Higher spin entanglement entropy at finite temperature with chemical potential

Abstract

(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image)

Abstract

It is generally believed that the semiclassical AdS₃ higher spin gravity could be described by a two dimensional conformal field theory with ......-algebra symmetry in the large central charge limit. In this paper, we study the single interval entanglement entropy on the torus in the CFT with a ...... deformation. More generally we develop the monodromy analysis to compute the two-point function of the light operators under a thermal density matrix with a ...... chemical potential to the leading order. Holographically we compute the probe action of the Wilson line in the background of the spin-3 black hole with a chemical potential. We find exact agreement.

Published for SISSA by Springer Received: May 4, 2016 Accepted: June 26, 2016 Published: July 11, 2016

Higher spin entanglement entropy at nite temperature with chemical potential

Bin Chena,b,c,d and Jie-qiang Wua

aDepartment of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University,

Beijing 100871, P.R. China

bCollaborative Innovation Center of Quantum Matter,

5 Yiheyuan Rd, Beijing 100871, P.R. China

cCenter for High Energy Physics, Peking University,

5 Yiheyuan Rd, Beijing 100871, P.R. China

dBeijing Center for Mathematics and Information Interdisciplinary Sciences, Beijing 100048, P.R. China

E-mail: mailto:[email protected]

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

Web End [email protected]

Abstract: It is generally believed that the semiclassical AdS3 higher spin gravity could be described by a two dimensional conformal eld theory with W-algebra symmetry in

the large central charge limit. In this paper, we study the single interval entanglement entropy on the torus in the CFT with a W3 deformation. More generally we develop the

monodromy analysis to compute the two-point function of the light operators under a thermal density matrix with a W3 chemical potential to the leading order. Holographically

we compute the probe action of the Wilson line in the background of the spin-3 black hole with a chemical potential. We nd exact agreement.

Keywords: AdS-CFT Correspondence, Conformal and W Symmetry, Higher Spin Gravity, Higher Spin Symmetry

ArXiv ePrint: 1604.03644

Open Access, c

[circlecopyrt] The Authors.

Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP07(2016)049

Web End =10.1007/JHEP07(2016)049

JHEP07(2016)049

Contents

1 Introduction 1

2 R enyi entropy at nite temperature 52.1 Conformal block from the monodromy 62.2 Conformal block at nite temperature 8

3 Correlation function at nite temperature with higher spin deformation 103.1 Picture-changing transformation 113.2 Monodromy problem 13

4 Monodromy analysis 174.1 Thermodynamics of the ensemble 174.2 Two-point function 18

5 Holographic computation 215.1 Wilson line probe action 215.2 Holographic correspondence for picture transformation 22

6 Conclusion and discussion 24

A W3 algebras 25

1 Introduction

The AdS/CFT correpondence provides a new tool to study the entanglement entropy. It was proposed by Ryu and Takayanagi that the entanglement entropy in the conformal eld theory (CFT) with a gravity dual can be evaluated by the area of a minimal surface in the bulk [1]

SHEE = A4GN : (1.1)

The Ryu-Takayanagi(RT) formula (1.1) de nes the holographic entanglement entropy (HEE), which implies a deep and intriguing relation between the entanglement and the quantum gravity. The HEE could be understood as a generalized gravitational entropy [2], as suggested by the similarity of (1.1) with the Bekenstein-Hawking entropy of the black hole.

On the other hand, the holographic entanglement entropy opens a new window to study the AdS/CFT correspondence. Especially in the AdS3/CFT2 correspondence, the semiclassical AdS3 gravity is dual to the large c limit of the two-dimensional conformal eld theory. In this context, under reasonable assumptions the Ryu-Takayanagi formula

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JHEP07(2016)049

has been derived in both the bulk [3] and the CFT [4] in AdS3/CFT2. On the CFT side, the partition function of the n-sheeted Riemann surface could be simpli ed in the large c limit. In fact, under this limit, the conformal block of multi-point functions could be dominated by the vacuum block, which allows one to solve the conformal block in the leading order using the monodromy techniques. On the dual bulk side, loosely speaking, the classical handle-body solution ending on the n-sheeted Riemann surface could be constructed, and its on-shell action reproduces the leading order CFT partition function [5]. Moreover, it has been shown that the 1-loop correction to the RT formula in the bulk is captured exactly by the next-leading order contribution in the CFT partition function [6{17]. This is due to the fact that the 1-loop partition function of any handle-body con guration [18, 19] could be reproduced by the CFT partition function [20].

Furthermore the study of the entanglement entropy sheds light on the correspondence between the higher spin(HS) gravity and the CFT with W symmetry. In the rst order

formulation of the AdS3 gravity, the theory could be rewritten in terms of the Chern-Simons(CS) theory with the gauge group SL(2; R) [notdef] SL(2; R) [21]. By generalizing the

gauge group from SL(2; R) to SL(N; R), the higher spin theory up to spin N in AdS3 could be constructed in the Chern-Simons formulation. The construction could be extended to the full higher spin algebra hs[ ]. More interestingly, by imposing the generalized Brown-Henneaux asymptotic boundary condition, the asymptotic symmetry group of the higher spin theory turns out to be generated by the WN algebra. This suggests that the higher spin AdS3 gravity could be dual to a 2D CFT with W-algebra [22, 23]. One typical feature

of the higher spin gravity is the loss of the di eomorphism invariance. As a result, the usual geometrical notion like the horizon, the singularity and the area make no much sense. As a result, the RT formula (1.1) may not be able to compute the HEE in a higher spin theory. More precisely, if one focus on the vacuum of the dual CFT, then the dual con guration is still gravitational and the higher spin elds appear only as the uctuations around the classical con gurations. In this case, one can still applies the RT formula and the higher spin uctuations contribute only at the next-leading order [12, 13, 16]. However, if one considers the highly excited states with W charge, then the dual con guration could be a higher spin black hole. Now the RT formula does not apply and one has to nd a new way to compute HEE.

One promising proposal for HEE in the higher spin AdS3 gravity is to use the Wilson line [24, 25]. As the theory is de ned in the framework of Chern-Simons theory, it is natural to consider the Wilson line operator de ned in terms of the gauge potential. By considering the Wilson line which ends on the branch points of the interval, it was proposed that the probe action of the Wilson line captures the entanglement entropy. As a consistency check, the entropy of the HS black hole has been reproduced. Furthermore, it was shown in [26] that the WL evaluated in a general asymptotically AdS background captures correctly the correlation function in the dual CFT. This puts the WL proposal on a rmer footing.

Let us review the work in [26] in more details. On the dual eld theory, the four-point correlation function, involving two heavy and two light operators, was considered. The heavy operator corresponds to the higher spin black hole or the conical defect with the higher spin hair. The light operator with conformal dimension c could be taken to

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be the twist operator in the n ! 1 limit. Therefore this four-point correlation function

encodes the single-interval entanglement entropy of a highly excited states. From the operator product expansion (OPE), this four-point function can be decomposed into the contributions from the propagating states in di erent modules. The contribution from each module is called the conformal block. For a CFT with W symmetry, the states in

the theory are classi ed by the representations of W algebra. Considering the large c

limit of the CFT with a sparse light spectrum, only vacuum conformal block dominates the four-point function. By studying the monodromy problem of a di erential equation, the classical order of the conformal block can be computed. Remarkably, as shown in [26] explicitly this W3 vacuum block can be computed even more e ciently by the bulk WL in

the AdS3 background corresponding to the heavy operator.

However, there are two subtleties in the study in [26]. First of all, the higher spin black hole solution usually contains two terms:

a = azdz + azd

z; (1.2)

for the holomorphic boundary condition [38], and

a = azdz + atdt; (1.3)

for the canonical boundary condition [27]. The rst term az contains the charge, while the second term az or at contains the chemical potential [28]. In the higher spin black hole solution, the asymptotic condition is di erent from the one for pure AdS. The boundary condition corresponds to the higher spin deformation in the eld theory. As shown in [27], there are two kinds of deformations to the CFT: the canonical deformation and the holomorphic deformation, corresponding to di erent asymptotic boundary conditions. For the holographic entanglement entropy we should evaluate the probe action of the WL in terms of the gauge potential, whose boundary condition should be in accord to the deformation in the eld theory. However, in [26], the chemical potential has been turned o in the holographic calculation.1 In other words, only the gauge potential including only az was discussed. Correspondingly, there is no deformation in CFT side such that the monodromy analysis is easy to do. Generically speaking, when there is a deformation in CFT, the entanglement entropy is hard to compute [29{31].

The second subtle point in [26] is that the higher spin black hole microstate was regarded to be created by a heavy operator in the CFT. On the other hand, it is quite often to use CFT at a nite temperature to represent a black hole. At the leading order, both pictures could be indistinguishable, but not at the quantum level [32]. It would be interesting to study the HEE in a higher spin black hole background in the nite temperature picture. This is the issue we want to address in this paper.

In this paper, we study the single-interval entanglement entropy at a nite temperature and with a higher spin chemical potential. We use a thermal density matrix with a nite

1This defect is not important for the entanglement entropy. When we evaluate the entanglement entropy in canonical deformation the two twist operator are at the same time such that the chemical potential does not make e ect. But for a general correlation function of two operators at di erent time the chemical potential make a di erence. We will go back to this problem later.

