. However, there is discrepancy for scalars. As noticed in [20], such discrepancy results from the boundary contributions to the modular Hamiltonian. Interestingly, we nd the following combined holographic relations

2fb(1) 3fc(1) = c [parenleftbigg]

1 + 1

12t2 +

1 15t4

[parenrightbigg]

(3.10)

2fb(1) + 92fa(1) = c [parenleftbigg]

4 + 1

12t2 +

1 15t4

[parenrightbigg]

(3.11)

are satised by all free CFTs including scalars. We conjecture these are universal laws for all CFTs in four dimensions. As mentioned in the Introduction, eq. (3.10) and eq. (3.11) are not independent, which can be derived from each other by applying eq. (1.7).

In the notation of [12], our conjecture (3.10) becomes

CD(1) 36hn(1) =

5 CT

1 + 112t2 +1 15t4

, (for 4d CFTs), (3.12)

where CT = 40

4 c for 4d. As the quantities hn and CD have natural denitions in all dimensions. It is expected that one can generalize our results to general dimensions. We will perform this analysis in the next section.

4 Universality of HRE in general dimensions

In this section, we study hn(n) and CD(n) of holographic Rnyi entropy for CFT in general d-dimensions. We rstly consider the 3d case and then discuss the case in higher dimensions. We nd that in general dimensions there are indeed similar holographic universal

laws expressing hn(1) and CD(1) in terms of a linear combination of CT , t2 and t4. And for all the examples we have checked, these holographic laws are obeyed by free fermions, but are violated by free scalars. Similar to what we did above for four dimensions, we are also able to nd a specic relation involving linearly the quantities hn(1), CD(1), CT , t2 and t4, which applies to free fermions, free scalars and strongly coupled CFTs with holographic dual. We conjecture that this relation holds for general CFTs.

To proceed, we apply the holographic approach developed in [10, 13] to derive hn(n)

and CD(n) for general higher curvature gravity. This procedure treats the extrinsic curvature perturbatively. For our purpose, we only need to consider the linear order of the extrinsic curvature below. Inspired by [10, 13], we consider the following bulk metric

ds2bulk = dr2

f(r) + f(r)F (r)d2 (4.1)

+r2

d2 + (ij + 2k(r) Kaijxa)dydyj +4d 2k(r)iKaxaddy + O(2)

where Kaij is the traceless part of extrinsic curvature and we have kKaij = jKaik+O(K2) for consistency [10]. According to [10, 13, 42], hn and CD(n) can be extracted from the boundary stress tensor

hTab(x)in =

JHEP12(2016)036

(d 1)ab dxaxb2[parenrightbigg]+ ,

hTai(x)in =

xaxb

2 iKb

knd 2

+ , (4.2)

hTij(x)in =

12 gnij + kn

Kaijxa

[parenrightbig]

+ ,

where

kn k1 =

(d 1) (d2 1)

hn2n. (4.3)

The boundary stress tensor in general higher curvature gravity has been calculated in [32], yielding that

hTiji =

dfd CT h(d)ij =

2 (d + 1)

2 2 CDn

3d 4

d 2

hn2n, and gn g1 =

d16GN 1 + 4(d 2)c(2)1

h(d)ij, (4.4)

where

fd = 2d + 1 d 1

(d + 1)d/2 (d/2), (4.5)

CT = fd

16GN 1 + 4(d 2)c(2)1

. (4.6)

and h(d)ij appears in the Fe erman-Graham expansion of the asymptotic AdS metric

ds2 = dz2z2 +

1z2 (g(0)ij + z2g(1)ij + + zdh(d)ij + )dyidyj. (4.7)

Notice that the stress-tensor eq. (4.4) contains contributions from the g(0)ij in even dimensions [26]. These contributions reect the presence of conformal anomalies. However, as argued in [10], these terms do not a ect CD(n) and hn.6 So we have ignored them in the present paper. Note also that we use a seemingly di erent stress tensor T@Mij eq. (2.48) in section 2. Actually, the stress-tensor eq. (2.48) is equivalent to eq. (4.4) up to a rescaling and some functions of g(0)ij [26]

hTiji = lim z0

1zd2 T@Mij. (4.8)

If we take the stress tensor eq. (4.4) instead of eq. (2.48) in the procedure of section 2, we get the same results for fb(n) and fc(n). The interested reader is referred to appendix A for the proof of the equivalence. Now let us focus on the stress tensor eq. (4.4) from now on.

