Holographic description of 2D conformal block in semi-classical limit

Abstract

Abstract

In this paper, we study the holographic descriptions of the conformal block of heavy operators in two-dimensional large c conformal field theory. We consider the case that the operators are pairwise inserted such that the distance between the operators in a pair is much smaller than the others. In this case, each pair of heavy operators creates a conical defect in the bulk. We propose that the conformal block is dual to the on-shell action of three dimensional geometry with conical defects in the semi-classical limit. We show that the variation of the on-shell action with respect to the conical angle is equal to the length of the corresponding conical defect. We derive this differential relation on the conformal block in the field theory by introducing two extra light operators as both the probe and the perturbation. Our study also suggests that the area law of the holographic Rényi entropy must holds for a large class of states generated by a finite number of heavy operators insertion.

Published for SISSA by Springer

Received: September 16, 2016 Accepted: October 18, 2016

Published: October 20, 2016

Bin Chen,a,b,c Jie-qiang Wua and Jia-ju Zhangd,e

aDepartment of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, 5 Yiheyuan Rd, 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

dTheoretical Physics Division, Institute of High Energy Physics,Chinese Academy of Sciences, 19B Yuquan Rd, Beijing 100049, P.R. China

eTheoretical Physics Center for Science Facilities, Chinese Academy of Sciences, 19B Yuquan Rd, Beijing 100049, P.R. China

E-mail: mailto:[email protected]

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

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

Web End [email protected]

Abstract: In this paper, we study the holographic descriptions of the conformal block of heavy operators in two-dimensional large c conformal eld theory. We consider the case that the operators are pairwise inserted such that the distance between the operators in a pair is much smaller than the others. In this case, each pair of heavy operators creates a conical defect in the bulk. We propose that the conformal block is dual to the on-shell action of three dimensional geometry with conical defects in the semi-classical limit. We show that the variation of the on-shell action with respect to the conical angle is equal to the length of the corresponding conical defect. We derive this di erential relation on the conformal block in the eld theory by introducing two extra light operators as both the probe and the perturbation. Our study also suggests that the area law of the holographic R enyi entropy must holds for a large class of states generated by a nite number of heavy operators insertion.

Keywords: AdS-CFT Correspondence, Conformal Field Theory

ArXiv ePrint: 1609.00801

Open Access, c

[circlecopyrt] The Authors.

Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP10(2016)110

Web End =10.1007/JHEP10(2016)110

Holographic description of 2D conformal block in semi-classical limit

JHEP10(2016)110

Contents

1 Introduction 1

2 Gravity solution with conical defect 52.1 Holographic R enyi entropy 52.2 Holographic description of two dimensional conformal block 72.3 Singular behaviors 8

3 Conformal block in eld theory 103.1 Monodromy and geodesic length 113.2 Length of conical defect 153.3 Conformal block 15

4 Examples 184.1 Two-point function 184.2 Four-point function 18

5 Conclusions and discussions 20

A Four-point function in another way 23

1 Introduction

The recent renaissance of the conformal bootstrap sheds new light on conformal eld theories and the AdS/CFT correspondence [1]. In the dimension greater than three, the conformal bootstrap leads to some remarkable results, including the study of 3D Ising model. For the nice reviews, see [2{4]. In two dimensions, due to the fact that the 2D conformal symmetry is in nite dimensional, there are more exact and analytical results on the conformal blocks, especially in the large c limit. The 2D semi-classical conformal blocks play essential role in at least two research areas, one being the AGT relation [5] and the other being the AdS3/CFT2 correspondence.

In the AdS3/CFT2 correspondence, the quantum gravity with the Brown-Henneaux asymptotic boundary condition [6] is dual to a two dimensional conformal eld theory(CFT) with the central charge

2G: (1.1)

In the semi-classical limit, it is believed that the pure AdS3 gravity is dual to a large c CFT with sparse light spectrum [7, 8]. By the modular invariance and conformal bootstrap there are more constrains on the spectrum and the OPE coe cients [7{14]. In this limit, the correspondence is simpli ed. On the eld theory side, the contribution from the vacuum

{ 1 {

JHEP10(2016)110

cL = cR = 3l

module states dominates, and the ones from other states are non-perturbatively suppressed. As the vacuum module is universal for all the CFT, it is not necessary to know the explicit construction of the CFT. On the gravity side, one may just focus on the semiclassical solutions of pure AdS3 gravity and ignore the other possible ingredients [15]. The study of the partition function of the handlebody solutions supports this semi-classical picture. For example, the 1-loop partition function of the gravitational handlebody solution of any genus [16] has been reproduced exactly in CFT [17].

The semiclassical conformal block has been intensively studied in the AdS3/CFT2 correspondence. One well-studied conformal block is in the correlation function of two heavy operators and two light operators. In this setup, the long distance behavior of a particle scattering with a BTZ black hole [18, 19] has been reproduced in [20]. This suggests that one can use this system to study various problems in black hole physics [21{26]. The essence is that the presence of the heavy operators change the background geometry to a BTZ black hole. For other aspects on this type of conformal block, see [27{31].

Moreover, the semiclassical conformal block plays a key role in the recent study on the holographic entanglement entropy. It was conjectured [32, 33] that for a eld theory with holographic description the entanglement entropy is proportional to the area of minimal surface anchored at the entanglement surface

SRT = Area

4G ; (1.2) at the leading order. In the eld theory, it is convenient to use the replica trick to study the entanglement entropy. By de nition, the n-th R enyi entropy is de ned by

Sn =

1n 1

TrnA; (1.3)

where A is the reduced density matrix. By the path integral, the R enyi entropy Sn can be given by the partition function of the theory on an n-sheeted space obtained by pasting n sheets of space with each other along the entanglement surface. Assuming Sn can be continuously extended to non-integer, SEE can be captured by taking the n ! 1 limit

from SnSEE = log TrA log A = lim

n!1

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Sn: (1.4)

The path integral can be regarded as the integral over an n-copied theory with twist boundary condition on the elds along the entanglement surface. In two dimensional conformal eld theory the twisted boundary condition can be imposed by the twist operators at the branch points. Then the R enyi entropy equals to the multi-point correlation function of the twist operators

[angbracketleft]T (z1)T (z2) : : :[angbracketright]; (1.5) where the twist operator is of the conformal dimension hT = nc24(1

1n2 ). In conformal eld theory, the correlation function can be decomposed into the conformal blocks. For the CFT with sparse light spectrum, only vacuum module states have perturbative contribution in the large central charge limit. At the leading order, the conformal block can be determined by a monodromy problem [34, 35]. Usually, there is no analytic solution for the monodromy

{ 2 {

problem for general operators. However, when n ! 1, the monodromy problem can be

solved explicitly. In this way, the multi-interval entanglement entropy was studied in [7].