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JHEP07(2016)049

chemical potential to describe a higher spin black hole. There is moreover a canonical deformation term in the Hamiltonian, corresponding to the canonical boundary condition [27] in the higher spin black hole solution. Our approach is di erent from the one in [29]. Instead of expanding the density matrix perturbatively in terms of the chemical potential, we treat the density matrix in a more exact way. This is feasible because we are only interested in the leading order result and we focus on the entanglement entropy rather than the R enyi entropy. Therefore we can use the saddle point approximation without worrying about the backreaction. In our case, the entanglement entropy is encoded in the two-point function of the twist operators under the density matrix. More generically we may consider the two-point function of two light primary operators with both conformal dimension and the spin-3 charge. Instead of studying the deformed theory directly, we take a picture-changing transformation and set the theory to a non-deformed theory. Under this picture transformation the two primary operators are transformed into two descendent operators, with the density matrix being invariant. As the spacial direction is compact, the correlation function is de ned on a torus, we need to study the conformal block on the torus. We may insert a complete state bases at the thermal cycle. Basically the 2-point correlation on the torus reduces to a sum of four-point functions which could be decomposed into the contributions from di erent propagating modules. We call the contribution from each module as a generalized conformal block. We can still use the monodromy analysis to study the leading order of generalized conformal block. Due to the presence of the chemical potential, we have one more di erential equation, which correspond to evolving the operator by the higher spin charge. By solving the monodromy problem, we determine the leading-order correlation function of two general light operators under the thermal density matrix with the chemical potential.

Furthermore, we discuss the HEE by computing the probe action of the Wilson line in the background of the higher spin black hole with the chemical potential in the canonical boundary condition. We nd complete agreement with eld theory correlator. The agreement between the 2-point function on torus and its holographic computation via Wilson line not only holds for the twist operators, but also for more general operators at di erent time. On the bulk side, the picture-changing transformation could be understood as the gauge transformation between di erent boundary conditions.

The remaining of the paper is organized as follows. In section 2, we review the computation of the R enyi entropy at nite temperature. In particular, we give a derivation of the di erential equation proposed in [7] to study the conformal block on the torus. In section 3, we study the two-point function of light operators on the torus with a chemical potential. We discuss the picture-changing transformation and introduce an auxiliary periodic coordinate with which the two-point function is de ned with respect to a theory without deformation. We furthermore establish the di erential equations of the wavefunction and the monodromy condition. In section 4, we solve the monodromy problem to read the correlation function. In section 5, we compute the correlation function holographically by using the Wilson line proposal. In section 6, we ends with conclusion and discussion.

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2 R enyi entropy at nite temperature

Entanglement entropy measures the entanglement between the subsystem and its environment [33]. Assuming the whole system can be described by a density matrix , we can de ne a reduced density matrix for the sub-system A by tracing out the degrees of freedom in its environment Ac

A = trAc : (2.1)

The entanglement entropy of subsystem A is de ned to be the Von Neumann entropy of the reduced density matrix

SEE(A) = log A log A: (2.2)

Moreover we can de ne the R enyi entanglement entropy

Sn(A) =

1n 1

log trnA; (2.3)

which allows us to read the entanglement entropy

SEE = lim

n!1

Sn; (2.4)

if n can be analytic extended to non-integer and the limit n ! 1 can be well taken.

The entanglement entropy and R enyi entropy can be computed by using the replica trick [34]. The n-th R enyi entropy is given by

Sn =

1n 1

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log trnA =

1n 1

log Zn

Zn1

; (2.5)

where Zn is the partition function on an n-sheeted space-time connected with each other at the boundary of sub-region A. In the path integral formalism, the partition function Zn can be taken in another way: the eld theory on the n-sheeted space-time is replaced by n-copies of the original theory on one-sheet spacetime with appropriate twisted boundary condition on the elds at the entangling surface. The entangling surface is at the boundary of sub-region A at a xed time, so is a surface of co-dimension 2. Circling around the entangling surface, the i-th copy of the eld is connected with the (i + 1)-th one. Speci cally, in two dimensional case, the entangling surface shrinks to some branch points, and the boundary condition on the elds at the branch points requires the introduction of the twist elds. The partition function can be computed by inserting the twist and anti-twist operators at the branch points in a orbifold CFT2

Zn Zn1

= [angbracketleft]T (z1;

z1)T (z2;

z2) : : : T (z2N;

z2N)[angbracketright]; (2.6)

where N is the number of the interval. In eq. (2.6) , the correlation can be de ned not only at the zero temperature but also at a nite temperature and even with a chemical potential as well.

2The twist operator description can also be extended to higher dimensions, see [35].

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In our case, we consider a two-point function on a torus, even with a higher spin current deformation. The correlation function can be decomposed into the generalised conformal block, whose leading order could be computed by using the monodromy analysis. Before going into the details, we would like to give some general comments on the accuracy of the calculation. For a multi-point correlation function, we can use the operator product expansion (OPE) recursively and the correlation function can be decomposed into the contributions from di erent propagating modules. If the correlation function is de ned on a higher genus Riemann surface, we need to cut the Riemann surface open at some cycles and inserting the states in di erent modules. We may just consider one module propagating in each OPE channel or one module at each cycle. This allows us to de ne generalized conformal block, or conformal block on the torus. The multi-point function is a summation of the generalised conformal blocks. We have di erent ways to take the operator product expansion and cutting the cycles. For a theory with crossing symmetry and modular invariance, all of these expansions equal to each other, but with di erent convergent rates. It is believed that for a large c theory, there is one kind of expansion in which the contribution from the vacuum module dominates and the contribution from other modules is non-perturbatively suppressed in the large c limit. Therefore even if we do not know the exact construction of the CFT dual to the AdS3 gravity, we can still compute the correlation function reliably from the vacuum block as long as we nd the proper channel. For di erent locations of the inserted operators and di erent Riemann surfaces, we may need to use di erent channels to expand the correlation function such that the vacuum module dominates in the expansion. There could be a phases transition at the parameter space when the expansion channel change. This e ect is already known for Hawking-Page transition and in holographic entanglement entropy calculation [4, 15].

In the next subsections, we rst review the conformal block of the four-point function on a complex plane as in [26]. Then we turn to the nite temperature case and show how to derive the partition function on a torus. Our discussion clari es the proposal in [7].

2.1 Conformal block from the monodromy

Let us rst consider the four-point function in full complex plane to show the general idea of the conformal block. For simplicity, we assume the symmetry of the theory is only generated by the Virasoro algebra. Our discussion follows [4, 36]. For the higher spin case, see ref. [27].

For a general four-point function, we can evaluate it by inserting an identity operator

h1(z1)2(z2)3(z3)4(z4)[angbracketright] =X~

h1(z1)2(z2) [notdef] ~[angbracketright][angbracketleft]~ [notdef] 3(z3)4(z4)[angbracketright]; (2.7)

where the states ~ are normalized and orthogonal to each other. For a conformal eld theory, the states can be classi ed by the representations of the conformal symmetry. The four-point function can be written as

h1(z1)2(z2)3(z3)4(z4)[angbracketright] =X X~

h1(z1)2(z2) [notdef] ~ [angbracketright][angbracketleft]~ [notdef] 3(z3)4(z4)[angbracketright] =

X

F (xi):

JHEP07(2016)049

(2.8)

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The function F (xi) is called the conformal block, which is the conformal partial wave

related to the representation . The semi-classical limit is de ned by taking ; c ! 1

with the ratio

c being xed. It is believed that under this limit the conformal block can be approximated to be

F (xi) e

c6 f(xi); (2.9)

where f(xi) depends only on c.

To determine the function f(xi), the standard way is to solve the monodromy problem. We rst introduce a null state

|[angbracketright] =

L2 32(2 + 1)L21 [notdef]^ [angbracketright]; (2.10)

where = 1

16

5 c +

p(c 1)(c 25)

: (2.11)

In the large c limit, ! 12

92c and the null states goes to

|[angbracketright] =

L2 +c6L21 [notdef]^

: (2.12)

Inserting the null state into the correlation funciton and de ning

(z) =

P~

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h1(z1)2(z2) [notdef] ~ [angbracketright][angbracketleft]~ [notdef]

^(z)3(z3)4(z4)[angbracketright]

P~

h1(z1)2(z2) [notdef] ~ [angbracketright][angbracketleft]~ [notdef] 3(z3)4(z4)[angbracketright]

; (2.13)

T (z) =

P~

h1(z1)2(z2) [notdef] ~ [angbracketright][angbracketleft]~ [notdef]

^

T (z)3(z3)4(z4)[angbracketright]

P~

h1(z1)2(z2) [notdef] ~ [angbracketright][angbracketleft]~ [notdef] 3(z3)4(z4)[angbracketright]

; (2.14)

we nd that the decoupling of the null state leads to

[prime][prime] (z) + 6

c T (z) (z) = 0; (2.15)

in the large c limit, where by using the Ward identity, the stress tensor is of the form

T (z) =

Xihi(z zi)2+ 1z zi@@zi log F: (2.16)

In eq. (2.13), there is a term [angbracketleft]~ [notdef]

^(z)3(z3)4(z4)[angbracketright]. Because that

^ is a null state, it leads to a di erential equation. Solving the di erential equation, we can get a monodromy when z moves around z3 and z4. As the monodromy is the same for all of the ~ , this indicates that

(z) in (2.13) has such a monodromy as well. With this monodromy condition and (2.15), we can solve all the coe cients in the stress tensor (2.16) and x the conformal block up to a constant. For the entanglement entropy, we need to consider vacuum conformal block. In this case the monodromy around z3 and z4 is trivial.

