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Published for SISSA by Springer

Received: October 16, 2015

Accepted: November 15, 2015 Published: November 30, 2015

A. Liam Fitzpatrick,a,b Jared Kaplanc and Matthew T. Waltersd

aStanford Institute for Theoretical Physics, Stanford University,

Via Pueblo, Stanford, CA 94305, U.S.A.

bSLAC National Accelerator Laboratory,

Sand Hill Road, Menlo Park, CA 94025, U.S.A.

cDepartment of Physics and Astronomy, Johns Hopkins University, Charles Street, Baltimore, MD 21218, U.S.A.

dDepartment of Physics, Boston University,

Commonwealth Avenue, Boston, MA 02215, U.S.A.

E-mail: mailto:[email protected]

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

Web End [email protected]

Abstract: We show that in 2d CFTs at large central charge, the coupling of the stress tensor to heavy operators can be re-absorbed by placing the CFT in a non-trivial background metric. This leads to a more precise computation of the Virasoro conformal blocks between heavy and light operators, which are shown to be equivalent to global conformal blocks evaluated in the new background. We also generalize to the case where the operators carry U(1) charges. The re ned Virasoro blocks can be used as the seed for a new Virasoro block recursion relation expanded in the heavy-light limit. We comment on the implications of our results for the universality of black hole thermality in AdS3, or equivalently, the eigenstate thermalization hypothesis for CFT2 at large central charge.

Keywords: AdS-CFT Correspondence, Conformal and W Symmetry

ArXiv ePrint: 1501.05315

Open Access, c

[circlecopyrt] The Authors.

Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP11(2015)200

Web End =10.1007/JHEP11(2015)200

Virasoro conformal blocks and thermality from classical background elds

JHEP11(2015)200

Contents

1 Introduction 1

2 Review of CFT2 and Virasoro conformal blocks 32.1 Relation to AdS/CFT 52.2 Virasoro blocks at large central charge 5

3 Heavy operators and classical backgrounds for the CFT 63.1 Stress tensor correlators and the background metric 63.2 Heavy-light Virasoro blocks from a classical background 103.3 Order of limits at large central charge 123.4 Recursion formula 123.5 Extension to U(1) 13

4 AdS interpretation 154.1 Matching of backgrounds 154.2 Extension to U(1) 174.3 Matching of correlators 18

5 Implications for thermodynamics 195.1 Far from the black hole: the lightcone OPE limit 205.2 Implications for entanglement entropy 205.3 General kinematics and a connection with AdS locality 21

6 Future directions 23

A Details of recursion formula 25

B Periodicity in Euclidean time and thermodynamics 27

1 Introduction

In gravitational theories the localized high-energy states are black holes, characterized by a universal Hawking temperature. More generally, in any su ciently complex theory, it has been hypothesized [1, 2] that high-energy microstates behave like a thermal background. We will derive more precise versions of these statements in 2d Conformal Field Theories (CFT2), which are believed [3{5] to also describe all consistent theories of quantum gravity in 3d Anti-de Sitter (AdS3) spacetime.

States in CFT2 can be decomposed into irreducible representations of the Virasoro symmetry algebra. Correlation functions can be written as a sum over the exchange of

{ 1 {

JHEP11(2015)200

OH(1)

OL(z)

JHEP11(2015)200

OL(1)

OH(0)

Figure 1. This gure suggests an AdS picture for the heavy-light CFT correlator | a light probe interacting with a heavy particle or black hole. We will show that the Virasoro blocks can be computed in this limit by placing the CFT2 in an appropriately chosen 2d metric, which is related to the at metric by a conformal transformation.

these irreducible representations, which are individually called conformal partial waves or Virasoro conformal blocks. The conformal blocks are atomic ingredients in the CFT bootstrap [6{8], and are particularly crucial for obtaining analytic constraints on CFTs [9{25]. Via the AdS/CFT correspondence, one can interpret the Virasoro blocks as the exchange of a sum of AdS wavefunctions. The states in the Virasoro block correspond with the wavefunctions of some primary object in AdS3, plus any number of AdS3 gravitons. In other words, the Virasoro blocks capture quantum gravitational e ects in AdS3, providing a way to precisely sum up all graviton exchanges. The Virasoro blocks have also been used to study entanglement entropy [26{35] and scrambling [36] in AdS/CFT, both in the vacuum and in the background of a heavy pure state [37]. There has also been signi cant recent progress [38{40] in using modular invariance to understand the spectrum of CFT2 at large central charge.