Holographically, the R enyi entropy is computed by the partition function of the gravitational solution whose asymptotic boundary is the n-sheeted Riemann surface [36, 37]. The solution can be given by extending the Schottky uniformization of the boundary Riemann surface into the bulk. Remarkably the uniformization is determined by the same monodromy problem in the eld theory. Simply speaking, the leading order contribution to the conformal block is captured by the on-shell regularized action of the gravitational con guration, governed by the same monodromy problem [38, 39].

For the holographic entanglement entropy (HEE), we need to take n ! 1 limit, then

the twist operators become light so we can ignore the back-reaction to the background. In this case, the twist operators only detect the geodesic in the bulk but never change the geometry, such that the HEE is given by the length of the geodesics, reproducing the RT formula. Similar discussion has been applied to the correlation function of two heavy and two light operators, to the thermalization e ect and to the higher spin system. The recent investigation shows that the single-interval entanglement entropy for the states built from nite number of heavy operator insertion is always given by the Ryu-Takayanagi formula [40]. The picture is that the insertion of the heavy operators changes the geometry, which is determined by the boundary stress tensor of the heavy operators, and the bulk geodesic in the modi ed spacetime ending at the branch points gives the HEE.

For the n-th R enyi entropy with n > 1, it has less been discussed, but it is more interesting in the sense that it encodes the spectral information of the reduced density matrix and may reveals more information of the theory. On the eld theory side, the twist operator becomes heavy so one has to consider the conformal block of the heavy operators. On the bulk side, the background is modi ed by the back-reaction and one has to nd the new semi-classical con guration. In general, it is di cult to determine the on-shell Zograf-Takhtadzhyan action in the bulk. But for the cases of two short intervals on the complex plane and a single interval on the torus, one can compute the action perturbatively and nd good agreement with the CFT computation [41{46]. Very recently, Xi Dong studied the gravity dual of the R enyi entropy and proposed that the holographic R enyi entropy satis ed an area law reminiscent of the Bekenstein-Hawking entropy formula and the RT formula [47]

n2 @ @n

n 1n Sn[parenrightbigg]= Area[notdef]n(Cosmic brane)4G ; (1.6)

where the cosmic brane has the tension Tn = n14nG. The relation (1.6) could be taken as an one-parameter generalization of the RT formula. When n ! 1, the cosmic brane becomes

tensionless and could be taken as a probe without back-reacting the background. In this case, the con guration of the cosmic brane is determined to be a minimal surface such that the relation (1.6) reduces to the RT formula (1.2).

In this paper, we try to extend the relation (1.6) to general conformal block of heavy operators. Actually there have already been some discussions on the holographic description for the conformal blocks [48{51]. It has been found a relation between the conformal block and the Witten diagram. The similar idea has also been used in the bulk operator

{ 3 {

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O2 ON-1

ON-1

O2

O1

[1 ]

[1 ]

ON

[1 ]

[1 ]

O1

ON

Figure 1. For simplicity, we only consider the OPE channel in which the heavy operators contract pairwise.

realization [52]. However only the conformal block for global conformal symmetry [49, 52] and Virasoro conformal block for the light operators [50] have been discussed. In those cases, there is no back-reaction from the inserting operators so that the probe picture makes perfect sense. In [53], the holographic description of of the 1-point toroidal block in the semiclassical limit has been discussed. In this work we try to study the holographic description of the conformal block for the heavy operators in the complex plane, in which the back-reaction to the background has to be considered. The operators we consider here are heavy in the sense that their conformal weight is of the same order as c.

To simplify the situation, we focus on the case that the heavy operators Oi(zi) and

Oi(z[prime]i) are of the same type are inserted in pairs such that the operators in a pair are near to each other, and the pairs sit far apart. We suggest that the leading order Virasoro conformal block should be dual to the on-shell action for 3D gravity with conical defects

F = [angbracketleft]O1(z1)O1(z[prime]1) : : : ON(zN)ON(z[prime]N)[angbracketright] = eIon-shell: (1.7)

In general, the conformal weights of the operators could be di erent, not necessarily to be the one for the twist operator. The correlation function could be expanded in di erent OPE channels, so in terms of di erent conformal block. We consider the channels shown in gure 1, in which the pair of heavy operators fuse into the states in the vacuum module.

In principle, the conformal block could be described holographically by the on-shell action of the gravitational con guration with the boundary stress tensor. However, such a picture is useless for our purpose. Instead, as suggested by the study of the R enyi entropy, the e ect of an operator pair is to create a conical defect in the bulk. Each conical defect ends at the insertion points of the operators Ok(zk) and Ok(z[prime]k) with the conical angle

k = 2 1 1 nk

; (1.8)

which is related to the conformal dimension of the inserted operators

hk = c

24

1 1 n2k

: (1.9)

Note that nk can be a non-integer so the conformal weights of the heavy operators are quite general. We take the above form for the conformal weight in order to compare with the

{ 4 {

JHEP10(2016)110

discussion for the twist operators in using the replica trick. The on-shell action on the right hand side of (1.7) includes the contributions from the AdS3 gravity, the conical defects and the boundary terms. Taking a variation with respect to the conical angle on (1.7) on the gravity side, which gives the length of the conical defect, we nd the following relation

n2j

@@nj log F =

Lj4G + fj; 8j (1.10)

where Lj is the length of the conical defect which connects the operators Oj(zj) and Oj(z[prime]j). fj is a re-normalization constant, which depends only on nj but not on the locations of operators. The relation (1.10) is a generalization of the relation (1.6).

On the eld theory side, in order to prove the relation, we have to face the problems of de ning properly the length Lj and the variation with respect to the conformal dimension.

These two problems can be solved once in a while. The essential point is to introduce two light operators of exactly the same type in the system. As discussed in [40], the geodesic length homologous to an interval can be de ned to be the expectation value of two extra light operators at the end points of the interval. We propose that the length Lj is de ned to be the limit of the geodesic length, when the light operators are moved to the insertion points of the corresponding heavy operators. Moreover, the light operators and the heavy operators may form a composite operator for the observer far from them. By considering the response of the system with respect to the bound pair of the heavy and the light operators, we read the variation of the conformal block with respect the conformal dimension, and prove the relation (1.10) in the eld theory.

In section 2, we brie y review the discussion in [47] and extend it to the holographic description of general conformal block in 2D large c CFT. We nd the relation (1.10). In section 3, we use the monodromy trick to prove this relation in the eld theory. We rst review the discussion in [40] on the expectation value of two extra light operators in the system with 2N heavy operators. With that result, we can give a eld theory interpretation for the equation (1.10) and prove it. In section 4, we apply our treatment to two well-studied examples, including the two-point function and the four-point function, and nd consistent agreement. In section 5, we end with brief discussion and conclusion. In appendix A we present another computation on the four-point function.

2 Gravity solution with conical defect

In this section, we discuss the conical defect in the AdS3 and the on-shell action. After a brief review of the gravity dual of the R enyi entropy following [47], we discuss how to extend the study to the case of general conformal block. In particular, we discuss carefully the singular behaviors near the conical defect and asymptotic boundary, and how to regularize them in di erent coordinates.