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2.2 Conformal block at nite temperature

In this subsection, we discuss the conformal block at a nite temperature. The thermal density matrix is

thermal = e2iL02i L0; (2.17)

where

L0 =

1

2

Z

2

^

T (w)dw;

0

L0 =

1

2

Z

^~

T (

w)d

2

w: (2.18)

Consider a multi-correlation function of the primary elds j with the conformal dimension hj on a torus. The torus is characterized by the moduli , and is doubly periodic

z z + 2; z z + 2: (2.19)

By inserting a complete set of states we change the correlation function on the torus to a summation of the correlator on a cylinder

[angbracketleft]

Yjj(zj)[angbracketright] [notdef] = Tr

e2iL02i L0

0

Yjj(zj)

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=

Xh,k,k[angbracketleft]h; k; k [notdef]

Yjj(zj) [notdef] h; k; k[angbracketright]e2i(h+kc24 )2i(h+k c24 ); (2.20)

where the index h denotes the di erent primary modules propagating on the torus and k and

k denote the descendants in that module. Under the conformal transformation the di erent modules do not mix with each other so we de ne a nite conformal block which only sum over the states in one module

F(; h; zj; hj; hp,r) =

Xk

hh; k [notdef]

Yjj(zj) [notdef] h; k[angbracketright]hp,re2i(h+kc24 ); (2.21)

where we only consider holomorphic part. It is a multi-point conformal block on the cylinder, with two descendants operators at the past in nity and the future in nity respectively, and hp,r denote the conformal dimension of the propagators.

Let us focus on the holomorphic sector and derive the Ward identity on the conformal block following the paper [37]. Consider the correlation function

Xk

hh; k [notdef]

^

T (z)

Yjj(zj) [notdef] h; k[angbracketright]hp,re2i(h+kc24 ): (2.22)

Even though we only sum over one module on the torus, the above function should be periodic in both direction. The periodic condition z ! z + 2 is trivial. For z ! z + 2,

we have

^

T (z)

Xk[notdef] h; k[angbracketright][angbracketleft]h; k [notdef] e2i(h+kc24 ) = Xk[notdef] h; k[angbracketright][angbracketleft]h; k [notdef] T (z + 2)e2i(h+kc24 ): (2.23)

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Therefore we nd that

Xk

hh; k [notdef]

^

T (z)

Yjj(zj) [notdef] h; k[angbracketright]hp,re2i(h+kc 24 )

=

Xj

hh; k

Yjj(zj) [notdef] h; k[angbracketright]hp,re2i(h+kc24 ) (2.24)

+

Xj

Xn

hj4 sin2 12(z zj + 2n) X k

Xn12 cot12(z zj + 2n)@ @zj

Xk

hh; k

Yjj(zj) [notdef] h; k[angbracketright]hp,re2i(h+kc24 ) + f();

where

z zj2 [notdef] + 122 1();

Xn12 cot12(z zj + 2n) =12 (z zj [notdef] ) 122 1()(z zj): (2.25)

To x the f(), we need to take an integral along the spacial direction. Considering that

Z

2

0 dz

Xn

14 sin2 12(z zj + 2n)= 1 42 }

^

T (z) = 42L0 +

c62: (2.26)

we get

f() = 2i @

@ F(): (2.27)

Similarly, by inserting the null state [notdef][angbracketright] in the correlation function as in (2.10), we

have a di erential equation

3 2(2 + 1)

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@@z2 + 2 1() +

Xj

1 42 }

z zj2 [notdef] + 122 1()

hj

+

Xj

1

2

z zj2 [notdef] 122 1()(z zj) @ @zj

+2i @

@

X

hh; k [notdef] (z)

Yjj(zj) [notdef] h; k[angbracketright]e2i(h+kc24 ) = 0; (2.28)

where (z) is the vertex operator for the state [notdef]

k

^[angbracketright]. De ning the function

Pk

hh; k [notdef] (z)

Qj j(zj)(zj) [notdef] h; k[angbracketright]e2i(h+kc 24 )

Qj j(zj)(zj) [notdef] h; k[angbracketright]e2i(h+kc24 ) (2.29)

which is assumed to be order c0, and taking the large c limit, we have the equation

c 6

Pk

hh; k [notdef]

@@z2 +

Xj

1 42 }

z zj2 [notdef] + 122 1()

hj

+

1

2

z zj2 [notdef] 122 1()(z zj)

@@zj log F

+ 2i @

@ log F = 0 (2.30)

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This is the equation (14) given in [7]. Here we give a eld theory derivation for that equation.

We need to x the monodromy condition around the propagator with

M = lim

c!1

0

B

B

B

@

1

C

C

C

A

1 24hp,rc [parenrightBig]12

[parenrightBigg] 0

0 e

i 1

1 24hp,rc [parenrightBig]12

e

i 1+

[parenrightBigg]

; (2.31)

and the monodromy around the spacial cycle

M = lim

c!1

0

@

ei(1 24hc )

12 0

0 ei(1 24hc )

12

1

A: (2.32)

The matrix denote the transformation for two independent solutions of the equation (2.30) up to conjugate, when the argument moves around the two cycles.

3 Correlation function at nite temperature with higher spin deformation

We now turn to compute the entanglement entropy at a nite temperature with a nite chemical deformation. The entanglement entropy of a single interval could be read from the two-point function of two primary twist operators in this system. The spatial direction of the torus is L=2 L=2. We de ne

L0 =

1

2

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Z

^

L 2 T (z)d;

W0 = 1 2

Z

^

L 2 W (z)d;

L0 =

1

2

Z

^

L 2 T (

z)d;

W0 = 1 2

Z

z)d; (3.1)

where the integral is over the real axis L=2 L=2, and the Hamiltonian for the

non-deformed theory is

H0 = L0 +

L0;

^

L 2 W (

L0: (3.2)

For a theory with a higher spin current deformation, the modi ed Hamiltonian is

H = H0

2i

W0 +

P0 = L0

2i

W0: (3.3)

In a deformed theory the system is evolved by this Hamiltonian.

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3.1 Picture-changing transformation

In the modi ed system, we can de ne the Euclidian version of the two-point function at a nite temperature with a non-zero potential conjugate to the momentum

1Z Tr (r1(tE,1; 1)r2(tE,2; 2)) ; (3.4)

where

= e H+i P ;

r(tE; ) = etEHr(0; )etEH = etEHeiP0r(0; 0)eiP0etEH: (3.5)

The superscript r denote that the operator is evolved by the modi ed Hamiltonian. In the nite temperature system, the operator is doubly periodic

r(tE + ; + ) = r(tE; + L) = r(tE; ): (3.6)

For the discussion we can also de ne an operator evolved by the original Hamiltonian without deformation, as

(z;

z) = eizL0eiz L0(0; 0)eizL0eizL0; (3.7)

where we have introduced the complex coordinate

z = + itE;

z = itE: (3.8)

If we regard the W0

JHEP07(2016)049

W0 term in eq. (3.3) as an interaction, then the operator r in (3.5) is the operator in the Hamiltonian picture and the operator in (3.7) is the operator in the interaction picture. We can recombine the chemical potential and the inverse temperature as a parameter in the complex coordinate

2 = + i ; 2

= i : (3.9)

Note that here we do not normalize the spatial direction so that the complex quantity is not the moduli of the torus.

From (3.2) and (3.3), we can rewrite the density matrix as

= e2iL0+2i W0e2i L02i W0: (3.10)

The operators in the Hamiltonian picture and interaction picture are related to each other as

r(; ) = e

2i

tEW0e

2i

2i

tE W0eizL0eiz L0r(0; 0)eizL0eiz L0e

2i

tEW0e

tE W0

2i

= e

2i

tEW0e

tE W0(z;

z)e

2i

tEW0e

tE W0; (3.11)

where we have used the relation

[iL0; (z;

z)] =

2i

@@z (z;

z);

L0; (z; z)] = @@ z (z; z): (3.12)

{ 11 {

[i

The relation (3.12) can be proved as follows. By the path integral we have

[iL0; (z)] =

i 2

Z

^

L 2 T (z[prime])dz[prime]; (z)

=

i 2

I

dz[prime] ^

T (z[prime])(z;

z)

= i

2

I

dz[prime]

1

Xm=1^

Lm(z[prime] z)m+2 (z)

= (

^

L1)(z) = @(z): (3.13)

Thus, we see that the translation along z is induced by the conserved charge L0. Similarly

we may consider the evolution with respect to the charge W0 as well

[iW0; (z)] =

i 2

JHEP07(2016)049

Z

^

L 2 W (z[prime])dz[prime]; (z)

=

i 2

I

dz[prime] ^

W (z[prime])(z;

z)

=

i 2

I

dz[prime]

1

Xm=1^

Wm(z[prime] z)m+3 (z)

= ( ^

W2)(z): (3.14)

By introducing two other auxiliary coordinates y;

y, we can de ne

(z; y;

z;

y) eiW0yei W0y(z;

z)eiW0yei W0y; (3.15)

then we have

r(tE; ) = (z; y;

z;

y) (3.16)

with

y = 2

tE

y = 2

tE: (3.17)

To compute the single-interval entanglement entropy, 1 and 2 are taken to be the twist and anti-twist operators at the branch points respectively. Both operators are primary. With (3.16), we can write (3.4) as

Tr ((z1; y1;

z1;

y1)(z2; y2;

z2;

y2)) : (3.18)

The operator (zi; yi;

zi;

yi) can be regarded to be a descendant operator inserted at (zi;

zi),

which is evolved by the non-deformed Hamiltonian H0.