We will present a new method for computing the Virasoro conformal blocks in the limit of large central charge

c = 3

2G 1; (1.1) which is often called the semi-classical limit, and for correlation functions

[angbracketleft]OH1(0)OH2(1)OL1(1)OL2(z)[angbracketright] with

hHc ; hL; H; L xed ;

hH1,L1 hH2,L2

2 ; (1.2) where hHi and hLi are the holomorphic dimensions of OHi and OLi. As suggested in

gure 1, via AdS/CFT we would naturally interpret correlators in this limit in terms of

{ 2 {

hH,L

hH1,L1 + hH2,L22 ; H,L

the motion of a light probe interacting with a heavy object or a BTZ black hole [41]. In the AdS description, the heavy operator produces a classical background in which the light probe moves.

We will argue that such an interpretation also exists in the CFT, and that it can be used as the basis of a precise new computational method. Speci cally, we will show that the computation of Virasoro conformal blocks can be trivialized by placing the CFT in a non-trivial background metric, related to the at metric by a conformal transformation.

Our main result is that the full Virasoro conformal block for a semi-classical heavy-light correlator is equal to a global conformal block with the light operators evaluated in a new set of coordinates, OL(z) = (w[prime](z))hLOL(w(z)). That is,

V(c; h; hi; z) = h(w[prime](1))hL1 (w[prime](z))hL2 G(h; h[prime]i; w); (1.3)

where

w = z with =

r1 24hH c ;

and h[prime]L = hL; h[prime]H = hH; [prime]L = L; [prime]H = H

; (1.4)

and G is the global conformal block [42{44]

G(h; hi; x) = (1 x)h2hL2F1(h 2 H; h + 2 L; 2h; 1 x): (1.5) This result con rms and extends the analysis of [25] beyond the range of validity of the older methods we used there.1 Note that is related to the Hawking temperature TH of a BTZ black hole by

TH = [notdef] [notdef]2 (1.6)

when hH > c

24 .

In the next section we will brie y review conformal blocks and the computation of the

Virasoro blocks at large central charge. Then in section 3 we explain our main result, the new recursion relations that use it as a seed, and an extension to include U(1) currents. In section 4 we discuss the AdS interpretation of our results, while in section 5 we discuss the implications for thermodynamics. We conclude with a discussion of future directions in section 6.

2 Review of CFT2 and Virasoro conformal blocks

In any CFT, correlation functions can be written as a sum over exchanged states

[angbracketleft]OH(1)OH(1)OL(z;

z)OL(0)[angbracketright] = [angbracketleft]OH(1)OH(1)

{ 3 {

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!OL(z; z)OL(0)[angbracketright]: (2.1)

1In [25] we obtained the Virasoro blocks as e c6 f+g, where we computed f but g was left undetermined; in this work we fully determine g up to 1/c corrections, which might be computed using recursion relations [45, 46].

| [angbracketright][angbracketleft] [notdef]

AOLOL

*OHOH

0@ X

{mi,ki}

Lk1 m1 Lkn mn|hihh|Lknmn Lk1m1 N{mi,ki}

Figure 2. This gure suggests how the exchange of all Virasoro descendants of a state [notdef]h[angbracketright]

corresponds to the exchange of [notdef]h[angbracketright] plus any number of gravitons in AdS3. This is su cient to build

the full, non-perturbative AdS3 gravitational eld entirely from the CFT2.

We can organize this into a sum over the irreducible representations of the conformal group, which are called conformal blocks. When working in radial quantization we diagonalize the dilatation operator D and angular momentum generators, so the conformal blocks are labeled by a scaling dimension and by angular momentum quantum numbers.

In the case of CFT2 the full conformal algebra is two copies of the Virasoro algebra

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

c12n(n2 1) n,m (2.2) one for the holomorphic (Ln) and one for anti-holomorphic (

Ln) conformal transformations.

The parameter c is the central charge of the CFT. It does not appear in commutation relations of the global sub-algebra,

L[notdef]1; L0; and

L[notdef]1;

L0; (2.3)

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= SO(3; 1) algebra. Because of the structure of (2.2), in the limit of c ! 1 with all other parameters held xed the Virasoro conformal algebra

essentially reduces to the global conformal algebra, as we will review below.