Comparing eq. (4.4) with eq. (4.2), one can read out hn(n) and CD(n). Let us take Einstein gravity as an example. The solution is given by

f(r) = r2 1

Mrd2 , F (r) = 1, (4.9)

k(r) = r2 1r +

nrd + O

[parenleftbigg]

1 rd+1

JHEP12(2016)036

. (4.10)

From the above equations, one can easily obtain

h(d)ij =

1 2

[bracketleftbigg] [parenleftbigg]

1dM + g0

, (4.11)

where g0 and k0 are constants which are not important.7 Comparing eqs. (4.4), (4.11) with the last equation of (4.2), one obtains [10, 13]

hnn =

M8GN , (4.12)

CDn =

ij +

2dM + 2n + k0[parenrightbigg] Kaijxa

2(d 2)(n 1) M

16GN . (4.13)

Now let us turn to discuss the general higher curvature gravity (2.45). In general, it is di cult to nd the black hole solutions for higher derivative gravity. For simplicity, we work in the perturbative framework of the coupling constants c(n)i. Remarkably, we nd the solutions behaving as

f(r) = r2 1

Merd2 + O [parenleftbigg]

d (d + 1)(d 1)d/22 (d/2)

1 rd

, F (r) = 1 + O[parenleftbigg]1 r2d

, (4.14)

k(r) = r2 1r +

nrd + O

[parenleftbigg]

1 rd+1

. (4.15)

6One can easily check that g(0)ij is independent of Me and n. Thus, the contributions to the stress-tensor eq. (4.4) from g(0)ij do not a ect CD(n) and n.

7From eqs. (4.2), (4.4), (4.11), we can derive kn and gn, which have a linear dependence on the constants g0 and k0 appearing in eq. (4.11). However, we are interested of CD(n) and hn instead of kn and gn. Since CD(n) and hn are functions of (knk1) and (gng1) from eq. (4.3). They do not depend on g0 and k0 instead.

Here e denotes e ective. Using the above solutions, we can work out h(d)ij in the Fe erman-Graham expansion. Interestingly, it takes exactly the same form as that of Einstein gravity eq. (4.11), only replacing M and n by the e ective counterparts Me and n. Comparing eqs. (4.4) with eq. (4.2), we nally obtain

CT = 2n

d2n

d + 1

3 r5(8c(3)7 4c(3)8 + 5) + 4r2(c(3)8 44c(3)7) + 144c(3)7[parenrightBig](n 1)2

8r6

+O(n 1)3, (4.18) F (r) = 1

9(12c(3)7 + c(3)8)

2r6 (n 1)2 + O(n 1)2. (4.19)

One can see that these solutions obey the behaving (4.14) and the e ective mass is given by

Me = (n 1) +

916(n 1)2

Mefd , (4.16)

CT =

(d 2)(n 1)

Me 2

, (4.17)

where fd and CT are given by (4.5) and (4.6). hn and CD were rst obtained for Einstein Gravity in [13] and for Gauss-Bonnet Gravity in [10]. Here we derive them for the general higher curvature gravity. It is remarkable that, when expressed in terms of Me and the ns, the coe cients hn and CD take on these very simple universal forms (4.16), (4.17). As a rst check, our formulae agree with those of [10, 13] for Einstein gravity and Gauss-Bonnet Gravity. The holographic relations (4.16) and (4.17) are one of the main results we obtain for general dimensional CFTs. It should be mentioned that GN and c(n)i appearing in this section are actuallyN and(n)i dened in the action (2.45). For simplicity, we have ignored the notation~.

4.1 CFTs in three dimensions

In this section, we use the formulas obtained in the above section to study the universal behaves of hn(1) and CD(1) for 3d CFTs. We need to solve the E.O.M of general higher curvature to get the e ective mass Me and n. Note that the Gauss-Bonnet term is a total derivative in four-dimensional spacetime. Without loss of generality, we can set c(2)1 = 0.