Up to now, we have transformed the correlation function of two primary operators at a nite temperature in a deformed theory to the correlation function of two descendant operators in a non-deformed theory under a density matrix at a nite temperature and with a nite chemical potential. We can regard this transformation to be a picture-changing transformation in quantum theory. In eq. (3.3), 2i W0 + 2i

W0 can be regarded as

an interaction term. In eq. (3.15) (zi; yi;

zi;

yi) on the left hand side can be regarded to be in the Heisenberg picture and its evolution is respect to the Hamiltonian with interaction, and the operator (z;

z) on the the right hand side can be regarded to be in the interaction picture. In the Heisenberg picture we compute the correlation function of two primary operators in the deformed theory, while in the interaction picture we compute the

{ 12 {

correlation function of two descendant operators in non-deformed theory. In section 5, we will show that the picture transformation here corresponds to the gauge transformation in the bulk theory. Di erent pictures here correspond to di erent boundary conditions in the bulk solutions.

With the relations (3.12) and (3.15) it is easy to prove

eiL0z1eiW0y1(z; y)eiL0z1eiW0y1 = (z + z1; y + y1): (3.19)

Furthermore using (3.6) or (3.10) and (3.18), we have

h(z + 2i; y + 2i ) : : :[angbracketright][notdef], = [angbracketleft](z; y) : : :[angbracketright][notdef], ; (3.20)

which is a generalized version of cyclic boundary condition in the thermal direction. However one should be aware that this periodicity is only true for a complete theory. That means we need to sum over all contributions from di erent channels with proper combination between the holomorphic and anti-holomorphic part. If we consider only one conformal block, the periodicity may break down. For example in [3, 7] the second order di erential equation has been de ned for the wavefunction of a multi-point function as eq. (2.15) and eq. (2.30). However, it was shown that the solution is not single-valued along the non-trivial cycle. Furthermore because the conformal block only contain holomorphic part, even in trivial cycle, the conformal block may have an extra phase, as shown in [7].

3.2 Monodromy problem

In this subsection, we will show how to expand the correlation function (3.18) in terms of the generalized conformal block and set up the monondromy condition to compute the generalized conformal block from propagating vacuum module states. In the semi-classical limit, we assume that the propagating vacuum module dominates the contribution. In a theory with W3 symmetry, the vacuum module include the excitations of Virasoro

generators and W3 generators acting on the vacuum.By the path integral the function (3.18) can be normalized to be

C2 = Tre2iL0+2i W02i L02i W0(z1; y1;

z1;

y1)(z2; y2;

z2;

y2)

JHEP07(2016)049

Tre2iL0+2i W02i

L02i

W0

= [angbracketleft]e2i W02i W0(z1; y1;

z1;

y1)(z2; y2;

z2;

y2)[angbracketright]2,2

: (3.21)

The correlation functions in the second line are de ned on a torus with (L; 2) being its periods. The e2i W02i W0 is a non-local operator inserting at a time slice. In this correlation function, the local operator can be continuously deformed in any contour away from the locations of the other operators, and the expectation value is continuously changed along this contour. When the contour crosses a non-local operator, the situation becomes subtle. In the case at hand, the non-local operators induce a jump of the operators. More precisely, we have the relation

e2iz1L0(z; y) = (z z1; y)e2iy1L0;

e2iy1W0(z; y) = (z; y y1)e2iy1W0: (3.22)

{ 13 {

he2i W02i

W0[angbracketright]2,2

(a) Numerator (b) Denominator

Figure 1. The correlation functions with a non-local operator inserting at the dashed line. The

operator can be moved to another time slice. The local operators are set below the nonlocal

operator.

This means that crossing an operator e2iz1L0 has the e ect of evolving z1 along the z

direction and crossing an operator e2iy1W0 has the e ect of evolving y1 in the y direction.

These equation can be written in the path integral formalism as

he2iz1L0lower(z; y) [notdef] [notdef] [notdef] [angbracketright] = [angbracketleft]upper(z z1; y)e2iy1L0 [notdef] [notdef] [notdef] [angbracketright];

he2iy1W0lower(z; y) [notdef] [notdef] [notdef] [angbracketright] = [angbracketleft]upper(z; y y1)e2iy1W0 [notdef] [notdef] [notdef] [angbracketright]; (3.23)

where the subscript \lower" or \upper" denotes the operator is below or above the nonlocal operators. Because the operator W0 commutes with the Hamiltonian, the inserted

non-local operator W0 can be moved to any imaginary time slice if the movement dont

touch other operators. Therefore, the correlation functions in the numerator and the denominator in eq. (3.21) are represented respectively as in (1a) and (1b) in gure 1.

As in previous section, both the numerator and denominator of (3.21) are the correlation functions on the torus. By inserting a complete basis in thermal cycle, they can be decomposed into contributions from the states in di erent modules. Furthermore for the two operators in the numerator we can take an OPE and the expansion can be decomposed into the contribution from di erent modules. For each choice of modules in the OPE and in thermal cycle, the contribution de nes a generalised conformal block. The numerator and denominator can be written as a summation of generalised conformal blocks from di erent modules. In the large central charge limit, we assume that the generalised conformal block from the vacuum module dominates the contribution and the ones with other modules are non-perturbatively suppressed. We note that the jump from the crossing a non-local operator (3.22), (3.23) remains in the conformal block.

To determine this conformal block, we may use the monodromy analysis as before. However, due to the presence of W3, there is one more di erential equation to consider.

Introduce a primary state [notdef]

^[angbracketright] such that

L0 [notdef]

^[angbracketright] = [notdef] [angbracketright]; W0 [notdef]

^[angbracketright] =

13 [notdef] [angbracketright]: (3.24)

{ 14 {

JHEP07(2016)049

In its descendants there are null states at level 1, 2, 3. In large c limit, they are

(L1 + 2W1) [notdef]

^[angbracketright] = 0;

^[angbracketright] = 0;

L31 +24c L2L1 +12c L3 +24c W3 [notdef]^= 0: (3.25)

Inserting the null states into the correlation function, we get three di erential equations on the correlation function involving the operator ^ corresponding to the state [notdef]

^[angbracketright]. In

particular the third equation can be transformed to

[prime][prime][prime] (z; y) + 24

c T (z; y) [prime](z; y) +

12c T [prime](z; y) (z; y) +

(L21 W2 +

16c L2 [notdef]

JHEP07(2016)049

24c W (z; y) (z; y) = 0; (3.26)

where the prime denotes the derivative with respect to z and

(z; y) = [angbracketleft]e2i W02i W0

^(z; y) : : :[angbracketright] he2i W02i

W0 : : :[angbracketright]

; (3.27)

and similarly

T (z; y) = [angbracketleft]e2i W02i W0

^

T (z; y) : : :[angbracketright]

he2i W02i

W0 : : :[angbracketright]

;

W (z; y) = [angbracketleft]e2i W02i W0

^

W3(z; y) : : :[angbracketright]

he2i W02i

: (3.28)

The ellipsis in (3.27) denotes other local operators at (zi; yi). Unlike the Virasoro case without deformation, the functions T (z; y) and W (z; y) can not be determined simply by imposing the doubly periodic condition.

With the sl(3; R) algebra the equation (3.26) can be rewritten in a compact form

@ (z; y)@z = a(z; y) (z; y); (3.29)

where

a(z; y) = L1 + 6c T (z; y)L1 6c W (z; y)W2

=

W0 : : :[angbracketright]

0

B

@

0 12cT (z; y) 24cW (z; y)

1 0 12cT (z; y)

0 1 0

1

C

A

; (3.30)

(z; y) =

0

B

@

[prime][prime](z; y) + 2kT (z; y) (z; y) [prime](z; y)

(z; y)

1

C

A

: (3.31)

From the de nition of ^(z; y), we have

@ ^(y; z)

@y = eiyW0[iW0;

^(z)]eiyW0 = eiyW0(

^

W2 ^(z))eiyW0; (3.32)

{ 15 {

where ( ^

W2 )(z) is the corresponding vertex operator for the state ^

W2 [notdef] [angbracketright]. Using the

relation (3.25), we have

@ (z; y)

@y = b(z; y) (z; y); (3.33)

where

b =

0

B

@

1

C

A

4c T (z; y) 4cT [prime](z; y) 24cW (z; y)

4c T [prime][prime](z; y) + 144c2T (z; y)2 0 8cT (z; y)

4c T [prime](z; y) 24cW (z; y)

1 0 4

c T (z; y)

: (3.34)

If the insertion of the operator ^ is away from the position of the non-local operator, the equations (3.29) and (3.33) can be solved formally by introducing an evolution operator,

(z; y) = U(z; y; z0; y0) (z0; y0); (3.35)

where

U(z; y; z0; y0) = P exp Z

(z,y)

(z0,y0) a(z; y)dz + b(z; y)dy

; (3.36)

is a path-ordered integral on a contour in the two-dimensional complex plane. The consistency condition for the path-ordered integral is

@@y a(z; y) +

@@z b(z; y) [a(z; y); b(z; y)] = 0; (3.37)

or explicitly as

JHEP07(2016)049

@T

@y + 2

@W

@z = 0;

96c T

@T

@3T

@z3 6

@W

@y +

@z = 0; (3.38)

which can be derived directly by W algebra in large c limit. In eq. (3.35), z and y have

to be regarded as two independent complex coordinates, representing the evolution by L0

and W0 respectively.