The Virasoro conformal blocks are most naturally labeled by holomorphic and anti-holomorphic dimensions, h and

h, which are the eigenvalues of L0 and

L0. The quantities

h and

h are restricted to be non-negative in unitary CFTs. The scaling dimension is h +

h and angular momentum is h

which together form an SL(2; C)=Z2

h. Any CFT2 correlation function can be written as a sum over Virasoro conformal blocks, with coe cients determined by the magnitude of matrix elements

[angbracketleft]OHOH[notdef]h;

h[angbracketright] and [angbracketleft]h;

h[notdef]OLOL[angbracketright] (2.4)

where [notdef]h;

h[angbracketright] is a primary state. Primaries are lowest weight under Virasoro, so Ln[notdef]h;

h[angbracketright] = 0

h[angbracketright],

and these matrix elements are just the 3-pt correlation functions of primary operators, which are otherwise known as operator product expansion (OPE) coe cients.

The Virasoro generators Ln appear directly in the Laurent expansion of the holomorphic part of the stress energy tensor

T (z) =

Xnzn2Ln; (2.5)

{ 4 {

for n > 0, and similarly for

Ln. By the operator state correspondence Oh,

h(0)[notdef]0[angbracketright] = [notdef]h;

with T (z) = 2Tzz(z) and similarly for the anti-holomorphic

Ln, and Tzz. The Virasoro algebra can be derived directly from the singular terms in the OPE of T (z)T (0). To take the adjoint of an operator we perform an inversion z ! 1=

z, because we are working

z);

in radial quantization. This means

[T (z)] = 1

z4 T (1=

z); (2.6)

which leads to the relation Ln = Ln for the Virasoro generators. Here we have assumed

that the CFT2 lives in the plane, with metric ds2 = dzd

z, as is conventional.

2.1 Relation to AdS/CFT

Via the AdS/CFT correspondence, the CFT stress tensor (and the Ln with n 2) creates

gravitons in AdS3. The central charge determines the strength of gravitational interactions via the relation c = 3

2G , where G is the AdS Newton constant. In global coordinates, the AdS Hamiltonian is the dilatation operator

D = L0 +

L0; (2.7)

and so CFT scaling dimensions determine AdS energies, with the Planck scale corresponding to a dimension

JHEP11(2015)200

c12 in the CFT.

Correlation functions in the CFT can be given an AdS interpretation, as indicated in gure 1. The Virasoro conformal blocks encapsulate all gravitational e ects in AdS3,

as suggested by gure 2. As we reviewed extensively in [25], CFT states with dimension hH / c correspond to AdS3 objects that produce either de cit angles or BTZ black holes.

We will study the Virasoro blocks in the limit

hH;

hH / c hL;

hL; (2.8)

which we will refer to as the heavy-light semi-classical limit. In this limit AdS gravity is weakly coupled, but the heavy operator OH has a Planckian mass. The operator OL

creates a state that can be thought of as probing the heavy AdS de cit angle or black hole. We emphasize that our computations will all be completely independent of AdS, but AdS/CFT provides the most natural interpretation of and motivation for our results.

2.2 Virasoro blocks at large central charge

Here we review the well-known fact that the Virasoro blocks simplify dramatically if we take the limit c ! 1 with other operator dimensions xed, reducing to representations of

the much smaller global conformal group. Speci cally, consider the correlator

[angbracketleft]OL(1)OL(0)OL(1)OL(z;

z)[angbracketright] (2.9)

for the primary operator OL with dimensions hL;

hL in the limit that

c ! 1 with hL;

hL xed: (2.10)

{ 5 {

In this case the holomorophic Virasoro blocks are simply

Gh(z) = (1 z)h2hL2F1 (h; h; 2h; 1 z) + O

1 c

; (2.11)

where we assume that h, the intermediate holomorphic dimension, is also xed as c ! 1.

That is, the Virasoro conformal block reduces to a conformal block for the global conformal algebra, generated by L[notdef]1 and L0 alone, in the large c limit.

To see why this is the case, consider the projection operator Ph onto the states in

Virasoro conformal block. The basis of states

Lk1m1 [notdef] [notdef] [notdef] Lknmn[notdef]h[angbracketright] (2.12)

is approximately orthogonal to leading order in 1=c. Thus we can approximate the projection operator onto the conformal block as

X{mi,ki[notdef]

Lk1m1 [notdef] [notdef] [notdef] Lknmn[notdef]h[angbracketright][angbracketleft]h[notdef]Lknmn [notdef] [notdef] [notdef] Lk1m1 N{mi,ki[notdef]

: (2.13)

In this expression we are summing over all possible states created by acting on the primary state [notdef]h[angbracketright] with any number of Virasoro generators, and dividing by their normalization.