After some calculations, we derive

f(r) = r2 1 +

n 1

JHEP12(2016)036

3(8c(3)7 4c(3)8 + 5)

8 (n 1)2 + O(n 1)2. (4.20)

Note that we have used the conditions f(rH) = F (rH) = 0 and T = 1

2n to x the

constants of integration for f(r) and F (r), with rH given by

rH = 1

n 1

2 +

9c(3)7(n 1)2

2 + O(n 1)3. (4.21)

Solving k(r) up to order O(n 1)2, we obtain

k(r) = r2 1r +

nr3 + O

[parenleftbigg]

1 r4

[parenrightbigg]

(4.22)

n = n 1

6 +

19c(3)7

19c(3)8

41 144

!(n 1)2 + O(n 1)3 (4.23)

Substituting eqs. (4.20), (4.23) into eqs. (4.16), (4.17), we obtain

hnCT =

243(n 1)

JHEP12(2016)036

311520(420 + t4)(n 1)2 + O(n 1)3, (4.24)

CT =

2240(100 t4)(n 1)2 + O(n 1)3. (4.25)

where we have used [32] t4 = 720(2c(3)7 c(3)8).

Now let us compare our holographic results with those of free CFTs. hn for free fermions and free scalars are calculated in [20, 36, 4345]. And it is proved in [43, 44] that CD = d (d+12)(

22(n 1)

2 )d1hn for free fermions and scalars in three dimensions. For free Dirac fermions, we have [45]

CT = 3162 , t4 = 4, hn(1) =

128, hn(1) =

1380, (4.26)

which exactly match the holographic results eqs. (4.24), (4.25). However, similar to the case of 4d CFTs, mismatch appears for free scalars. According to [20, 43], it is

CT = 3

162 , t4 = 4, hn(1) =

128, hn(1) =

332, CD(1) =

960 , CD(1) =

1780, (4.27)

for free complex scalars. It is found in [20, 45] there is discrepancy for hn(1). Here we note further that there is a discrepancy in CD(1) too. Similar to the 4d case, we nd a combination of hn(1) and CD(1),

CD(1) 16hn(1) =

3 CT

332, CD(1) =

960 , CD(1) =

1 + t4 30 [parenrightbigg]

(4.28)

which is obeyed by free scalars, free fermions and CFTs with gravity dual. In addition to free CFTs and strongly coupled CFTs with gravity dual, it is interesting to investigate whether the universal law (4.28) is obeyed by more general CFTs.

4.2 CFTs in higher dimensions

Let us go on to discuss hn and CD in higher dimensions. Similar to the cases of 3d CFTs and 4d CFTs, we need to solve the E.O.M in the bulk to get the e ective mass and n.

Then we can derive hn and CD from the general formula eqs. (4.16), (4.17).

By solving the E.O.M for the general higher curvature gravity (2.45), we obtain

f(r) = r2 1

MEin

rd2 +

c(2)1f1(r) + c(3)7f7(r) + c(3)8f8(r)

1 + 4(d 2)c(2)1

(n 1)2 + O(n 1)3

= r2 1

Merd2 + O [parenleftbigg]

1 rd

, (4.29)

F (r) = 1 + c(3)7F7(r) + c(3)8F8(r)1 + 4(d 2)c(2)1

(n 1)2 + O(n 1)3 = 1 + O [parenleftbigg]

1 r2d

, (4.30)

k(r) = r2 1r +

nrd + O

[parenleftbigg]

1 rd+1

, (4.31)

JHEP12(2016)036

where

Me =

2d 1

(n 1) +

(2d 3)(2d 1)

(d 1)3

(n 1)2

+c(2)1m1 + c(3)7m7 + c(3)8m81 + 4(d 2)c(2)1

(n 1)2 + O(n 1)3, (4.32)

and

n = 1 + 1 d(d 1)

(n 1)

4d3 8d2 + d + 2 2d2(d 1)3

(n 1)2

(n 1)2 + O(n 1)3. (4.33)

Here f1(r), f7(r), f8(r), F7(r), F8(r), m1, m2, m3, b1, b7, b8 are determined by the E.O.M. We

have worked out the solutions case by case up to d = 9. Please refer to the appendix for these solutions. In summary we obtain:

hnCT = 2

+c(2)1b1 + c(3)7b7 + c(3)8b81 + 4(d 2)c(2)1

2 +1 (d2)

(d + 2)(n 1) +

hn(1)

2CT (n 1)2 + O(n 1)3, (4.34)

CT =

22d + 1(n 1) +

CD(1)

2CT (n 1)2 + O(n 1)3, (4.35)

with h

n(1)

CT and C

D(1)

CT given by

hn(1)