Now let us discuss the monodromy condition. In the case at hand, there are two types of expansions: one of them is the operator product expansion of two operators, and the other one is for inserting a complete bases along the thermal circle. Correspondingly, we need to impose the monodromy condition on two circles, the thermal circle and the circle enclosing the two operators. At each of the circle there is a monodromy condition. Because we only keep the vacuum module in the OPE of two operators, the monodromy around the the circle enclosing the two operators is trivial. The monodromy along the thermal cycle is more subtle. By inserting a complete set of state basis the torus is cut open and becomes a cylinder, we can take a conformal transformation

w = e

2

z; (3.39)

which maps the cylinder to the full complex plane. The operators are related by

^(z) =

@w @z

2h^ (w) =

2

2he2 hz ^ (w): (3.40)

{ 16 {

The monodromy in the w coordinate is trivial. While in the z coordinate, there is an extra phase e

2

hz from the conformal transformation. When the conformal dimension of ^ is a half-integer, the monodromy around the thermal circle is -1, as in [7]. When the conformal dimension of ^ is an integer, as in our case, the monodromy around the thermal circle must be trivial

(z + 2)upper = (z)lower: (3.41)

Considering the relation (3.23), we have the monodromy condition around the thermal cycle

(z + 2; y + 2 )lower = (z; y)lower: (3.42)

4 Monodromy analysis

In this section, we use the monodromy condition to compute the correlation function (3.21) on the torus. Firstly we study the function in the denominator [angbracketleft]e2i W02i W0[angbracketright]2, which is just the partition function of a higher spin black hole. We discuss the expectation values of the stress tensor and the higher spin charge. Then we compute the two-point function in the numerator. In the discussion, we assume the conformal dimension of the operator is of order c but still light compared to the charge of the higher spin black hole so that we can ignore the backreaction to the background.

4.1 Thermodynamics of the ensemble

In this subsection, we show that the monodromy condition can determine the thermodynamics of the ensemble with the higher spin deformation. The thermodynamics of higher spin black hole was studied holographically in [38] and in [27] from the point of view of canonical deformation. A eld theory derivation was presented in [39] by using the perturbation expansion. Here we give another eld theory derivation for the thermodynamics. In our derivation the relation with the holographic study becomes more clear.

In this case, because of the translation invariance, the matrices a and b are constant-valued

a0 =

0

B

@

: (4.2)

Here the subscript 0 denotes the expectation value with no operator insertion. Because the operators L and W commute, the matrices a0 and b0 also commute with each other.

Then the monodromy condition (3.42) can be written as

exp[2 b0 + 2a0] = 1: (4.3)

{ 17 {

JHEP07(2016)049

0 12c[angbracketleft]T [angbracketright]0 24c[angbracketleft]W [angbracketright]0 1 0 12c[angbracketleft]T [angbracketright]0 0 1 0

4 c

1

C

A

; (4.1)

b0 =

0

B

@

1

C

A

hT [angbracketright]0 24c[angbracketleft]W [angbracketright]0

144 c2

hT [angbracketright]20

0 8c[angbracketleft]T [angbracketright]0 24c[angbracketleft]W [angbracketright]0

1 0 4

c

hT [angbracketright]0

This is exactly the monodromy condition suggested in [38]. Now we derive it from the eld theory. With this monodromy condition we can easily solve the expectation value and derive the thermodynamics law as in [38]. Here we omit the details.

For our later study, we diagonalize the matrices a0 and b0 by

a0 = M

0

B

@

1 0 0

0 2 0

0 0 3

1

C

A

M1; (4.4)

b0 = M

0

B

@

13 ( 2 + 3)2 + 23 2 3 0 0

0 13 ( 1 + 3)2 + 23 1 3 00 0 13 ( 1 + 2)2 + 23 1 2

1

C

A

M1; (4.5)

JHEP07(2016)049

where

M =

0

B

@

1

C

A

3

4 21 14 22 14 23

3

4 22 14 21 14 23

3

4 23 14 21 14 22

1 2 3

1 1 1

: (4.6)

Here 1; 2; 3 are three roots of the following cubic equation

3 + 24c [angbracketleft]T [angbracketright]0 +

24c [angbracketleft]W [angbracketright]0 = 0; (4.7)

with 1 > 2 > 3.

4.2 Two-point function

In this subsection, we evaluate the correlation function (3.21) by imposing the monodromy condition. The conformal dimensions and the higher spin charges of two operators are respectively (h1; q1) (h2; q2). As we are interested in the entanglement entropy, we set

h1 = h2;

q1 = q2; (4.8) such that the operators can fuse to the vacuum module. We only consider light operators with 1 h; q c, so that we can use the saddle point approximation and ignore their

back reaction to the background. When we calculate the entanglement entropy, in n ! 1

limit, the twist operator satisfy the light operator condition.

Because h and q are much smaller than a0 and b0 which are the charges with no operator inserting, we can take a linear perturbation about the solution a0; b0 as

a = a0 + a1;

b = b0 + b1; (4.9)

where

a1 =

0

B

@

0 12c(T [angbracketleft]T [angbracketright]0) 24c(W [angbracketleft]W [angbracketright]0) 0 0 12c(T [angbracketleft]T [angbracketright]0)

0 0 0

1

C

A

;

b1 =

0

B

@

4c (T [angbracketleft]T [angbracketright]0) 4cT [prime] 24c(W [angbracketleft]W [angbracketright]0)

4c T [prime][prime] + 288c2[angbracketleft]T [angbracketright]0(T [angbracketleft]T [angbracketright]0)

0 8c(T [angbracketleft]T [angbracketright]0)

4c T [prime] 24c(W [angbracketleft]W [angbracketright]0)

0 0 4

c (T [angbracketleft]T [angbracketright]0)

1

C

A

:

{ 18 {

De ne

U0(z; y) = exp[a0(z z0) + b0(y y0)];

U(z; y) = U0(z; y)U1(z; y): (4.10)

The di erential equations (3.29) and (3.33) can be rewritten as

@@z U1(z; y) = U10(z; y)a1(z; y)U0(z; y);

@@y U1(z; y) = U10(z; y)b1(z; y)U0(z; y): (4.11)

The equations can be solved by

U1(z; y) = P exp Z

{z,y[notdef]

{z0,y0[notdef]

[U10(z; y)a1(z; y)U0(z; y)dz + U10(z; y)b1(z; y)U0(z; y)dy]: (4.12)

Because of the consistency relation (3.37) and (3.38) we can continuously deform the contour as long as the contour is away from the singular points. The singular points of a(z; y) and b(z; y) can only appear at the locations of the operators, z1 and z2. By OPE we also have

T (z; y1)

h1(z z1)2

+ r1

JHEP07(2016)049

z z1

;

T (z; y2)

h2(z z2)2

+ r2

z z2

;

W (z; y1)

q1(z z1)3

+ p1

(z z1)2

+ s1

z z1

;

W (z; y2)

q2(z z2)3

+ p2

(z z2)2

+ s2

z z2

: (4.13)

To the linear order the monodromy condition can be written as

I

U10(z; y)a1(z; y)U0(z; y)dz + U10(z; y)b1(z; y)U0(z; y)dy = 0: (4.14)

This condition should satisfy for both the contour around the two operators and the contour around the thermal circle. Now we choose a special contour as follows

(z; y) =

8

>

>

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

>

>

:

(z2 ; y2 + t1(y1 y2)) 0 < t1 < 1

(z2 + (z1 z2 + 2 )t2; y1) 0 < t2 < 1

(z1 + e2i t3; y1) 0 < t3 < 1 (z1 + + (z2 z1 2 )t4; y1) 0 < t4 < 1

(z2 ; y1 + t5(y2 y1)) 0 < t5 < 1

(z2 e2it6; y2) 0 < t6 < 1:

(4.15)

The integrals from t1, t5 and t2, t4 are canceled with each other. The integrals from t3 and t6 can be evaluated by using the residue theorem. Then the monodromy condition (4.14) leads to

Xi=1,2Reszi(D(z; yi)1M1a0MD(z; yi)) = 0; (4.16)

{ 19 {

where

D(z; y)=Diag

e 1z(13 ( 2+ 3)2+ 23 2 3)y; e 2z(13 ( 1+ 3)2+ 23 1 3)y; e 3z(13 ( 1+ 2)2+ 23 1 2)y :

It is easy to solve these equations. The solution is

r1 = r2 = m1h1 + m2q1; s1 = s2 = n1h1 + n2s2; (4.17)

where

m1 = 12(K1 K2)

m2 = 32(K1 + K2)

n1 = 12(K3 K4)

n2 = 32(K3 + K4)

with

K1 = a3 ( 1 2) 3 + a1 ( 2 3) 1 + a2 ( 3 1) 2a3 ( 1 2) + a1 ( 2 3) + a2 ( 3 1)

K2 = a2a3 ( 2 3) 1 + a1a2 ( 1 2) 3 + a3a1 ( 3 1) 2 a2a3 ( 2 3) + a1a2 ( 1 2) + a3a1 ( 3 1)

K3 = a3 ( 1 2)

JHEP07(2016)049

2

3 23 13 21 13 22

+a1 ( 2 3)23 21 13 22 13 23

+a2 ( 3 1)23 22 13 21 13 23

a3 ( 1 2)+a1 ( 2 3)+a2 ( 3 1)

K4 =

a2a3 ( 2 3)

2 213 223 2 3 3

+a1a2 ( 1 2)

2 233 213 2 2 3

+a3a1 ( 3 1)

2 223 213 233

a2a3 ( 2 3) + a1a2 ( 1 2) + a3a1 ( 3 1)

a1 = exp

(z1 z2) 1 + (y1 y2)

23 21 13 22 1 3 23

a2 = exp

(z1 z2) 2 + (y1 y2)

23 22 13 21 1 3 23

a3 = exp

23 23 13 21 1 3 22 :

Using eq. (3.12) we nd that the holomorphic part of the correlator (3.21) obeys the equation

@@z1 log C2(z1; y1; z2; y2) = r1

@@y1 log C2(z1; y1; z2; y2) = s1: (4.18)

Finally, we obtain

log C2(z1; y1; z2; y2) (4.19)

=

1

2 log[a3( 1 2) + a1( 2 3)+a2( 3 1)][a11( 2 2) + a12( 3 1)+a13( 1 2)]h1

3

(z1 z2) 3 + (y1 y2)

2 log[a3( 1 2)+a1( 2 3)+a2( 3 1)][a11( 2 2) + a12( 3 1)+a13( 1 2)]1q1:

We have similar result for the anti-holomorphic part.