Using the Virasoro algebra from equation (2.2) it is easy to see that the normalization

hh[notdef]LnLn[notdef]h[angbracketright] =

JHEP11(2015)200

2nh + c12n(n 1)(n + 1)

(2.14)

for n > 0, and so for n 2 the normalization is of order c. The Lm in the numerator of Ph can produce powers of h and hL via the relation

[Lm; O(z)] = hi(1 + m)zmO(z) + z1+m@zO(z) (2.15) for primary operators O, but the numerator of Ph will be independent of c. This means

that the normalizations N{mi,ki[notdef] are proportional to a polynomial in c if any mi 2, and

that there are no factors in the numerator to make up for this suppression, so terms with mi 2 make negligible contributions to the conformal block at large c. The projector

reduces to

Xk Lk1[notdef]h[angbracketright][angbracketleft]h[notdef]Lk1 hh[notdef]Lk1Lk1[notdef]h[angbracketright]

+ O

1 c

; (2.16)

which just produces the 2d global conformal block for the holomorphic sector. The result is equation (2.11). Readers interested in a detailed review of all these statements can consult appendix B of [25].

3 Heavy operators and classical backgrounds for the CFT

3.1 Stress tensor correlators and the background metric

The large central charge analysis above becomes invalid if the dimensions of the various operators are not xed as c ! 1. When the dimensions become large, the numerator of

{ 6 {

OL(z)

OH(0)

OL(1)

OH(1)

!z ! w = z

OH (0)

OL(w)

OL(1w)

OH(1)

Figure 3. This gure suggests the conformal mapping from z to w coordinates, so that stress-tensor exchange between heavy and light operators has been absorbed by the new background metric. Note that we have placed the heavy operators at 0 and 1 for simplicity, although we place

one of them at 1 to emphasize a particular OPE limit in section 3.

Ph in equation (2.13) can compensate for the powers of c in the denominator. The problem is that one obtains contributions

[angbracketleft]OHOHLk1m1 [notdef] [notdef] [notdef] Lknmn[notdef]h[angbracketright] (3.1)

that are proportional to some polynomial in hH, due to the relation of equation (2.15), and hH / c. Since the Ln are just Laurent coe cients in the expansion of the stress tensor, we

can reformulate the problem in terms of the correlators

[angbracketleft]OHOHT (x1) [notdef] [notdef] [notdef] T (xk)Oh(0)[angbracketright]: (3.2)

The issue is that this correlator will involve powers of the dimension hH, and we would like to take hH / c as c ! 1.

The crucial idea behind our method is that we can eliminate all dependence on hH by

performing a conformal transformation to a new background. A well-known feature of the CFT2 is that the stress tensor does not transform as a primary operator. Physically, this can be regarded as a consequence of the fact that T picks up a vacuum expectation value (VEV) when the CFT is placed in a non-trivial 2d background metric g[notdef] . We will choose

g[notdef] so that the VEV of T cancels its large matrix element with OH. The relevant mapping

is illustrated in gure 3.Recall that in a CFT, conformal transformations act on primary operators as

O(z) ! O(w)

JHEP11(2015)200

dz dw

O(z(w)) (3.3)

under a conformal transformation z ! w(z). The conformal transformation rule for the

stress tensor T (z) is

T (z) ! T (w) =

2T (z(w)) + c12S(z(w); w): (3.4)

The rst term is the transformation rule for a primary operator, while the second term

S(z(w); w)

z[prime][prime][prime](w)z[prime](w) 32(z[prime][prime](w))2

(z[prime](w))2 (3.5)

is the Schwarzian derivative.

dz dw

{ 7 {

A physical way to think about the transformation rule (3.4) is that T obtains a VEV in a background metric. We can use the result that in a general background

ds2 = e2(z,z)dzd

z (3.6)

the stress tensor picks up the vacuum expectation value [47, 48]

hT [notdef] [angbracketright] =

r[notdef]r + r[notdef]r g[notdef] r r + 12r r : (3.7)

After the conformal transformation z ! w(z) the CFT lives in a background metric

ds2 = w[prime](z)dz

dz: (3.8)

By using w[prime](z) = e2 and the normalization T (z) = 2Tzz(z), we obtain the Schwarzian derivative in equation (3.5) from the general formula for the VEV.