2.1 Holographic R enyi entropy

The holographic entanglement entropy could be taken as a generalized version of black hole entropy [54]. Actually the rst proof for the holographic entanglement entropy was based

{ 5 {

JHEP10(2016)110

on the topological black hole entropy [55]. From the replica trick, the R enyi entropy can be transformed to the partition function on the branched cover Mn, an n-sheeted space pasting n-copies of the spacetime along the branch cut

Sn = 1 1 n

(log Z(Mn) n log Z(M1)) (2.1)

From the AdS/CFT correspondence, the eld theory partition function can be evaluated by the gravity partition function with the n-sheeted surface as its asymptotic boundary. At the leading order, the partition function equals to the exponential of the on-shell bulk action

Z(Mn) = eI[Bn]: (2.2)

The gravity solution Bn is smooth and respect the Zn replica symmetry. Taking a quotient of the bulk con guration by the replica symmetry, we may nd a bulk solution as a orbifold

^

Bn = Bn=Zn: (2.3)

The xed point of the Zn symmetry is a co-dimension two surface homologous to the entangling region in the boundary. This is a conical defect with a conical angle

= 2 1 1 n

: (2.5)

The on-shell action of the bulk solution Bn equals to the one from n-copies of the solution ^

BnI[Bn] = nI[ ^

Bn]; (2.6)

whereI[ ^

Bn] =

The integral region ^

Bn include the bulk region except the conical defect. As argued in [54], there is no singularity around the xed point in the bulk solution Bn, so we should not include the contribution from the conical defect in ^

Bn. On the other hand, we may introduce

the cosmic brane in the action

1 16G

JHEP10(2016)110

: (2.4)

Equivalently, the conical singularity can be provided by a cosmic brane in the bulk with the tension

Tn = 1 4G

1 1 n

1 16G

[integraldisplay]

Bn dd+1xLg + boundary term: (2.7)

1 1n[parenrightbigg] [integraldisplay]d + boundary term: (2.8)

The integral region B includes all of the bulk region, both the smooth part and singular part. d denotes the integral over the cosmic brane. The integral d gives the area of the brane, such that the contribution from the cosmic brane cancels the contribution from the conical singularity in the rst term. In other words, the conical singularity can be understood as the backaction of the cosmic brane on the background. From this action,

{ 6 {

I[ ^

Bn] =

ZB dd+1xLg +1 4G

we can nd the classical solution of the gravity and the embedding of the cosmic brane. In the case that there exist multiple solutions, the one with the least action dominates the path integral and gives I[Bn]. In the limit n ! 1, the brane tension is vanishing and the

brane can be taken as a probe such that its trajectory is a minimal surface, which leads to the RT formula.

By taking a variation of the action with respect to the conical angle, we get the area of the conical defect

I[ ^

Bn] = I[

^

Bn]

X X +

1 4G

n n2

[integraldisplay]

4G : (2.9)

Here X denotes all the degrees of freedom including the metric and the embedding of the cosmic brane. Because the elds in the system satisfy the equations of motion, the rst part in the variation of the action vanish, and the remaining part is proportional to the area of the cosmic brane. In terms of the R enyi entropy, it is easy to get

n2 @

@n

n 1n Sn[parenrightbigg]= Area[notdef]n(Cosmic brane)4G ; (2.10)

as shown in [47].

2.2 Holographic description of two dimensional conformal block

Now we try to use the technic discussed above to study the conformal block in 2D large c CFT

F = [angbracketleft]O1(z1)O1(z[prime]1)O2(z2)O2(z[prime]2) : : : ON(zN)ON(z[prime]N)[angbracketright]; (2.11)

where

hk = c

24

1 1 n2k

: (2.12)

We emphasize that, the nk is not necessarily an integer but just a convenient way to describe the conformal dimension. In the large c CFT, there are only vacuum module states propagating in the intermediate channels as shown in gure 1.

In 2D CFT, the R enyi entropy could be computed by the multi-point function of the twist operators in a orbifold CFT. The conformal weight of the twist operator is1

hT =

c 24

d = n

n2

Area

JHEP10(2016)110

1 1 n2

; (2.13)

so without taking n ! 1 limit the twist operator is absolutely heavy. As shown in the

last subsection, the cosmic brane ending on the branch points back-react the background spacetime. On the other hand, the conformal block for the light operators is computed by the geodesic Witten diagram [49, 50]. This can be easily understood in the two-point function case. The particle corresponding to the light operator propagates along the geodesics as a probe. When the dimension of the operator becomes large, the massive particle leads

1Note that the central charge of the orbifold CFT is c = nc0, where c0 is the central charge of original CFT.

{ 7 {

to a conical defect. In general, when all the operators are heavy, it is di cult to determine the full back-reacted geometry. In this paper, we consider a special but interesting case. We would like to study the conformal blocks of 2N heavy operators, with each pair of the operators Oi(zi) and Oi(z[prime]i) being near to each other and di erent pairs are far apart. In this case, each pair of the operators may create a cosmic brane, homologous to the interval between two operators. On the CFT side, this means that in solving the monodromy problem to get the uniformization we should impose trivial monodromy condition around the cycle enclosing the pair of the operators. We propose that the conformal block is equal to the on-shell gravitational action with a set of conical defects, each being homologous to the interval between zi and z[prime]i. The conical angle for each defect equals

k = 2 1 1 nk

: (2.14)

Each conical defect is created by propagating a massive particle with mass

Tk = 1 4G

1 1 nk

: (2.15)

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The conformal block can be evaluated as

F = eIon-shell; (2.16)

where

[integraldisplay]

1 1nk[parenrightbigg] [integraldisplay]d ; (2.17)

where Lg is the Lagrangian for the AdS3 gravity, including the Einstein-Hilbert term and

a negative cosmological constant. As derived in previous sub-section, when we take a variation with respect to the conical defect

n2j

@@nj log F =

Ion-shell =

1 16G

d3xLg + boundary terms +

N

Xk=11 4G

Lj4G + fj; (2.18)

where Lj is the length of the j-th conical defect which connect the operators Oj(zj) and Oj(z[prime]j). fj is a renormalization constant, which depends only on nk, independent of the locations of operators. As we will explain in the next subsection, the length Lj depends on the regularization scheme. Di erent regularization equals to each other up to a term fj.

2.3 Singular behaviors

In the AdS3 metric with a conical defect, there are two kinds of singularities: conical singularity and the IR divergence in the asymptotic boundary. The IR divergence in the asymptotic boundary is more common to us. For example, when we study the holographic entanglement entropy or the on-shell action, the geodesic length or volume is always divergent close to the asymptotic boundary. So we can take an IR cut-o , which corresponds to the UV cut-o in the eld theory. Di erent cut-o s can be imposed by choosing di erent metrics for the same conformal structure [56].