CT =

2 +1 d

(d 1)3d(d + 1) (d + 3)

2d5 9d3 + 2d2 + 7d 2 [parenrightbig]

(4.36)

+(d 2)(d 3)(d + 1)(d + 2)(2d 1)t2 + (d 2) 7d3 19d2 8d + 8

[parenrightbig]

CD(1)

CT =

42 d + 1

1 d2 + dd2 d (d 2)(d 3)(d 1)2dt2 (d 2) 3d2 7d 8 (d 1)2d(d + 1)(d + 2)t4

. (4.37)

Note that the coe cients of t2 and t4 (t2) in hn(1) and CD(1) (4.36), (4.37) vanish when d = 2 ( d = 3). This is the expected result, which can be regarded as a check of our general formula (4.36), (4.37). One can also check that the general formulas (4.36), (4.37) reproduce

the results of 3d and 4d CFTs. We remark that the holographic formula of hn(1) (4.36) agrees with the those of [20, 45], which are derived by using three-point functions of stress tensor. As they have checked, the relation (4.36) for hn(1) works well for free fermions (up to d = 12) but not for free scalars (d > 2).

Before we end this section, let us make some comments about the possible universal relation between CD and hn. For general dimensions, the generalization of the 4d conjecture (1.8) is the statement [12]:

CD(n) = d

d + 1 2

[parenrightbigg] [parenleftbigg]

d1 hn(n). (4.38)

This relation can be motivated by the observation that if one assume (4.38) holds for free fermions and conformal tensor elds, one can prove CD(1) of these elds exactly match the holographic formula (4.37). Turning the logic around, if one assume free fermions and conformal tensor elds obey the holographic formulas (4.36), (4.37),8 one can prove that the weaker relation

CD(1) = d

d + 1 2

d1 hn(1) (4.39)

holds in general dimensions. In proving these, we have found useful the relations (4.6), (3.5), (3.6) and that

A = 1

S3d

[bracketleftbigg]

d3(d 1)3

[parenrightbigg] [parenleftbigg]

JHEP12(2016)036

d3d 3

, (4.40)

B =

1 S3d

(d 2)d3(d 1)3ns + d22f +(d 2)d3d 3t

, (4.41)

C =

1 S3d

(d 2)2d24(d 1)3ns + d24f +(d 2)d3 2(d 3)t

, (4.42)

where Sd = 2d/2/ (d2),f = tr(1)nf = 2[d/2]nf, nf is the number of Dirac fermion, tr is the Dirac trace andt denotes the number of degrees of freedom contributed by the (n 1)-form in even dimensions d = 2n [19]. However incompatiblity arises in the

scalar sector as before. Indeed using (4.6), (3.5), (3.6) and (4.40)(4.42) in the holographic formulas (4.36), (4.37), we nd for a free theory with ns scalars,

CD(1) d [parenleftbigg]

d + 1 2

[parenrightbigg] [parenleftbigg]

d1 hn(1) =(d 2)42d

d 2

216(d 1)3ns 6= 0. (4.43)

Thus problems only appear for scalars. This equation shows that the relation (4.39) and the holographic formulas (4.36), (4.37) cannot both be satised at the same time by free scalars.

8This is indeed the case at least in three dimensions for free fermions.

That the relation (4.38) is not compatible with the holographic results can also be seen from the consideration of the positivity constraints [19, 46] for CFTs in general dimensions:

Scalar Constraint : 1 + d 3

d 1

t2 + d2 d 4 d2 1

t4 0, (4.44)

Vector Constraint : 1 + d 3

2(d 1)

2 d2 1

t4 0, (4.45)

Tensor Constraint : 1

1d 1

2 d2 1

t4 0. (4.46)

These constraints are consequences of the requirement of the positivity of the energy uxes. Now it is easy to compute from (4.36), (4.37) that

CD(1) d [parenleftbigg]

d + 1 2

JHEP12(2016)036

[parenrightbigg] [parenleftbigg]

d1 hn(1)

1 + d 3d 1t2 + d2 d 4d2 1t4[parenrightbigg] 0, (4.47)

where in the last step we have used the unitarity constraint CT 0 and the scalar con

straint (4.44). This shows that, unless d = 2 or if the scalar constraint is saturated,9 the relation (4.39) and our holographic results (4.36), (4.37) cannot both be satised at the same time.