Now let us demonstrate how we can make use of these results. We would like to study the correlator

[angbracketleft]OH(1)OH(1)T (z)Oh(0)[angbracketright] = CHHh

c 12

hH(1 z)2

JHEP11(2015)200

+ h

; (3.9)

where CHHh is an OPE coe cient. Note that we are now placing the second heavy operator at z = 1 rather than z = 0 as before. This will cause less clutter in many expressions since the OPE limit will correspond to an expansion in powers of z rather than powers of 1 z.

If we perform a conformal transformation to w(z) de ned by

1 w = (1 z) with = r

1 24

(1 z)z2

hHc ; (3.10)

then in the new coordinate system

[angbracketleft]OH(1)OH(1)T (w)Oh(0)[angbracketright] = CHHh

h 1 z(w) z2(w)

: (3.11)

The key point is that all hH dependence has been removed through a cancellation between the Schwarzian derivative and the primary transformation law. The conformal transformation itself could have been determined a priori by writing the result for a transformation parameterized by a general w(z) and then solving a di erential equation demanding that all hH dependence be eliminated. In fact, one can generalize the procedure to nd more complicated conformal transformations that cancel the contribution from more than two heavy operators.

We have seen how operators transform, but now we would like to study the states that they create, especially the Virasoro descendants or gravitons created by the stress tensor. We can expand T (w) in the new coordinate w

T (w) =

Xnw2nLn: (3.12)

{ 8 {

The Virasoro algebra can be derived entirely from the singular terms in the T (z)T (0) OPE, and these terms are preserved by conformal transformations. An important consequence is that the new generators Ln expanded in w still satisfy the usual Virasoro algebra from

equation (2.2), with Ln ! Ln, and also the relation (2.15) when Ln act on conformally

transformed operators O(w).

The Ln are a complete basis of Virasoro generators, so one can write Lm as a linear

combination of the Ln and vice versa. A non-trivial special feature of the Ln generators is

Ln |h[angbracketright] = 0 for n 1 (3.13) when the state [notdef]h[angbracketright] is primary. Consequently, all states in the representation are generated

by acting with Lns, n > 0. These statements depend on the fact that the inverse

transformation z(w) has an analytic Taylor series expansion in w when expanded about the origin, as can be seen from equation (3.10). Therefore, because correlators of the form

h [notdef]T (z)[notdef]h[angbracketright] are regular as z ! 0 for all [angbracketleft] [notdef], they are also regular in w coordinates [angbracketleft] [notdef]T (w)[notdef]h[angbracketright]

as w ! 0. Equation (3.13) follows directly.

We can also be more explicit and use the transformation rule from equation (3.4) to write the w-expansion of T (w) directly in terms of T (z(w)), which gives the relation

Xnwn2Ln =1 (1 w)1

JHEP11(2015)200

Xm[z(w)]m2Lm hH 2(1 w)2: (3.14)

All terms on the right-hand side have a series expansion in positive integer powers of w, so when we combine terms proportional to wn2, we will include only Lm with m n.

This provides an alternate and more explicit derivation of equation (3.13).

The conformal transformation z(w) has a branch cut running to in nity, so it cannot be expanded in a Taylor series around w = z = 1. As a result the Ln do not have

well-behaved adjoints with respect to inversions. In particular, unlike the case of the usual Virasoro generators

Ln [negationslash]= Ln (3.15) so the Ln do not act to the left on ket states [angbracketleft]h[notdef] in a simple way. Nevertheless, we may

formally de ne the w-vacuum state [angbracketleft]0w[notdef], chosen to satisfy

h0w[notdef]Ln = 0 (3.16) for all n 1, and we can normalize it so that [angbracketleft]0w[notdef]0[angbracketright] = 1. Similarly, for any primary state |h[angbracketright] there exists a ket [angbracketleft]hw[notdef] de ned through

hhw[notdef] = limw!1[angbracketleft]0w[notdef]w2hO(w): (3.17)

Since the new Virasoro generators Ln act on operators in the new metric such as O(w) in

exactly the same way that the conventional generators act on operators in a at metric, matrix elements can be easily calculated. The [angbracketleft]hw[notdef] state satis es

hhw[notdef]L0 = [angbracketleft]hw[notdef]h and [angbracketleft]hw[notdef]h[angbracketright] = 1 (3.18) so for most purposes we can use [angbracketleft]hw[notdef] in place of [angbracketleft]h[notdef], and Ln in place of Ln. We will use

this construction in the next section to create a modi ed version of the Virasoro conformal block projector from equation (2.13), utilizing the Ln in place of the Ln generators.