{ 8 {

We may use the Banodos metric

ds2 = l2

dv2v2 6c T (z)dz2 6c

T ( z)d z2 +[parenleftbigg]1v2 + v236c2 T (z)

T ( z)

; (2.19)

and choose the IR cut-o at v = . The dual eld theory now lives on a space with a at metric

ds2 = dzd

dzd z

z: (2.20)

For a smooth stress tensor, the metric is well-de ned. However if there are operators inserting in the eld theory, the stress tensor have singularities and there are singularites in the bulk as well. For example, if there is a heavy operator inserting at the origin,

T (z) = c

24

1 1n2[parenrightbigg]1 z2

T (

z) = c

24

1 1n2[parenrightbigg]1 z2 ; (2.21)

there is a singularity in the metric when z ! 0 for any v. In terms of a new set of

coordinates

z = ei

z = ei ; (2.22)

the metric can be written as

ds2 = l2

dv2v2 +

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1v v 4

1 1n2[parenrightbigg]1 2

2d2 +

1v +v 4

1 1n2[parenrightbigg]1 2

2 2d 2

: (2.23)

It is clear for a xed radius v, when we take goes to zero there is always a singularity. To understand this metric, we take a further coordinate transformation

r =

1 n

2n ( 1n 1)v2

1 n

42 + v2( 1n 1)2

;

4n

1n +1

u = v

42 + v2( 1n 1)2

: (2.24)

In terms of (r; u), the metric can be written as

ds2 = l2 u2

du2 + dr2 + r2n2 d 2[parenrightbigg]: (2.25)

This is a conical defect. The coordinates (u; r; ) is the Poincare coordinates, with u being the radial direction and u = 0 being the asymptotic boundary. Around the conical defect, even though the metric is still non-smooth, it is continuous and less singular than (2.23) when ! 0 or r ! 0.

Let us consider the coordinate transformation (2.24) more carefully. If we keep v xed and take ! 0, both u and r go to zero. That means for any xed v the coordinates close

{ 9 {

to = 0 only describe the region close to the end of conical defect. Because of the IR divergence, the metric around that part is always singular.

To take a proper IR cut-o , we need to use both the coordinates (2.23) and (2.25). We call the coordinates (; v) in (2.23) canonical coordinates, because the metric in terms of them has right asymptotic metric ds2 = dzd

z. However it is singular around the conical defect ! 0. We call the coordinates (r; u) in (2.25) regular coordinates, because the

metric in them is regular at the conical defect, but its asymptotic condition is not correct. To take an IR cut-o for the system, we can still choose v = in (2.23) as usual for the region away from the conical singularity. However, for the region close to the conical singularity, we need to use the coordinates (2.25) to take an IR cut-o . We choose u = u0 for the IR cut-o around the conical singularity, which gives a nite cut for the length of the conical defect. The IR cut-o v = in (2.23) and u = u0 in (2.25) should connect with each other around the conical singularity.

At the end of this section, we introduce another regularization for the IR divergence which is more convenient for eld theory description. From the coordinate transformation (2.24), if we keep xed and take v to zero, we get (r =

1n ; u ! 0). If is also close

to zero, it describe the end of the conical defect as well. For the length of conical defect, it can be evaluated by the geodesic length whose ending point is close to the conical defect. In the next section, we will mainly use this way to regularize the IR divergence. As we will see, this regularization has a proper eld theory description in terms of a two-point function.

3 Conformal block in eld theory

In this section, we try to give a eld theory description of the result2

n2j

1

2hl log [angbracketleft]

where is a light operator with the conformal dimension

1 hl c: (3.3) With this quantity, we can evaluate the length Lj of the conical defect by moving u1 and u[prime]1 close to the locations zj and z[prime]j of the heavy operators Oj

Lj = lim u1 ! zj u[prime]1 ! z[prime]j

2We set the AdS radius to be unit l = 1, then we have the relation

14G =

which often makes the formulas simpler.

JHEP10(2016)110

Lj4G + fj: (3.1)

As in [40], the holographic geodesic length can be de ned through the correlation function of two light operators

L =

@@nj log F = n2j

@@nj log[angbracketleft]O1(z1)O[prime]1(z[prime]1) : : : ON(zN)O[prime]N(z[prime]N)[angbracketright] =

(u1)(u[prime]1)O1(z1)O1(z[prime]1) : : : ON(zN)ON(z[prime]N)[angbracketright]

hO1(z1)O1(z[prime]1) : : : ON(zN)ON(z[prime]N)[angbracketright]

; (3.2)

(L + cut-o ): (3.4)

c 6 ,

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O2 ON-1

ON-1

O2

O1

[1 ]

[1 ]

ON

[1 ]

[1 ]

O1

ON

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(a) Conformal block

O2 ON-1

ON-1

O2

O1

[1 ]

[1 ]

[1 ]

ON

[1 ]

[1 ]

[1 ]

O1

ON

(b) With two extra operator

Figure 2. We only consider vacuum module contribution in the correlation function.

In (3.4), we remove the divergent terms. We will see that the cut-o terms do not depend on the locations of the operators. In (3.2), we only consider the contribution from the vacuum module states in each channel as shown in gure 2. The numerator and denominator correspond to gure 2(a) and gure 2(b) respectively. For simplicity, we show that the light operators s are near the pair of the operators O1s in the gure. They can actually move around to any other pair of the heavy operators.

3.1 Monodromy and geodesic length

In this subsection, we brie y review how to the use the monodromy trick to calculate the geodesic length (3.2). The discussion follows the work in [40]. The rst step is to introduce a degenerate representation with a null state [7]

|~[angbracketright] =

L2 32(2h + 1)L21[parenrightbigg][notdef]^ [angbracketright]; (3.5)

where in the large c limit

h =

1

2

92c: (3.6)

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We de ne the quantities

(z) [angbracketleft]

^(z)(u1)(u[prime]1)O1(z1)O1(z[prime]1) : : : ON(zN)ON(z[prime]N)[angbracketright]

h(u1)(u[prime]1)O1(z1)O1(z[prime]1) : : : ON(zN)ON(z[prime]N)[angbracketright]

;

T (z) [angbracketleft]

^

T (z)(u1)(u[prime]1)O1(z1)O1(z[prime]1) : : : ON(zN)ON(z[prime]N)[angbracketright]

h(u1)(u[prime]1)O1(z1)O1(z[prime]1) : : : ON(zN)ON(z[prime]N)[angbracketright]

;

0(z) [angbracketleft]

^(z)O1(z1)O1(z[prime]

1) : : : ON(zN)ON(z[prime]N)[angbracketright] hO1(z1)O1(z[prime]1) : : : ON(zN)ON(z[prime]N)[angbracketright]

;

T0(z) [angbracketleft]

^

T (z)O1(z1)O1(z[prime]1) : : : ON(zN)ON(z[prime]N)[angbracketright]

hO1(z1)O1(z[prime]1) : : : ON(zN)ON(z[prime]N)[angbracketright]