{ 20 {

5 Holographic computation

5.1 Wilson line probe action

The holographic computation of the correlation function (3.21) is to use the Wilson line proposal. The action of the Wilson line probe should give the function C2. The general framework for de ning and computing the probe action can be found in [24].

To calculate the two-point function holographically we need the at connection for the spin-3 black hole

A = eL0(a + d)eL0 (5.1)

A = eL0(a + d)eL0 (5.2)

with

a = L1 + 6c [angbracketleft]T [angbracketright]0L1 6c [angbracketleft]W [angbracketright]0W2

dy (5.3)

W2 +12c [angbracketleft]~T[angbracketright]0W0 +36c2 [angbracketleft]~T[angbracketright]20W2 +12c [angbracketleft] [angbracketright]0L1 d y; (5.4)

where y = 2 tE. The terms proportional to dy in a;

a show that we are actually considering the black hole solution with a chemical potential in the canonical boundary condition. Correspondingly, the dual CFT is canonically deformed by the spin-3 current. Here we use y instead of t just to show the relation with the eld theory analysis more clearly.

To calculate the action of the Wilson line probe, we introduce

L = eL0e(azz+ayy); (5.5)

R = ezz+yyeL0 (5.6)

such that

A = LdL1;

A = R1dR: (5.7)

Then the probe action is de ned by the diagonalized matrix

H [similarequal] (LiL1fR1fRi)

= diag

t1e40; t2; 1t1t2 e40 ; (5.8)

where the subscripts i; f denote the endpoints of the Wilson line at the boundary; t1e40; t2; e40=t1t2 are the eigenvalues of the matrix LiL1fR1fRi and 0 is the IR cut-o

{ 21 {

JHEP07(2016)049

dz

W2 + 12c [angbracketleft]T [angbracketright]0W0 +36c2 [angbracketleft]T [angbracketright]20W2 +12c [angbracketleft]W [angbracketright]0L1

L1 +6c [angbracketleft]~T[angbracketright]0L1 6c [angbracketleft] [angbracketright]0W2 d z

for the boundary. By direct calculation we get

t1 = 4

e 1(zfzi)+(23 2 3 13 ( 2+ 3)2)(yf yi)( 1 2)( 1 3)+ e 2(zfzi)+(23 1 3 13 ( 1+ 3)2)(yf yi) ( 2 1)( 2 3)

+e 3(zfzi)+(

2

3 1 2 13 ( 1+ 2)2)(yf yi)

[notdef]

e 1(zfzi)+(

2

3

2

3+ 13 (

2+

3)2)(yf yi)

(

1

( 3 1)( 3 1)

2)(

1

3)

+e 2(zfzi)+(

2

3

1

3+ 13 (

1+

3)2)(yf yi)

(

2

2

3

1

2+ 13 (

1+

2)2)(yf yi)

(

3

1)(

2

3) +

e 3(zfzi)+(

1)(

3

2)

;

e 1(zfzi)+(23 2 3+ 13 ( 2+ 3)2)(yf yi)( 1 2)( 1 3)+ e 2(zfzi)+(23 1 3+ 13 ( 1+ 3)2)(yf yi) ( 2 1)( 2 3)

+e 3(zfzi)+(

2

3 1 2+ 13 ( 1+ 2)2)(yf yi)

JHEP07(2016)049

t1t2 = 4

[notdef]

e 1(zfzi)(

2

3

2

3+ 13 (

2+

3)2)(yf yi)

(

1

( 3 1)( 3 1)

2)(

1

3)

+e 2(zfzi)(

2

3

1

3+ 13 (

1+

3)2)(yf yi)

(

2

2

3

1

2+ 13 (

1+

2)2)(yf yi)

(

3

1)(

2

3) +

e 3(zfzi)(

1)(

3

2)

The action of the probe is given by [26]

Iprobe = Tr

log(H)

h12 L0 +3q12 W0 : (5.9)

It is straightforward to check that

C2 = eIprobe: (5.10)

The agreement between the correlation function in the eld theory and its holographic computation is remarkable. The correlation function of two light operators is de ned on the torus, and there is no restriction on the locations of the operators. When one considers the entanglement entropy, the operators are set to the same time slice. In this case, the correlation function is independent of the chemical potential and reduces to the one found in [26]. In other words, the holographic computation of the entanglement entropy is not sensitive to the choice of the gauge potential with or without the chemical potential. This is not the case if one considers more general two-point function on the torus.

5.2 Holographic correspondence for picture transformation

On the eld side, we can transform two primary operators in the deformed theory into two descendant operators in a non-deformed theory by the picture-changing transformation. In this subsection, we would like to discuss the holographic correspondence of the picture-changing transformation. We suggest that the picture-changing transformation in the eld theory correspond to a time-dependent gauge transformation on the gauge potential in the bulk.

Let us focus on a simple case. We assume that the state in the eld theory is translational invariant. In a canonical deformed theory its holographic dual is just like the higher spin black hole as in [27, 39].

A = eL0(a + d)eL0; (5.11)

{ 22 {

where

dt: (5.12)

where [angbracketleft]T [angbracketright]0 and [angbracketleft]W [angbracketright]0 are constants. However, the state we consider here can be any

translation invariant state, not necessarily the thermal state, so [angbracketleft]T [angbracketright]0 and [angbracketleft]W [angbracketright]0 can take

any values.

In the eld theory we have W symmetry so that we can take a symmetry transformation

on the state

[notdef]

[angbracketright] = ei W0 [notdef] O[angbracketright]: (5.13) In the gravity side, the transformation can be written as

= U1(a + d)U; (5.14)

where

U = exp

(W2 +12c [angbracketleft]T [angbracketright]0W0 +36c2 [angbracketleft]T [angbracketright]20W2 +12c [angbracketleft]W [angbracketright]0L1)

dt: (5.18)

For di erent s, it de ne a di erent picture in the eld theory, and it corresponds to di erent asymptotic boundary condition. Speci cally, when s = , the gauge transformation cancel the chemical potential term in (5.12), and we get the gauge connection used in [26].

3In [27], they regard the source term as a gauge eld, and the time dependent transformation as the gauge transformation.

{ 23 {

a =

L1 + 6c [angbracketleft]T [angbracketright]0L1 6c [angbracketleft]W [angbracketright]0W2

+

dz

W2 + 12c [angbracketleft]T [angbracketright]0W0 +36c2 W2 +12c [angbracketleft]W [angbracketright]0L1

; (5.15)

which is an asymptotic symmetry in the bulk. This asymptotic symmetry was derived in [22] for asymptotic AdS boundary condition and was extended to the canonical deformed boundary condition in [42]. In this simple case we can give the transformation explicitly. Furthermore, taking (5.12) into (5.14), we see that the gauge transformation keeps a invariant. This corresponds to the fact that the state [notdef]O[angbracketright] is an eigenstate of the

symmetry generator W0.

In the above discussion, we take the gauge transformation parameter to be constant. We can furthermore extend it to be time dependent such that it correspond to a picture-changing transformation in the eld theory.3 We take = ts, then the states transform as

[notdef]

[angbracketright] = eitsW0 [notdef] O[angbracketright]; (5.16)

and the corresponding operators transform as

~

(t) = eitsW0(t)eitsW0; (5.17)

which is exactly the picture-changing transformation. Taking the parameter = ts into (5.14), we get

= a s

JHEP07(2016)049

W2 + 12c [angbracketleft]T [angbracketright]0W0 +36c2 [angbracketleft]T [angbracketright]20W2 +12c [angbracketleft]W [angbracketright]0L1

As we shown before, the probe action of the WL reproduces exactly the correlation function of two light operators located on the torus. One interesting question is if it is possible to read the correlation function from the gauge potential without chemical potential. The question is related to the holographic computation of two-point function of descendent operators. If these two operators are at the same slice, the direct computation of the probe action gives the correct answer. But if the operators are at di erent time slices, then one has to develop the WL proposal to address this issue.

6 Conclusion and discussion

In this article, we studied the entanglement entropy on a torus in the large c CFT with

W current deformation. More generally, we discussed the two-point function of the light

operator with h; q c under a thermal density matrix with a chemical potential. Due

to the presence of the deformation, the correlation function seems to be hard to compute. However, in the large c limit, if we accept that the vacuum module dominates the propagation, the problem is still tractable. First of all, under the limit, the W0 charge commutes

with the Hamiltonian so that we may apply a picture-changing transformation to turn o the deformation. Moreover, just as the Hamiltonian induce the translation of the time, the spin-3 operator induce the translation along an auxiliary coordinate. This leads us to nd another di erential equation on the wavefunction of the two-point function such that the monodromy problem could be well-de ned. By imposing the monodromy condition, the two-point function could be determined at the linear order. Holographically we computed the probe action of the Wilson line in the background of spin-3 black hole with the chemical potential, and we found perfect agreement with eld theory result.