{ 9 {

3.2 Heavy-light Virasoro blocks from a classical background

We would like to compute the conformal blocks for the semi-classical heavy-light correlator

Vh(z) = [angbracketleft]OH1(1)OH2(1) [Ph] OL1(z)OL2(0)[angbracketright] (3.19) where Ph projects onto the state [notdef]h[angbracketright] and all of its Virasoro descendants. We will choose

the normalization of Vh(z) so that its leading term is zh2hL in an expansion around z = 0.

We now allow hH1 [negationslash]= hH2 and hL1 [negationslash]= hL2 for maximal generality, but we will need to take

H = hH1 hH2 with [notdef] H[notdef] c (3.20) and we also de ne L = hL1 hL2. This computation is naively much more di cult than

the one in section 2.2, because the Virasoro generators act on OH to produce factors of

hH / c. This means that the contribution from states containing Ln with n > 1 in Ph no

longer vanish, since the factors of hH in [angbracketleft]OH1(1)OH2(1)Ln[notdef]h[angbracketright] compensate for factors of

c in the normalization. However, we will make use of the conformal transformation from the previous section to simplify the computation, basically turning it into a recapitulation of the global conformal case.

Let us begin by transforming the light operators in the correlator to w coordinates, so we want to compute the Virasoro blocks for

Vh(w) = [angbracketleft]OH1(1)OH2(1) [Ph] OL1(w)OL2(0)[angbracketright]: (3.21) This formula di ers from Vh(z) by the Jacobian factor (z[prime](w))hL

1 (z[prime](0))hL2 , which we will restore below. We claim that in the heavy-light semi-classical limit

Ph,w

X{mi,ki[notdef]

Lk1m1 [notdef] [notdef] [notdef] Lknmn[notdef]h[angbracketright][angbracketleft]hw[notdef]Lknmn [notdef] [notdef] [notdef] Lk1m1 N{mi,ki[notdef]

is a projector onto the irreducible representation of the Virasoro algebra with primary

|h[angbracketright]. We have used because this basis is not orthogonal at higher orders in 1=c, though

one can easily write down a formula without approximations by using the matrix of inner

products of states instead of just using the normalizations N{mi,ki[notdef]. We need to show that

the normalizations for Ln are identical to those for Ln, that this is a projection operator

satisfying (Ph,w)2 = Ph,w, and that it is complete, i.e. that all states in the representation

are included with proper normalization. The state [angbracketleft]hw[notdef] was de ned in equation (3.17).

If Ph,w is a projector, then its completeness follows because the Ln form a basis for

the Virasoro algebra. The fact that Ph,w is projector, and the equivalence of normalization

factors, follows from the general identi cation of matrix elements

hhw[notdef]Lknmn [notdef] [notdef] [notdef] Lk1m1[notdef]h[angbracketright] = [angbracketleft]h[notdef]Lknmn [notdef] [notdef] [notdef] Lk1m1[notdef]h[angbracketright] (3.23) for arbitrary mi, ki, and n. This identi cation holds because the Lm and Lm both satisfy

the Virasoro algebra, so we can use their commutation relations to move mi < 0 to the left

and mi > 0 to the right. Since both generators annihilate [notdef]h[angbracketright] when mi > 0 and [angbracketleft]hw[notdef] and

{ 10 {

JHEP11(2015)200

(3.22)

hh[notdef] when mi < 0, the full computation is determined by the Virasoro algebra. Note that

despite the presence of the adjoint state [angbracketleft]hw[notdef] we have not made use of Ln.2

Thus we can compute the heavy-light Virasoro block using

Vh(w) = [angbracketleft]OH1(1)OH2(1) [Ph,w] OL1(w)OL2(0)[angbracketright]: (3.24)

The matrix elements

hhw[notdef]Lknmn [notdef] [notdef] [notdef] Lk1m1OL1(0)OL2(w)[angbracketright] (3.25) are identical to the matrix elements of light operators in a at background, except now they depend on w instead of z. This again follows because the Ln satisfy the Virasoro

algebra, and act according to equation (2.15) except with Ln ! Ln and O(z) ! O(w).