: (3.7)

Inserting the null state into the correlation function,

h~(z)(u1)(u[prime]1)O1(z1)O1(z[prime]1) : : : ON(zN)ON(z[prime]N)[angbracketright] = 0;

h~(z)O1(z1)O1(z[prime]1) : : : ON(zN)ON(z[prime]N)[angbracketright] = 0; (3.8)

we get

@2 (z) + 6c T (z) (z) = 0;

@2 0(z) + 6c T0(z) 0(z) = 0; (3.9)

where

T (z) =

n

Xk=1

JHEP10(2016)110

[bracketleftbigg]

hk(z zk)2

+ k

+ [prime]kz z[prime]k [bracketrightbigg]

z zk

+ hk

(z z[prime]k)2

+ hl

(z u1)2

+ ~

1z u1

+ hl

(z u[prime]1)2

+ ~

[prime]1z u[prime]1

;

T0(z) =

n

Xk=1

[bracketleftbigg]

hk(z zk)2

+ k,0 z zk

+ hk

(z z[prime]k)2

+ [prime]k,0z z[prime]k [bracketrightbigg]

: (3.10)

Here, the accessory parameters k, [prime]k, ~

1 and ~

[prime]1 depend on the locations zks, z[prime]ks, u1 and u[prime]1 of all the operators, while k,0 and [prime]k,0 depend only on zks and z[prime]ks. There are two independent solutions (+) and () to the di erential equations (3.9). In terms of the quantities

a(z) = 0 6cT (z)

1 0

!;

a0(z) = 0 6cT0(z)

1 0

!;

v(z) = @ (+)(z) @ ()(z)

(+)(z) ()(z)

!;

v0(z) = @ (+)0(z) @ ()0(z)

(+)0 (z) ()0(z)

!; (3.11)

{ 12 {

the equation (3.9) can be recast into a compact form

@v(z) = a(z)v(z);

@v0(z) = a0(z)v0(z): (3.12)

Because that the conformal dimension hl is much smaller than the others, we can take a perturbation with respect to hl. By imposing the trivial monodromy condition around the cycle enclosing only (u1) and (u[prime]1), we can easily get

~

1 = 2h

tr 1 0

0 0

!v0(u1)v0(u[prime]1)1

tr 0 1

0 0

JHEP10(2016)110

!v0(u1)v0(u[prime]1)1;

!v0(u1)v0(u[prime]1)1

tr 0 1

0 0

[prime]1 = 2h

tr 0 0

0 1

~

!v0(u1)v0(u[prime]1)1: (3.13)

Taking into the Ward identity

@@u1 log[angbracketleft](u1)(u[prime]1)O1(z1)O1(z[prime]1) : : :[angbracketright] = ~

1;

@ @u[prime]1

log[angbracketleft](u1)(u[prime]1)O1(z1)O1(z[prime]1) : : :[angbracketright] = ~

[prime]1; (3.14)

we nd

log [angbracketleft](u1)(u[prime]1)O1(z1)O1(z[prime]1) : : :[angbracketright]

hO1(z1)O1(z[prime]1) : : :[angbracketright]

= 2hl log tr

0 1 0 0

!v0(u1)v0(u[prime]1)1 + g(z1; z[prime]1; : : : zN; z[prime]N); (3.15)

where the function g depends only on zj and z[prime]j but not on u1 or u[prime]1. The monodromies around other cycles are very hard to solve even perturbatively, so we cannot x the function g directly. However, by taking the small interval limit we can show that it is actually a constant. In the small interval limit with u1 ! u[prime]1, the ~

1 and ~

[prime]1 asymptotically behave as

~

1 =

2hl u1 u[prime]1

+ O(u1 u[prime]1);

~

+ O(u1 u[prime]1): (3.16)

Taking them into the stress tensor and keeping zs far from u1 and u[prime]1

|z u[prime]1[notdef] [notdef]u1 u[prime]1[notdef]; (3.17)

{ 13 {

[prime]1 = 2hl

u1 u[prime]1

we get

T (z) =

Xj

[bracketleftbigg]

hj(z zj)2

+ j

+ [prime]jz z[prime]j [bracketrightbigg]

+ O(u1 u[prime]1): (3.18)

That means if we take u1 close to u[prime]1 and keep z far from them, the system can be e ectively described by the insertion of 2N heavy operators. There is still a second order di erential equation and the cycles around zk and z[prime]k are still of trivial monodromy. Actually this is exactly the original system shown in gure 2(a), so the accessory parameters should be the same

lim

u1!u[prime]1

j = j,0; lim

u1!u[prime]1

z zj

+ hj

(z z[prime]j)2

JHEP10(2016)110

[prime]j = [prime]j,0: (3.19)

On the other hand, we may take a small interval expansion for the relation (3.15) and nd

log [angbracketleft](u1)(u[prime]1)O1(z1)O1(z[prime]1) : : :[angbracketright]

hO1(z1)O1(z[prime]1) : : :[angbracketright]

= 2h log(u1 u[prime]1) + O(u1u[prime]1) + g(z1; z[prime]1; : : : zN; z[prime]N):(3.20)

From the Ward identity, we get

j j,0 =

@@zj log [angbracketleft]

(u1)(u[prime]1)O1(z1)O1(z[prime]1) : : :[angbracketright]

hO1(z1)O1(z[prime]1) : : :[angbracketright]

= @

@zj g + O(u1 u[prime]1) (3.21)

Comparing (3.21) with (3.19), we can see

@@zj g = 0; for all j. (3.22)

Similarly we also have

@ @z[prime]j

g = 0; for all j. (3.23)

That means g is a constant. By choosing appropriate normalization of the light operators, g can be set to zero. Then the geodesic length read by the light operators is just

L = log tr 0 1

0 0

!v0(u1)v0(u[prime]1)1: (3.24)

Taking into (3.15), and by the Ward identity, we have

j j,0 = 2hl

@@zj log tr[parenleftBigg][parenleftBigg]

0 1 0 0

!v0(u1)v0(u[prime]1)1;

[prime]j [prime]j,0 = 2hl

@ @z[prime]j

log tr 0 1

0 0

!v0(u1)v0(u[prime]1)1: (3.25)

{ 14 {

3.2 Length of conical defect

The length of the conical defect Lj is de ned by the limit (3.4). This requires us to consider a geodesic which is close to the conical defect. However because of singularities, when we move the geodesic close to the conical defect, there appears divergence, which should be regularized properly. To deal with the divergence, we de ne a matrix for each kind of the operator

M(j) = @ (j,+)0 @ (j,)0

(j,+)0 (j,)0

!; (3.26)

where (j,+)0 and (j,)0 satisfy the di erential equation (3.9) and have the asymptotic behaviors as

(j,+)0 = (z zj)

1

2 + 12n (1 + O(z zj));

(j,)

0 = (z zj)

1

2

12n (1 + O(z zj)): (3.27)