Our treatment in the eld theory could be applied to other deformation, as long as they are conserved. In the large c limit, the other higher spin currents could be studied straightforwardly. This may help us to understand the W conformal block on the torus [40].

Our study keeps to the leading order of 1c expansion. It would be interesting to discuss the next leading order e ect, especially considering the fact that the nite size correction to the entanglement entropy appears only at the next-leading order [14, 41]. On the bulk this corresponds to the 1-loop quantum correction to the holographic entanglement entropy.

Our study supports the Wilson line proposal to compute the holographic entanglement entropy. With canonical boundary condition, the probe action of the Wilson line computes the HEE even with non-zero chemical potential. This suggests that in the probe limit, the Wilson line proposal can be applied to more general bulk con guration [43, 44] . On the other hand, how to determine the 1-loop quantum correction in the Wilson line proposal is an interesting question [45].

Acknowledgments

The work was in part supported by NSFC Grant No. 11275010, No. 11335012 and No. 11325522. We would like to thank W. Song, A. Castro L. Hung and T. Takayanagi for helpful discussions. Wu would like to thank YITP for hospitality, where the nal stage

{ 24 {

JHEP07(2016)049

of this work was nished. Wu was supported by Short-term Overseas Research Program from Graduate School of Peking University.

A W3 algebrasFor completeness we list the W3 algebras in this section as

T (z1)T (z2)

c=2 (z1 z2)4

+ 2T (z2) (z1 z2)2

@@z2 T (z2)

+ (z1 z2)

+ : : : ; (A.1)

JHEP07(2016)049

T (z1)W (z2)

3W (z2)

(z1 z2)2

@@z2 W (z2)

+ (z1 z2)

+ : : : ;

1N3 W (z1)W (z2)

1(z1 z2)6

3 c

@@z2 T (z2)

+ (z1 z2)3

9 10c

6c T (z2)

+ (z1 z2)4

@2@z22 T (z2)

+ (z1 z2)2

+ 96

c(5c + 22)

(z2)

(z1 z2)2

+ 1

5c

@@z32 T (z2)

z1 z2

+ 48

c(5c + 22)

@@z2 (z2)

z1 z2

+ : : : ;

where N3 = 5c6, and

(z) =: T (z)2 :

310@2T (z): (A.2)

From the OPE coe cient, we can derive the commutators

[Lm; Ln] = (m n)Lm+n +

c12m(m2 1) m+n,0;

[Lm; Wn] = (2m n)Wm+n; [Wm; Wn] =

112(m n)(2m2 mn + 2n2 8)Lm+n

40

5c + 22(m n) m+n

15!m(m2 1)(m2 4): (A.3)

Open Access. This article is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/

Web End =CC-BY 4.0 ), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

References

[1] S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT,

http://dx.doi.org/10.1103/PhysRevLett.96.181602

Web End =Phys. Rev. Lett. 96 (2006) 181602 [http://arxiv.org/abs/hep-th/0603001

Web End =hep-th/0603001 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0603001

Web End =INSPIRE ].[2] A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, http://dx.doi.org/10.1007/JHEP08(2013)090

Web End =JHEP 08 (2013) 090

[http://arxiv.org/abs/1304.4926

Web End =arXiv:1304.4926 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1304.4926

Web End =INSPIRE ].[3] T. Faulkner, The Entanglement Renyi Entropies of Disjoint Intervals in AdS/CFT,

http://arxiv.org/abs/1303.7221

Web End =arXiv:1303.7221 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1303.7221

Web End =INSPIRE ].[4] T. Hartman, Entanglement Entropy at Large Central Charge, http://arxiv.org/abs/1303.6955

Web End =arXiv:1303.6955 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1303.6955

Web End =INSPIRE ].[5] K. Krasnov, Holography and Riemann surfaces, Adv. Theor. Math. Phys. 4 (2000) 929

[http://arxiv.org/abs/hep-th/0005106

Web End =hep-th/0005106 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0005106

Web End =INSPIRE ].

5c

6

{ 25 {

[6] M. Headrick, Entanglement Renyi entropies in holographic theories, http://dx.doi.org/10.1103/PhysRevD.82.126010

Web End =Phys. Rev. D 82 (2010)

http://dx.doi.org/10.1103/PhysRevD.82.126010

Web End =126010 [http://arxiv.org/abs/1006.0047

Web End =arXiv:1006.0047 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1006.0047

Web End =INSPIRE ].[7] T. Barrella, X. Dong, S.A. Hartnoll and V.L. Martin, Holographic entanglement beyond

classical gravity, http://dx.doi.org/10.1007/JHEP09(2013)109

Web End =JHEP 09 (2013) 109 [http://arxiv.org/abs/1306.4682

Web End =arXiv:1306.4682 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1306.4682

Web End =INSPIRE ].[8] B. Chen and J.-J. Zhang, On short interval expansion of R enyi entropy, http://dx.doi.org/10.1007/JHEP11(2013)164

Web End =JHEP 11 (2013) 164

[http://arxiv.org/abs/1309.5453

Web End =arXiv:1309.5453 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1309.5453

Web End =INSPIRE ].[9] B. Chen, F.-y. Song and J.-j. Zhang, Holographic Renyi entropy in AdS3/LCFT2

correspondence, http://dx.doi.org/10.1007/JHEP03(2014)137

Web End =JHEP 03 (2014) 137 [http://arxiv.org/abs/1401.0261

Web End =arXiv:1401.0261 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1401.0261

Web End =INSPIRE ].[10] M. Beccaria and G. Macorini, On the next-to-leading holographic entanglement entropy in

AdS3=CF T2, http://dx.doi.org/10.1007/JHEP04(2014)045

Web End =JHEP 04 (2014) 045 [http://arxiv.org/abs/1402.0659

Web End =arXiv:1402.0659 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1402.0659

Web End =INSPIRE ].[11] Z. Li and J.-j. Zhang, On one-loop entanglement entropy of two short intervals from OPE of

twist operators, http://dx.doi.org/10.1007/JHEP05(2016)130

Web End =JHEP 05 (2016) 130 [http://arxiv.org/abs/1604.02779

Web End =arXiv:1604.02779 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1604.02779

Web End =INSPIRE ].[12] B. Chen, J. Long and J.-j. Zhang, Holographic R enyi entropy for CFT with W symmetry,

http://dx.doi.org/10.1007/JHEP04(2014)041

Web End =JHEP 04 (2014) 041 [http://arxiv.org/abs/1312.5510

Web End =arXiv:1312.5510 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1312.5510

Web End =INSPIRE ].[13] E. Perlmutter, Comments on R enyi entropy in AdS3/CFT2, http://dx.doi.org/10.1007/JHEP05(2014)052

Web End =JHEP 05 (2014) 052

[http://arxiv.org/abs/1312.5740

Web End =arXiv:1312.5740 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1312.5740

Web End =INSPIRE ].[14] B. Chen and J.-q. Wu, Single interval R enyi entropy at low temperature, http://dx.doi.org/10.1007/JHEP08(2014)032

Web End =JHEP 08 (2014)

http://dx.doi.org/10.1007/JHEP08(2014)032

Web End =032 [http://arxiv.org/abs/1405.6254

Web End =arXiv:1405.6254 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1405.6254

Web End =INSPIRE ].[15] B. Chen and J.-q. Wu, Holographic calculation for large interval R enyi entropy at high

temperature, http://dx.doi.org/10.1103/PhysRevD.92.106001

Web End =Phys. Rev. D 92 (2015) 106001 [http://arxiv.org/abs/1506.03206

Web End =arXiv:1506.03206 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1506.03206

Web End =INSPIRE ].[16] B. Chen, J.-q. Wu and Z.-c. Zheng, Holographic R enyi entropy of single interval on Torus:

With W symmetry, http://dx.doi.org/10.1103/PhysRevD.92.066002

Web End =Phys. Rev. D 92 (2015) 066002 [http://arxiv.org/abs/1507.00183

Web End =arXiv:1507.00183 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1507.00183

Web End =INSPIRE ].[17] J.-j. Zhang, Holographic R enyi entropy for two-dimensional N = (1; 1) superconformal eld

theory, http://dx.doi.org/10.1007/JHEP12(2015)027

Web End =JHEP 12 (2015) 027 [http://arxiv.org/abs/1510.01423

Web End =arXiv:1510.01423 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1510.01423

Web End =INSPIRE ].[18] X. Yin, Partition Functions of Three-Dimensional Pure Gravity, http://dx.doi.org/10.4310/CNTP.2008.v2.n2.a1

Web End =Commun. Num. Theor.

http://dx.doi.org/10.4310/CNTP.2008.v2.n2.a1

Web End =Phys. 2 (2008) 285 [http://arxiv.org/abs/0710.2129

Web End =arXiv:0710.2129 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0710.2129

Web End =INSPIRE ].[19] S. Giombi, A. Maloney and X. Yin, One-loop Partition Functions of 3D Gravity, http://dx.doi.org/10.1088/1126-6708/2008/08/007