However, the heavy matrix elements

[angbracketleft]OH1(1)OH2(1)Lk1m1 [notdef] [notdef] [notdef] Lknmn[notdef]h[angbracketright] (3.26) are non-trivial, and have no direct equivalent in a at background. They can be obtained from equation (3.11), and its generalization to multiple T (w) insertions, simply by expanding in w. In fact, in the case of the global generator L1 = @w, we need only compute

[angbracketleft]OH1(1)OH2(1)Lk1[notdef]h[angbracketright] = lim

w!0

@kw[angbracketleft]OH1(1)OH2(1)Oh(w)[angbracketright] = hcH1H2h lim

w!0

JHEP11(2015)200

@kw(1 w)h+ H/ (3.27) since L1 translates Oh in the w coordinate system. This di ers from the equivalent result

in a at metric only by the intriguing replacement H ! H= .

We have all the ingredients we need to compute the heavy-light Virasoro block at large central charge. Just as in section 2.2, because the normalizations N{mi,ki[notdef] are proportional

to positive powers of c when mi [negationslash]= [notdef]1, the Virasoro block reduces to

Vh(w) = [angbracketleft]OH1(1)OH2(1)Xk Lk1[notdef]h[angbracketright][angbracketleft]hw[notdef]Lk1 hhw[notdef]Lk1Lk1[notdef]h[angbracketright]! OL1

(w)OL2(0)[angbracketright]: (3.28)

Only the heavy side di ers from the usual global conformal block, due to the H rescaling.

Performing the sum, we nd that in the scaling limit of (1.2),

lim

c!1 V

(c; hp; hi; z) = (1 w)(hL+ L)(1

1 )

h2hL 2F1

h H ; h + L; 2h; w

; (3.29)

where we view w as a function of z through equation (3.10), and we have included the Jacobian factors w[prime](z)hLi from relating OLi(w) to OLi(z). The Virasoro block is normalized

so that the leading term at small z is zh2hL. This formula agrees with our less powerful results from [25], and we have checked up to the rst six terms that it matches a series expansion of the exact Virasoro block in this limit, computed using recursion relations derived by Zamolodchikov [45, 46].

2One may worry whether the state [angbracketleft]hw[notdef] is physical and thus whether our construction is meaningful.

Ultimately, one may view it simply as a convenience for constructing the projection operator Ph,w, whose action on any state can be de ned operationally by using (3.23). The projector thus de ned is easily seen to satisfy P2h,w = Ph,w, and also to project any state onto the basis Lk1m1 . . . Lknmn[notdef]h[angbracketright], which is all that is

required for the construction of the conformal blocks.

{ 11 {

3.3 Order of limits at large central charge

The key advance beyond [25] is that previously, only part of the conformal block was known in the heavy-light large c limit, whereas (3.29) contains the full answer at c ! 1. The

results in [25] apply to a di erent regime of parameter space, so let us clarify how those results relate to the present ones.

The monodromy method we employed in [25] formally only applies to the limit where all hi=c and hp=c are held xed while c ! 1. In this limit, the conformal blocks are

expected to grow like V ecf(z)g(z) for some function f that depends only on the ratios

hi=c; hp=c. Such methods cannot determine the c-independent prefactor g(z). Furthermore, it is very challenging to determine the function f(z) exactly for arbitrary hi=c; hp=c, and in practice additional approximations must be made (but see [49] for further ideas). Consequently, we made the further expansion of f in small hL=c and worked to linear order only, while working exactly as a function of hH=c.

In fact, it is not immediately obvious that the limit (1.2) considered in this paper has any overlap with that just described, i.e. that the limits

lim

hL/c!0

JHEP11(2015)200

c hL

lim

c!1,hL/c xed

1c log V

(z) zhp2hL

= lim

lim

c!1,hL xed

log V(z)

zhp2hL

(3.30)

(with hH=c and hp=hL xed on both sides) are equivalent. In particular, one might wonder why coe cients like

hHhp+h3p

c+h2p , which would give

hL!1

1 hL

hH c

hphL in the second limit but

hphL in the

rst limit, never appear.

To understand why the limits are equivalent, one can compare terms in log V in a series

expansion in z, each of whose coe cients is a rational function of hL and c. At each level, the Virasoro block projectors produce rational functions of c; hp where the denominator comes solely from inverting the Gram matrix. This means that the denominator is proportional to the Kac determinant,

Qm,n(hphm,n(c)), where hm,n(c) =

c24 ((m+n)21)+O(c0) and the product is taken over all m; n corresponding to the Gram matrix at that level.