The inverse of the matrix M(j) is

(M(j))1 = n[parenleftBigg][parenleftBigg]

(j,) @ (j,)

(j,+) @ (j,+) [parenrightBigg]

: (3.28)

JHEP10(2016)110

The geodesic length can be evaluated in a di erent way

L = log tr 0 1

0 0

!Mj(u1)(Mj(u1))1v0(u1)v0(u[prime]1)1Mj[prime](u[prime]1)(Mj[prime](u[prime]1))1

= log tr 0 1

0 0

!((Mj(u1))1v0(u1)v0(u[prime]1)1Mj[prime](u[prime]1)) (3.29)

+ log(n)(u1 zj)

1

2

1

2nj (u[prime]1 z[prime]j)

1

2

1

2nj + O((u1 zj)

1n ; (u[prime]1 z[prime]j)

1n ):

The rst term does not depend on u1 or u[prime]1 but only on zj and z[prime]j, because that M(j) and M(j[prime]) satisfy the equation (3.12) as well. The second term is a divergent term, which depends only on u1 zj and u[prime]1 z[prime]j but not on the locations of other operators. The last

term vanish when the geodesic is close to the conical defect. So the length of the conical defect Lj can be taken to be

Lj = log tr 0 1

0 0

!((Mj(u1))1v0(u1)v0(u[prime]1)1Mj[prime](u[prime]1)); (3.30)

which is nite and depends only on the locations zj and z[prime]j.

3.3 Conformal block

In this subsection, we give a proof for the relation (3.1). As we explained in the previous subsection, the conical defect length can be de ned as (3.30). Instead of directly taking a variation with respect to the conformal dimension, we consider the same system with two more light operators, discussed in the last subsection. Now we require the conformal

{ 15 {

weight of the light operators to be variable. By considering the composition of the heavy operator with the light operator, we can read the response of the conformal block with respect to the conformal weight.

The stress tensor of the system with two light operators is

T (z) =

Xk

[bracketleftbigg]

hk(z zk)2

+ k

+ [prime]kz z[prime]k [bracketrightbigg]

z zk

+ hk

(z z[prime]k)2

+ hl

(z u1)2

+ ~

1z u1

+ hl

(z u[prime]1)2

+ ~

[prime]1z u[prime]1

: (3.31)

We move u1 ! zj, u[prime]1 ! z[prime]j and observe the system away from u1, u[prime]1, zj and z[prime]j such that

|z zj[notdef] [notdef]u1 zj[notdef]; [notdef]z z[prime]j[notdef] [notdef]u[prime]1 z[prime]j[notdef]: (3.32)

In this limit, the system can be e ectively described by 2N heavy operators with the stress tensor

T (new)(z) =

Xk

[bracketleftbigg]

h(new)k (z zk)2

+ (new)k,0 z zk

+ h(new)k (z z[prime]k)2

+ [prime](new)k,0z z[prime]k [bracketrightbigg]

JHEP10(2016)110

; (3.33)

where the \new" conformal weight are

h(new)j = hj + hl 2hl[parenleftbigg]

1

2

1

2nj

;

h(new)k = hk; for k [negationslash]= j; (3.34)

and the \new" accessory parameters are

(new)k,0 =

[braceleftBigg]

~

1 + j; for k = j

k; for k [negationslash]= j

(3.35)

and similarly for [prime]. The di erences between the accessory parameters with and without the light operators are respectively

(new)k,0 k,0 = 2hl

@@zk Lj; for 1 k N

[prime](new)k,0 [prime]k,0 = 2hl

@ @z[prime]k

Lj; for 1 k N: (3.36)

To get the above relations , we consider a holomorphic function s(z) and take a contour integral around zj and u1

[contintegraldisplay]

dzs(z)T (z) = hjs[prime](zj) + hls[prime](u1) + js(zj) + ~

1s(u1)

1(u1 zj))s[prime](zj) + (~ 1 + j)s(zj): (3.37)

{ 16 {

= (hj + hl + ~

Considering the limit u1 ! zj and u[prime]1 ! z[prime]j

~

1 = 2hl

12 12nj[parenrightbigg]1u1 zj

1 + O((u1 zj)

1n )

[parenrightbig]

~

1 + j j,0 = 2hl

@@zj Lj + O(u1 zj; u[prime]1 z[prime]j)

k k,0 = 2hl

@@zk Lj + O(u1 zj; u[prime]1 z[prime]j); (3.38)

and taking into (3.37)

lim

j,0 2hl@@zj Lj[parenrightbigg]s(zj); (3.39)

we can easily read out the asymptotic condition close to zj as in (3.31). It is easy to read the asymptotic condition for other zks as well. Then we can nd the relations (3.34)

and (3.36).

To nd the variation of the conformal block, we requires the conformal weight of the light operators to be a small variable,

hl = c 24

1 1(1 + n)2[parenrightbigg]= c 12 n;

where n is a very small variable. Moreover, we may de ne the conformal weight of the composite operator formed by the heavy operator Ok and the light operator to be

h(new)k =

c 24

JHEP10(2016)110

T (z)s(z) =

hj + hl nj

s[prime](zj) +

u1!zj u[prime]1!z[prime]j

[contintegraldisplay]

1 1 (n(new)k)2

; (3.40)

analogous to the conformal weight of a single operator. Then we nd

n(new)j nj = n2j n;

(new)k,0 k,0 = 2hl

@@zk Lj; for 1 k N: (3.41)

That means

n2j @@nj k,0 =

@ @zk

Lj 4G;

n2j @@nj [prime]k,0 =

@ @z[prime]k

Lj4G: (3.42)

From the Ward identity,

k,0 = @

@zk log F; [prime]k,0 =

@ @z[prime]k

log F; (3.43)

and taking an integral, we get the relation (3.1).

{ 17 {

4 Examples

In this section, we illustrate the above discussion by two concrete examples. One is the two-point function, which can be determined by the conformal symmetry. The other one is the four-point function which has been discussed before in the literature. In both cases, we show that the probe method introduced in the last section gives the consistent results.

4.1 Two-point function

The simplest example is the two-point function of the twist operators, which give the single-interval R enyi entropy. More generally, we may consider two heavy operators with the conformal dimension h = c

24 (1

1n2 ) and

h = c

24 (1

1n2 ), whose two-point function is

xed by the conformal symmetry

1

hO(z1; z1)O(z2; z2)[angbracketright] =

1

(z1 z2)

c12 (1 1n2 )

c 6

1n log(z1 z2)(

z1

JHEP10(2016)110

(

z1

z2)

c12 (1 1n2 )

: (4.1)

On the bulk side, as suggested by the relation (1.6), we obtain

Area[notdef]n

4G = n2

@ @n[angbracketleft]O(z1;

z1)O(z2;

z2)[angbracketright] =

z2): (4.2)

On the other hand, in the eld theory we can use the correlation function of two light operators to de ne the geodesic length

Area

4G =

c12hl log [angbracketleft]

(u1;

u1)(u[prime]1;

u2)O(z1;

z1)O(z2;

z2)[angbracketright]

: (4.3)

where hl is the conformal dimension for . With the semiclassical approximation, the four-point function in the numerator equals

z1 z2 n

2hl(u1 z2)(1 n

hO(z1; z1)O(z2; z2)[angbracketright]

1)hl

1)hl(u[prime]1 z1)(

1 n

1)hl(u1 z1)(

1 n

1)hl(u[prime]1 z2)(

1 n

1

i2hl : (4.4)

Taking u1 = z1 + 1ei 1 u[prime]1 = z2 + 2ei 2 and setting 1; 2 ! 0, we can see that the four-

point function is proportional to (z1 z2)

2n hl times some coe cient related to the cut-o .