All in all, it is therefore interesting to look for a di erent relation between CD(1) and hn(1) like those of (1.12) for the 4d case and (1.21) for the 3d case, that would hold for all free theories as well as strongly coupled dual theories. To do so, we need the information of hn(1) and CD(1) of free scalars. hn(1) of free scalars is discussed in [37, 43, 45] in general dimensions. However, so far we do not know CD(1) in dimensions higher than four (d > 4).

On the other hand, if we assume (4.39) holds for free scalars in general dimensions as has been suggested in [12], then we obtain

CD(1) 2(d 1) [parenleftbigg]

d + 1 2

= CT 22(d 2)(d 1)2d(d + 1)

[parenrightbigg] [parenleftbigg]

d1 hn(1)

= 42(d 2) d(d + 1) CT

1 + d 3d(d 1)t2 + 4 d2 2d 2

(d 1)d(d + 1)(d + 2) t4

, (4.48)

which is such a universal law obeyed by free scalars, free fermions, free conformal tensor elds and CFTs with holographic dual. Please refer to the appendix for the derivation of eq. (4.48). As a quick check, eq. (4.48) reproduces (1.12) and (1.21) for 4d and 3d CFTs, respectively. It is interesting to nd out if the universal law (4.48) is indeed valid for general CFTs. We leave this interesting problem and related questions for future work.

In summary, our holographic results (4.36), (4.37) are obeyed by free fermions and conformal tensors but are violated by free scalars. According to [12], it seems that the free CFTs satisfy (4.39). However, as we have proven above, this relation does not agree with

9The relation between (4.39) and lower bound of unitarity constraint (which is equivalent to the scalar constraint ) is observed for Gauss-Bonnet gravity for d = 4, 5, 6 in [10]. Here we nd this is a universal property for general higher curvature gravity in general dimensions.

eqs. (4.36), (4.37). So neither the relation (4.39) nor the holographic relations (4.36), (4.37)) can be universally true for all CFTs. Instead, we nd that the suitably combined relation (4.48) is satised by free CFTs (including scalars) as well as by CFT with holographic duals, and stands a chance to be a universal relation satised by all CFTs.

5 Conclusions

In this paper, we have investigated the universal terms of holographic Rnyi entropy for 4d CFTs. Universal relations between the coe cients fa(1), fb(1), fc(1) in the logarithmic terms of Rnyi entropy and the parameters c, t2, t4 of stress tensor two-point and three-point functions are found. Interestingly, these relations are also obeyed by weakly coupled CFTs such as free fermions and vectors but are violated by scalars. Similar to the case of fa(1) [20], one expects that the discrepancy for scalars comes from the boundary contributions to the modular Hamiltonian. Remarkably, We have found that there is a combination of our holographic relations which is satised by all the free CFTs including scalars. We conjecture that this combined relation (1.14) is universal for general CFTs in four dimensional spacetime. For general spacetime dimensions, we obtain the holographic dual of hn and CD for general higher curvature gravity. Our holographic results together with the positivity of energy ux imply CD(1) d (d+12)(

2 )d1hn(1). And the equality is satised by free fermions and the conformal tensor elds if they obey the holographic universal laws. We also nd there are similar holographic universal laws of hn(1) and CD(1). By assuming (4.39) for free CFTs, we nd that for general dimensions, the relation (4.48) is obeyed by all the free CFTs as well as by CFTs with holographic duals. It is interesting to test these universal laws by studying more general CFTs. We leave a careful study of this problem to future work.

Acknowledgments

R. X. Miao thank Yau Mathematical Sciences Center for hospitality during the early stages of this work. In particular, R. X. Miao wish to thank Prof. W. Song, Q. Wen and J. F. X for helpful discussions and kind help during the stay at YMSC. This work is supported in part by the National Center of Theoretical Science (NCTS) and the grant MOST 105-2811-M-007-021 of the Ministry of Science and Technology of Taiwan.

A Equivalence between two stress tensors

In the analysis in the main text, we have considered in section 2 the Brown-York boundary stress tensor eq. (2.48) in section 2, and in section 4 the holographic stress tensor eq. (4.4). As we have mentioned in section 4, they are actually equivalent up to a rescaling and some functions of g(0)ij [26] that are irrelevant:

hTiji = lim z0

JHEP12(2016)036

1zd2 T@Mij. (A.1)

Here the l.h.s. is the holographic stress tensor and the r.h.s. is the Brown-York boundary stress tensor. In this appendix, we shall prove that, by applying the stress tensor eq. (4.4) instead of eq. (2.48) in the approach of section 2, we obtain the same results for fb(n) and

fc(n). This is can be regarded as a double check of our results.