Web End =JHEP 08

http://dx.doi.org/10.1088/1126-6708/2008/08/007

Web End =(2008) 007 [http://arxiv.org/abs/0804.1773

Web End =arXiv:0804.1773 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0804.1773

Web End =INSPIRE ].[20] B. Chen and J.-q. Wu, 1-loop partition function in AdS3/CFT2, http://dx.doi.org/10.1007/JHEP12(2015)109

Web End =JHEP 12 (2015) 109

[http://arxiv.org/abs/1509.02062

Web End =arXiv:1509.02062 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1509.02062

Web End =INSPIRE ].[21] E. Witten, (2 + 1)-Dimensional Gravity as an Exactly Soluble System, http://dx.doi.org/10.1016/0550-3213(88)90143-5

Web End =Nucl. Phys. B 311

http://dx.doi.org/10.1016/0550-3213(88)90143-5

Web End =(1988) 46 [http://inspirehep.net/search?p=find+J+%22Nucl.Phys.,B311,46%22

Web End =INSPIRE ].[22] A. Campoleoni, S. Fredenhagen, S. Pfenninger and S. Theisen, Asymptotic symmetries of

three-dimensional gravity coupled to higher-spin elds, http://dx.doi.org/10.1007/JHEP11(2010)007

Web End =JHEP 11 (2010) 007

[http://arxiv.org/abs/1008.4744

Web End =arXiv:1008.4744 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1008.4744

Web End =INSPIRE ].[23] M. Henneaux and S.-J. Rey, Nonlinear W1 as Asymptotic Symmetry of Three-Dimensional

Higher Spin Anti-de Sitter Gravity, http://dx.doi.org/10.1007/JHEP12(2010)007

Web End =JHEP 12 (2010) 007 [http://arxiv.org/abs/1008.4579

Web End =arXiv:1008.4579 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1008.4579

Web End =INSPIRE ].[24] M. Ammon, A. Castro and N. Iqbal, Wilson Lines and Entanglement Entropy in Higher Spin

Gravity, http://dx.doi.org/10.1007/JHEP10(2013)110

Web End =JHEP 10 (2013) 110 [http://arxiv.org/abs/1306.4338

Web End =arXiv:1306.4338 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1306.4338

Web End =INSPIRE ].[25] J. de Boer and J.I. Jottar, Entanglement Entropy and Higher Spin Holography in AdS3, http://dx.doi.org/10.1007/JHEP04(2014)089

Web End =JHEP 04 (2014) 089 [http://arxiv.org/abs/1306.4347

Web End =arXiv:1306.4347 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1306.4347

Web End =INSPIRE ].

{ 26 {

JHEP07(2016)049

[26] J. de Boer, A. Castro, E. Hijano, J.I. Jottar and P. Kraus, Higher spin entanglement and WN conformal blocks, http://dx.doi.org/10.1007/JHEP07(2015)168

Web End =JHEP 07 (2015) 168 [http://arxiv.org/abs/1412.7520

Web End =arXiv:1412.7520 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1412.7520

Web End =INSPIRE ].[27] J. de Boer and J.I. Jottar, Boundary conditions and partition functions in higher spin

AdS3/CFT2, http://dx.doi.org/10.1007/JHEP04(2016)107

Web End =JHEP 04 (2016) 107 [http://arxiv.org/abs/1407.3844

Web End =arXiv:1407.3844 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1407.3844

Web End =INSPIRE ].[28] M. Henneaux, A. Perez, D. Tempo and R. Troncoso, Chemical potentials in three-dimensional

higher spin anti-de Sitter gravity, http://dx.doi.org/10.1007/JHEP12(2013)048

Web End =JHEP 12 (2013) 048 [http://arxiv.org/abs/1309.4362

Web End =arXiv:1309.4362 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1309.4362

Web End =INSPIRE ].[29] J. Long, Higher Spin Entanglement Entropy, http://dx.doi.org/10.1007/JHEP12(2014)055

Web End =JHEP 12 (2014) 055 [http://arxiv.org/abs/1408.1298

Web End =arXiv:1408.1298 ]

[http://inspirehep.net/search?p=find+EPRINT+arXiv:1408.1298

Web End =INSPIRE ].

[30] S. Datta, J.R. David, M. Ferlaino and S.P. Kumar, Higher spin entanglement entropy from

CFT, http://dx.doi.org/10.1007/JHEP06(2014)096

Web End =JHEP 06 (2014) 096 [http://arxiv.org/abs/1402.0007

Web End =arXiv:1402.0007 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1402.0007

Web End =INSPIRE ].[31] S. Datta, J.R. David, M. Ferlaino and S.P. Kumar, Universal correction to higher spin

entanglement entropy, http://dx.doi.org/10.1103/PhysRevD.90.041903

Web End =Phys. Rev. D 90 (2014) 041903 [http://arxiv.org/abs/1405.0015

Web End =arXiv:1405.0015 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1405.0015

Web End =INSPIRE ].[32] A.L. Fitzpatrick, J. Kaplan, D. Li and J. Wang, On information loss in AdS3/CFT2, http://dx.doi.org/10.1007/JHEP05(2016)109

Web End =JHEP

http://dx.doi.org/10.1007/JHEP05(2016)109

Web End =05 (2016) 109 [http://arxiv.org/abs/1603.08925

Web End =arXiv:1603.08925 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1603.08925

Web End =INSPIRE ].[33] M.A. Nielsen and I.L. Chuang, Quantum computation and quantum information Cambridge

University Press, Cambridge U.K. (2010).[34] C. Holzhey, F. Larsen and F. Wilczek, Geometric and renormalized entropy in conformal

eld theory, http://dx.doi.org/10.1016/0550-3213(94)90402-2

Web End =Nucl. Phys. B 424 (1994) 443 [http://arxiv.org/abs/hep-th/9403108

Web End =hep-th/9403108 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/9403108

Web End =INSPIRE ].[35] L.-Y. Hung, R.C. Myers and M. Smolkin, Twist operators in higher dimensions, http://dx.doi.org/10.1007/JHEP10(2014)178

Web End =JHEP 10

http://dx.doi.org/10.1007/JHEP10(2014)178

Web End =(2014) 178 [http://arxiv.org/abs/1407.6429

Web End =arXiv:1407.6429 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1407.6429

Web End =INSPIRE ].[36] D. Harlow, J. Maltz and E. Witten, Analytic Continuation of Liouville Theory, http://dx.doi.org/10.1007/JHEP12(2011)071

Web End =JHEP 12

http://dx.doi.org/10.1007/JHEP12(2011)071

Web End =(2011) 071 [http://arxiv.org/abs/1108.4417

Web End =arXiv:1108.4417 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1108.4417

Web End =INSPIRE ].[37] T. Eguchi and H. Ooguri, Conformal and Current Algebras on General Riemann Surface,

http://dx.doi.org/10.1016/0550-3213(87)90686-9

Web End =Nucl. Phys. B 282 (1987) 308 [http://inspirehep.net/search?p=find+J+%22Nucl.Phys.,B282,308%22

Web End =INSPIRE ].[38] M. Gutperle and P. Kraus, Higher Spin Black Holes, http://dx.doi.org/10.1007/JHEP05(2011)022

Web End =JHEP 05 (2011) 022

[http://arxiv.org/abs/1103.4304

Web End =arXiv:1103.4304 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1103.4304

Web End =INSPIRE ].[39] P. Kraus and E. Perlmutter, Partition functions of higher spin black holes and their CFT

duals, http://dx.doi.org/10.1007/JHEP11(2011)061

Web End =JHEP 11 (2011) 061 [http://arxiv.org/abs/1108.2567

Web End =arXiv:1108.2567 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1108.2567

Web End =INSPIRE ].[40] K.B. Alkalaev and V.A. Belavin, Holographic interpretation of 1-point toroidal block in the

semiclassical limit, http://arxiv.org/abs/1603.08440

Web End =arXiv:1603.08440 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1603.08440

Web End =INSPIRE ].[41] J. Cardy and C.P. Herzog, Universal Thermal Corrections to Single Interval Entanglement

Entropy for Two Dimensional Conformal Field Theories, http://dx.doi.org/10.1103/PhysRevLett.112.171603

Web End =Phys. Rev. Lett. 112 (2014) 171603

[http://arxiv.org/abs/1403.0578

Web End =arXiv:1403.0578 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1403.0578

Web End =INSPIRE ].[42] G. Comp ere and W. Song, W symmetry and integrability of higher spin black holes, http://dx.doi.org/10.1007/JHEP09(2013)144

Web End =JHEP

http://dx.doi.org/10.1007/JHEP09(2013)144

Web End =09 (2013) 144 [http://arxiv.org/abs/1306.0014

Web End =arXiv:1306.0014 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1306.0014

Web End =INSPIRE ].[43] A. Castro, N. Iqbal and E. Llabr es, Eternal Higher Spin Black Holes: a Thermo eld

Interpretation, http://arxiv.org/abs/1602.09057

Web End =arXiv:1602.09057 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1602.09057

Web End =INSPIRE ].[44] E. Perlmutter, Bounding the Space of Holographic CFTs with Chaos, http://arxiv.org/abs/1602.08272

Web End =arXiv:1602.08272

[http://inspirehep.net/search?p=find+EPRINT+arXiv:1602.08272

Web End =INSPIRE ].

[45] M. Besken, A. Hegde, E. Hijano and P. Kraus, Holographic conformal blocks from interacting

Wilson lines, http://arxiv.org/abs/1603.07317

Web End =arXiv:1603.07317 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1603.07317

Web End =INSPIRE ].

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