Taking it into (4.3), it is easy to see that the eld theory de nition of the length (4.3) is equal to (4.2) up to a UV cut-o . Here we evaluate the geodesic length between u1 and u[prime]1, and we take u1 and u[prime]1 close to z1 and z2 respectively to evaluate the length between

O1 and O2. 1 and 2 can be regarded as the wave packet of the point particle, and they provide the UV regularization for the point particle.

4.2 Four-point function

Next we consider the four-point function of the heavy operators. The conformal block of four operators in a sphere has been discussed before, by using the brute-force expansion [35,

{ 18 {

[notdef] h(u1z1u1z2 )1n (u[prime]1z1 u[prime]1z2 )1 n

57, 58]. We assume that the operators can be divided into two pairs, in each pair the operators are of the same type and they are close such that the cross ratio of the insertion is small. In this case, we can take an expansion with respect to the cross ratio up to some orders. Consider the following four-point function

hO1(y)O1(y)O2(1)O2(1)[angbracketright]; (4.5)

where O1 and O2 have conformal dimensions h1 = c

24 (1

1n21 ) and h2 =

c24 (1

1n22 ) respec-

tively, and y 1. The stress tensor is now

T (z) = h1

(z + y)2 +

h1(z y)2

JHEP10(2016)110

+ 1

z + y

1z y

+ h2

(z + 1)2 +

h2(z 1)2

+ 2

z + 1

2z 1

; (4.6)

where

2 = h1 + h2 y 1: (4.7)

By solving the di erential equation

@2 (z) + 6c T (z) (z) = 0; (4.8)

and imposing the trivial monotromy condition around the cycle enclosing y and y, we

get the expansion for the accessory parameter 1

1 = c(n21 1)24n21

1y

c(n21 1)(n22 1)

18n21n22

y

c(n21 1)(n22 1)(11 n21 n22 + 49n21n22)

810n41n42

y3

(376 86n21 2n41 86n22 1052n21n22 86n41n22 2n42

86n21n42 + 3211n41n42)y5 + O(y7); (4.9)

and the asymptotic behaviors of the solutions near the insertion points

1(z) = (z y)

1

2 +

c(n21 1)(n22 1)

51030n61n62

1

2n1 (z + y)

1

2

1

2n1 f1(z; y; n1; n2);

2(z) = (z y)

1

2

1

2n1 (z + y)

1

2 +

1

2n1 f2(z; y; n1; n2); (4.10)

where

f2(z; y; n1; n2) = f1(z; y; n1; n2); (4.11)

and f1 can be expanded as

f1 =

1

Xk=0f1kzk: (4.12)

{ 19 {

The rst few coe cients in the expansion are of the following forms

f10 = 1;

f11 = 1 + n22 3n1n22

y + (n22 1)(13n21n22 + 8n21 n22 11)

135n31n42

y3

+ 1

8505n51n62

(1 + n22)(376 419n21 + 61n41 86n22

440n21n22 + 355n41n22 2n42 41n21n42 + 439n41n42)y5 + O(y7);

f12 = 1 n22 6n22

+ (2 + n21)(1 5n22 + 4n42)

135n21n22

y2

JHEP10(2016)110

+(43 + 23n21 + 2n41 5n22 + 73n21n22 + 31n41n22)(1 5n22 + 4n42)

8505n41n62

y4 + O(y6);

f13 = 1 + n42 30n1n42

y + (1 + n22)(15+12n2122n22+19n21n22+13n42+47n21n42)

1890n31n62

y3 + O(y5);

f14 = 1 + 10n22 11n42120n42

+ (2 + n21)(1 21n42 + 20n62)

1890n21n62

y2 + O(y4);

+ O(y3);

f16 = 1 + 49n22 + 259n42 309n625040n62

f15 = 1 21n22 + 21n42 + n62840n1n62

+ O(y2): (4.13)

With these coe cients we can read the length of cosmic brane ending on y and y:

Ly,y =

1n1 log y

2(1 + n22)

3n1n22

y2

2(11 5n21 10n22 20n21n22 n42 + 25n21n42)

135n31n42

y4

(376 + 308n21 28n41 + 462n22 + 336n21n22 294n41n22 84n42

588n21n42 777n41n42 2n62 56n21n62 + 1099n41n62)y6 + O(y8): (4.14) It is straightforward to check

n21

@ 1@n1 =

2 8505n51n62

1

2

@ @y

4G : (4.15)

The extra coe cient 12 is because that when we take a variation with respect to y,

both the initial and nal end points change so we have two parts of contributions to the variation of Ly,y.

In the appendix, we give a di erent computation of the four-point conformal block and the distance Ly,y, and nd complete agreement. This provides nontrivial support to our

computation of the cosmic brane length using the probe.

5 Conclusions and discussions

In this paper, we studied the holographic description of the conformal block of the heavy operators in 2D large c CFT. In general such conformal block is hard to compute. On

{ 20 {

Ly,y

the bulk side, the back-reaction to the background cannot be ignored. Technically, the di culty is related to the fact that the monodromy problem of the di erential equation involving the stress tensor is hard to solve. Moreover it is hard in practice to integrate the accessory parameters to get the ZT-action. In this work, inspired by the recent work on the gravity dual of the R enyi entropy [47], we proposed another di erential relation of the conformal block with respect to the conformal dimension of the operators. We considered a class of conformal block in which the heavy operators are inserted in pairs such that the operators in a pair are close to each other and the pairs are separated far apart. In this case, we found the relation

n2j

@@nj log F =

Lj4G + fj; for all j (5.1)

where F is the conformal block, nj is related to the conformal dimension of the operator

Oj, Lj is the length of the cosmic brane homologous to the interval between the pair of operators Oj(zj) and Oj(z[prime]j), and fj is a renormalization constant. We gave a eld theory derivation of the above relation by introducing two light operators as both the probe and the perturbation.