The key point in section 2 is that the change in the partition function is govern by the stress tensor one-point function

log Zn = 1

2 Z@M dx4Tij@M ij (A.2)

From eq. (A.1) and the asymptotic AdS metric in the FG gauge eq. (4.7), one can rewrite it in terms of hTiji and g(0)ij as

log Zn = 1 2

[integraldisplay]

dx4

pg(0)hT ijig(0)ij (A.3)

The boundary metric g(0)ij is given by (2.18) of [10]

ds2 = d2 + 1

2 d2 + [ij + 2

Kaijxa + Qabijxaxb]dydyj

[parenrightbig]

+ O(K2). (A.4)

Actually, we can ignore the Q terms above, since it is of order O(K2). For simplicity, we focus on the case of traceless extrinsic curvature Kii = 0 as in section2. Using eqs. (4.4), (4.11), we can derive the stress tensor in j components for 4d CFTs as

hTiji =

4f4 CT h(4)ij =

JHEP12(2016)036

4f4 CT

1 2

[bracketleftbigg] [parenleftbigg]

14Me + g0

ij +

12Me + 2n + k0[parenrightbigg] Kaijxa

[bracketrightbigg]

+ O(K2).

(A.5)

From the above two equations, we get

hTji =

4f4 CT h(4)j =

4f4 CT 2

[bracketleftbigg] [parenleftbigg]

14Me + g0

j +

12Me + 2n + k0 4g0[parenrightbigg] Kjxa

[bracketrightbigg]

+ O(K2).

(A.6)

12 (2 Kaijxa + Qabijxaxb) into eq. (A.3), we get

log Zn = 12 [integraldisplay]@M

dx4g0Tjg(0)ij (A.7)

Integrating eq. (A.7) and selecting the logarithmic divergent terms, we obtain

log Zn = log

Z dy22nf4 CT [bracketleftbigg] 3

14Me + g0

Substituting eq. (A.6) and g(0)ij =

[bracketrightbigg]

(A.8)

where we have used Cabab = 13Qiai in the above derivations. Using eq. (A.8) and Me(1) = 0, we obtain the logarithmic divergent terms of Rnyi entropy

Sn = log Zn n log Z1

1 n = log n

n 1

f4 CT [integraldisplay]

Cabab +

12Me + 2n + k0 4g0

tr K2

dy2

32MeCabab + (Me + 4(n 1))tr K2

[bracketrightbigg]

(A.9)

= log

Z dy2 [bracketleftbigg]fb(n)2 trK2 fc(n)2 Cabab[bracketrightbigg]. (A.10)

Notice that the constants g0 and k0 are canceled automatically in the above calcultions. Comparing eq. (A.9) and eq. (A.10) and using CT /f4 = c

22 , we nally obtain

fb(n) = n(4n 41 Me)

n 1

c, (A.11)

fc(n) =

3nMe 2(n 1)

c. (A.12)

Recall that Me and n are given by

Me =

3(n 1) +

JHEP12(2016)036

(336c(2)1 + 192c(3)7 96c(3)8 + 35) 27(1 + 8c(2)1)

(n 1)2 + O(n 1)3 (A.13)

n =

18 +

12 +

(600c(2)1+4224c(3)72112c(3)867)

432(1 + 8c(2)1)

(n1)2+O(n1)3. (A.14)

Substituting eqs. (A.13), (A.14) and c =

8GN (1 + 8c(2)1) into eqs. (A.11), (A.12), we reproduce the results (2.69) and (2.74) in section 2. So the stress tensor eq. (4.4) indeed yields the same results for 4d CFTs as the stress tensor eq. (2.48).

Substituting hn/(n 1) =

23 fc(n), CD/(n 1) = 162 fb(n) and c = 4CT /40 [12] into eqs. (A.11), (A.12), one can also reproduce hn and CD eqs. (4.16), (4.17) for 4d CFTs in section 4.

B Solutions in general higher curvature gravity

In this appendix we provide the solutions to E.O.M of the general higher curvature gravity (2.45), which are found to be useful for the derivations of holographic hn and CD in section 4.2. For simplicity, we work in the perturbative framework of the coupling constants c(n)i. To derive hn(1) and CD(1) in terms of CT , t2, t4, we can further set c(2)1 = 0 in dimensions except d = 4. The solutions for d = 3 and d = 4 are given in section 4.1 and section 2.2. Below we list the key results for d = 5, 6, 7, 8, 9 up to O(n 1)2.

5d CFTs:

f(r) = r2 1 +

n 1

2r3

3 3r7(104c(3)7 34c(3)8 + 7) 8r2(1044c(3)7 9c(3)8) + 7200c(3)7[parenrightBig]

64r10 (n 1)2 + O(n 1)3,

F (r) = 1

45(22c(3)7 + c(3)8)

4r10 (n 1)2 + O(n 1)3, (B.1)

k(r) = r2 1r +

1 r5

(n 1)(160 (n 1)(30840c(3)7 + 15990c(3)8 + 307))

3200 + O

[parenleftbigg]

1 rd+1

6d CFTs:

f(r) = r2 1 +

2(n 1)

5r4

3 r8(992c(3)7 232c(3)8 + 33) 80r2(328c(3)7 2c(3)8) + 23040c(3)7[parenrightBig]

125r12 (n 1)2 + O(n 1)3,

F (r) = 1

72(132c(3)7 + 5c(3)8)

25r12 (n 1)2 + O(n 1)3, (B.2)

k(r) = r2 1r +

1 r6

(n 1)(75 2(n 1)(10548c(3)7 + 5733c(3)8 + 73))2250 + O

[parenleftbigg]

1 rd+1

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7d CFTs:

f(r) = r2 1 +

n 1

3r5

+ r9(7320c(3)7+1320c(3)8143)+900r2(220c(3)7c(3)8)176400c(3)7[parenrightBig]

216r14 (n1)2+O(n1)3,

F (r) = 1

35(92c(3)7 + 3c(3)8)

6r14 (n 1)2 + O(n 1)3, (B.3)

k(r) = r2 1r +

1 r7

(n 1)((n 1)(192696c(3)7 109704c(3)8 989) + 504)

21168 + O

[parenleftbigg]

1 rd+1

8d CFTs:

f(r) = r2 1 +

2(n 1)

7r6

3 r10(5088c(3)7744c(3)8+65)504r2(284c(3)7c(3)8)+129024c(3)7[parenrightBig]

343r16 (n1)2+O(n1)3,

F (r) = 1

144(244c(3)7 + 7c(3)8)

49r16 (n 1)2 + O(n 1)3, (B.4)

k(r) = r2 1r +

1 r8

(n 1)((n 1)(194304c(3)7 115392c(3)8 773) + 392)

21952 + O

[parenleftbigg]

1 rd+1

9d CFTs:

f(r) = r2 1 +

n 1

4r7

3 r11(9464c(3)71162c(3)8+85)784r2(356c(3)7c(3)8)+254016c(3)7[parenrightBig]

512r18 (n1)2+O(n1)3,

F (r) = 1

189(39c(3)7 + c(3)8)

8r18 (n 1)2 + O(n 1)3, (B.5)

k(r) = r2 1r +

1 r9

(n 1)((n 1)(715608c(3)7 441234c(3)8 2279) + 1152)

82944 + O

[parenleftbigg]

1 rd+1

Using these solutions, we can derive Me and n from eqs. (4.14), (4.15) as

Me =

2d 1

(n 1) +

(2d 3)(2d 1)

(d 1)3

(n 1)2

+12(d 2) d3 6d2 + 11d 4

(d 1)3

c(3)7(n 1)2

3(d 2) 3d2 9d + 4

(d 1)3c(3)8(n 1)2 + O(n 1)3, (B.6)

and

n = 1 + 1 d(d 1)

(n 1)

4d3 8d2 + d + 2 2d2(d 1)3

(n 1)2

+6 d4 + 2d3 21d2 + 36d 16 (d 1)3d

c(3)7(n 1)2

3 4d4 17d3 + 35d2 40d + 16 2(d 1)3d

c(3)8(n 1)2 + O(n 1)3. (B.7)

Recall that we have

c(3)7 =

2 d2 + 3d + 2

t2 + (7d + 4)t4 12(d 1)d (d3 d2 10d 8)

, (B.8)

JHEP12(2016)036

[parenrightbig]

c(3)8 =

d2 + 3d + 2

t2 + (3d + 4)t4 3d (d4 2d3 9d2 + 2d + 8)

. (B.9)

Substituting eqs. (B.6)(B.9) into the holographic formula (4.16), (4.17), we can derive hn and CD eqs. (4.34)(4.17) in section 4.2.