Our discussion can be applied to more general conformal block. Actually, it is not necessary to require all the operators to be inserted in compact pairs. The above di erential relation make sense as long as the operator pair Ojs are much closer than the other operators such that the monodromy around the circle enclosing these two operators is trivial. For example, we may consider the case that the operators O1(z1) and O1(z[prime]1) form a compact pair, but other operators could be distributed in any way provided they are far from the pair, then the discussion in section 3 leads to the relation

n21

@@n1 log F =

JHEP10(2016)110

L14G + f1: (5.2)

This suggests that the area law of the holographic R enyi entropy (1.6) should be true for more general states, at least for the large class of states discussed in [40]. Certainly it would be interesting to study the holographic dual of a general conformal block of heavy operators in complex plane or on a torus.

One remarkable point is on the binding energy. In the eld theory, when the light operator combine with a heavy operator, for a distant observer they behave like a new composite heavy operator. However, the conformal dimension of the new operator is not just the sum of the conformal dimensions of two operators. There is anomalous dimension, as shown in the form of h(new)k. Naively one may expect there could be binding energy between two particles in the bulk corresponding to the heavy and light operators respectively. However this is not true. In fact, there is no binding energy in the gravity side. Considering the (3.41), the change of nj implies that

1 1 n(new)j[parenrightbigg]= 1 1nj[parenrightbigg]+

1 11 + n

: (5.3)

On the other hand, the mass of particle is

Tn = 1

4G

1 1 n

: (5.4)

{ 21 {

That means in this case there is no binding energy between a light particle and a heavy particle in the bulk, even though they are next to each other.

The issue of binding energy can be seen more clearly for two particles of general masses. For one particle one has the relation [13]

m = 1

4G 1 p1 8GM [parenrightbig]

: (5.5)

Here m is the local/proper mass of the particle, i.e. the energy seen by an observer near the particle, and M is the global/ADM mass, i.e. the energy seen by an distant observer. Due to the back-reaction of the heavy particle m [negationslash]= M. Also one has M = with being

the scaling dimension of the dual CFT scalar operator. If we put two particles with the proper masses m1 and m2 together and take them as one particle of mass m, then we have

m1 + m2 m = 0; (5.6) since there should not be any binding energy for the proper mass. But for the ADM mass there is binding energy

M1 + M2 M = 4Gm1m2: (5.7) Since M = , this is in accord with the anomalous dimension of the composite operator.

In a word, there is no binding energy for the proper mass, but there is binding energy for the ADM mass.

In this paper we have focused on the classical part of the conformal block, and it is certainly a very interesting question to consider the 1=c corrections to the relation (5.2). On the CFT side one can use the operator product expansion to do short interval expansion and get the subleading terms [42, 43, 46]. Naively the next leading order contribution should correspond to the quantum correction around the cosmic brane. However, we are not sure if we can get a di erential relation with a similar form as (5.2).

When the scaling dimension of the heavy operator satisfy > c=12, the back-reaction of the dual eld in the gravity side would create a black hole con guration rather than a geodesic with conical defect. In the CFT, the monodromy analysis could still be applicable. However, the naive analytic continuation n ! in leads to a equation with

imaginary part, which is unreasonable. Moreover, on the gravity side, the mass of dual particle becomes complex, which is unacceptable. And in this case, we are short of a clear holographic picture of the conformal block of multi-point heavy operators. We would like to leave this case for future study.3

Acknowledgments

We would like to thank Xi Dong, Wu-Zhong Guo, Muxin Han, Daniel Harlow, Feng-Li Lin, Tatsuma Nishioka, Wei Song, Tadashi Takayanagi, Herman Verlinde and Jianfei Xu for helpful discussions. B.C. and J.-q.W. were in part supported by NSFC Grant No. 11275010, No. 11335012 and No. 11325522. J.-j.Z. was in part supported by NSFC Grant No. 11222549 and No. 11575202.

3We would like to thank the anonymous referee for inspiring the discussions in the last three paragraphs of this section.

{ 22 {

= c

6

1

r1 12 c

JHEP10(2016)110

A Four-point function in another way

In the appendix we use another method to calculate the correlation function of four heavy operators and reproduce (4.14). For convenience, we rename O2 and O1 as and respectively, and put the four operators at points 1; 1; x and 0, which are di erent from (4.5). In

this setup, the cross ratio is simply x. The operators are heavy, and we have the conformal weights

h = c 24

1 1 n2

; h = c 24

1 1 n2

: (A.1)

We would like to compute the four-point function

h(1)(1) (x) (0)[angbracketright] =

1x2h f(x); (A.2)

with the function f(x) being invariant under a general conformal transformation.

In the space of the holomorphic sector of the vacuum conformal family we have the identity operator

P =XhGij(h)[notdef]h; i[angbracketright][angbracketleft]h; j[notdef]; (A.3)

with h = 0; 2; 3; 4; [notdef] [notdef] [notdef] being the level, i; j = 1; 2; [notdef] [notdef] [notdef] ; dim h, and Gij(h) being the inverse

matrix of [angbracketleft]h; i[notdef]h; j[angbracketright]. For examples, at level 0, 2 and 3, we have, respectively, the vacuum

state [notdef]0[angbracketright], L2[notdef]0[angbracketright], and L3[notdef]0[angbracketright]; at level 4 we have the states L4[notdef]0[angbracketright] and L2L2[notdef]0[angbracketright]. Inserting

the identity operator in the four-point function (A.2) we get the vacuum conformal block

h(1)(1)P (x) (0)[angbracketright] =

1 x2h

JHEP10(2016)110

XhxhGij(h)[angbracketleft][notdef]h+h[notdef]h; i[angbracketright][angbracketleft]h; j[notdef] h h[notdef] [angbracketright]: (A.4)

Taking the large c limit, we can get the conformal block order by order and then read the length

L =

6c n2 @n log[angbracketleft](1)(1)P (x) (0)[angbracketright] =

1n log x

6c n2 @n log f(x): (A.5)

At the end, we nd

L = 1n log x

(n2 1)x2

24n2n

(n2 1)x3

24n2n

(n2 1)(655n2n2 n2 + 5n2 11)x4

17280n4n3

(n2 1)(295n2n2 n2 + 5n2 11)x5

8640n4n3

(n2 1)x6 8709120n6n5

(269087n4n4

1414n4n2 + 7091n2n4 15568n2n2 n4 43n2 + 14n4 154n2 + 188)

(n2 1)x7 2903040n6n5

(81767n4n4 574n4n2 + 2891n2n4 6328n2n2 n4

43n2 + 14n4 154n2 + 188) + O(x8): (A.6)

{ 23 {

A conformal transformation can put the operators at 1, 1, y and y, and we have the

four-point function

h(1)(1) (y) (y)[angbracketright] =

1

22h(2y)2h

f

[parenleftbigg]

4y(1 + y)2

; (A.7)

with f(x) being the same function as the one in (A.2). Then we get the length

Ly,y =

1n log y

6c n2 @n log f[parenleftbigg]

4y(1 + y)2

: (A.8)

Taking the fact that n = n2, n = n1 into account, we see that this is just the length (4.14).

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.

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