C Universal laws in general dimensions

hn for free comformally coupled scalars in even-dimensional space-time are calculated in [45]

hn = (2)1d

d 1

(d4)/2

[parenrightbig]

Xj=0a(0)j,l1(2j d + 1)2jd/2(n2jd+1 n) [parenleftbigg]d2 j

(d 2j), (C.1)

where d = 2l + 2 and a(0)j,l1 are dened by

P (0)l1(t) =

Xj=0 a(0)j,l1tj

= lim

(4t)l

1 2 sinh

l exp

2 4t

. (C.2)

The rst few polynomials are given by

P (0)0(t) = 1,

P (0)1(t) = 1 + 2t

3 ,

P (0)2(t) = 1 + 2t + 16t2

15 ,

P (0)3(t) = 1 + 4t + 28t2

5 +

96t3

35 ,

P (0)4(t) = 1 + 20t

3 +

52t2

3 +

1312t3

63 +

1024t4

105 ,

P (0)5(t) = 1 + 10t + 124t2

3 +

5560t3

63 +

30656t4

315 +

10240t5

231 ,

P (0)6(t) = 1 + 14t + 84t2 + 2488t3

9 +

4736t4

9 +

30208t5

55 +

245760t6

1001 . (C.3)

From eq. (C.1), it is easy to derive hn(1) as

hn(1) = (2)1d

d 1

(d4)/2

Xj=0a(0)j,l1(2j d + 1)2(2j d)2jd/2 [parenleftbigg]d2 j

(d 2j), (C.4)

Unfortunately, now we do not have a formula of CD(n). It seems that CD(1) = d (d+12)(

2 )d1hn(1) holds for free scalars in general dimensions [12]. This is at least the case for d = 3 and d = 4. With this assumption, now we are ready to derive the universal law (4.48).

We require that the universal laws are obeyed by both the free scalars and the holo-graphic CFTs. From eqs. (C.4), (4.34), (4.37) and the assumption mentioned above, we nially obtain the universal law (4.48) in general dimensions. In the derivations, we have used the following useful formula

(d4)/2

Xj=0a(0)j,l14d(d + 2)(d 2j 1)2d+2j

JHEP12(2016)036

2 d+3

[parenrightbig]

2 (d 2j)

d 2

j + 1

2 = 1. (C.5)

Using eq. (C.3), we veried that this identity holds up to d = 16.

It should be mentioned that although we focus on even dimensions in the above discussions. We have checked that the universal law (4.48) produces correct result (1.21) in three dimensions. So it is expected that eq. (4.48) works well in general odd dimensions too.

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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Abstract

We consider higher derivative gravity and obtain universal relations for the shape coefficients (f _a , f _b , f _c) of the shape dependent universal part of the Rényi entropy for four dimensional CFTs in terms of the parameters (c, t ₂ , t ₄) of two-point and three-point functions of stress tensors. As a consistency check, these shape coefficients f _a and f _c satisfy the differential relation as derived previously for the Rényi entropy. Interestingly, these holographic relations also apply to weakly coupled conformal field theories such as theories of free fermions and vectors but are violated by theories of free scalars. The mismatch of f _a for scalars has been observed in the literature and is due to certain delicate boundary contributions to the modular Hamiltonian. Interestingly, we find a combination of our holographic relations which are satisfied by all free CFTs including scalars. We conjecture that this combined relation is universal for general CFTs in four dimensional spacetime. Finally, we find there are similar universal laws for holographic Rényi entropy in general dimensions.

Details

Title

Universality in the shape dependence of holographic Rényi entropy for general higher derivative gravity

Author

Chu, Chong-sun; Miao, Rong-xin

Pages

1-34

Publication year

2016

Publication date

Dec 2016

Publisher

Springer Nature B.V.

e-ISSN

10298479

Source type

Scholarly Journal

Language of publication

English

DOI

https://doi.org/10.1007/JHEP12(2016)036

ProQuest document ID

1848098286

Universality in the shape dependence of holographic Rényi entropy for general higher derivative gravity

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