At large c in either limit (hp xed or hp=c xed), the leading terms in the denominator all scale like the same power of s under c ! cs; hp ! hps, so coe cients like

hH c

hphL are

indeed impossible. In more physical terms, in this limit we can think of c and hp having mass dimension, so that the form of the result is constrained by dimensional analysis.

Consequently, the denominator is a polynomial in hp and c, and the leading term at large c with hp=c xed is

Qm,n hp

c24 ((m + n)2 1)

, and the hp is formally negligible if we x hp in the large c limit. Taking the logarithm of the Virasoro block mixes powers of z at many di erent orders, but the denominator of each resulting term is a product of the original denominators and thus does not change the fact that both limits have the same leading term. The leading term in the numerator must be the same because both the hp=c xed and hp xed limits exist. The power of c and hp of the leading term in the denominator xes the power of c and hp of the leading term in the numerator.

3.4 Recursion formula

Our result (3.29) for the heavy-light conformal block at c = 1 can be systematically

improved by adapting the recursion relations of [45, 46]. The basic idea of this method

{ 12 {

is to consider the conformal blocks as analytic functions of c. Poles in c arise at discrete values cm,n where the representation becomes null, parameterized by pairs of integers m; n satisfying m 1; n 2. The exact conformal block is a sum over its poles plus the value

at c = 1:

V(c; hi; hp; z) = V(1; hi; hp; z)+Xm,n

Rm,n(cm,n(hp); hi)c cm,n(hp) V(cm,n(hp); hi; hp+mn; z): (3.31)

The residues of the poles must be conformal blocks with arguments shifted as indicated above, leading to a recursion relation seeded by the c = 1 piece. One can treat the

conformal block as an analytic function of c with hi; hp xed, as in the original papers [45, 46], or one can treat it as an analytic function of c with hH=c; hL; L; H xed. In the former case, the c = 1 piece is just the global conformal block, but in the latter case

it is the semi-classical block (3.29). When applying the above formula in this case, it is important to remember that hH has been promoted to a function of c, i.e. hH(c) Hc

with H xed, so on the r.h.s. we have hH ! hH(cm,n) = Hcm,n. Explicit expressions

for the coe cients Rm,n were derived in [45, 46] and are given in appendix A, along with expressions for cm,n.

Starting from the c = 1 seed in the recursion formula, each subsequent term in the

sum involves conformal blocks with the weight of the exchanged primary operator increased by mn. Since each block is a sum over powers of w beginning with whp and increasing by integer powers, this means that only a nite number of terms in (3.31) can ever contribute to a given power of w. The recursion formula provides an e cient method for computing the exact series coe cients of the Virasoro blocks in an expansion in powers of w.

It is less clear what can be learned analytically in powers of 1=c. At leading order in 1=c, one may neglect the cm,n in the denominator and write

V(c; hi; hp; z) = V(1; hi; hp; z) (3.32)

cXm,n

Rm,n(cm,n(hp); hi)V(cm,n(hp); hi; hp + mn; z) + O(1=c2):

This expansion should contain universal information about quantum 1=mpl corrections in gravitational theories. To obtain non-perturbative information one would need to solve the recursion relation exactly for some w and study the behavior in 1=c.

3.5 Extension to U(1)

In the case where the theory contains additional conserved currents, the symmetry algebra of the theory is enlarged beyond the Virasoro algebra. The simplest such extension is to add a U(1) current J, enhancing the algebra to include a Kac-Moody algebra:

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

[Ln; Jm] = mJn+m;

[Jn; Jm] = nk n+m,0;

[Jn; (z)] = qzn(z); (3.33)

{ 13 {

JHEP11(2015)200

c12(n3 n) n+m,0;

where J(z) =

Pn zn1Jn and k is the level of the current. Now, we want to know the U(1)-extended Virasoro conformal blocks VT+J , i.e. the contribution from irreps of the

extended algebra. To simplify its computation, we choose a new basis of operators as

follows. We start with the well-known construction of the Sugawara stress tensor Tsug (see

e.g. [50])

T sug(z) =

Xn zn2Lsugn;

Lsugn = 1

2kXm 1

JmJnm + Xm 0

JnmJm

112n(n2 1) n+m,0;

[Lsugn; Jm] = mJn+m; (3.35)

and a short computation shows

[Ln; Lsugm] = (n m)Lsugn+m +

2k : (3.39) In the conformal block, all descendants of the exchanged state [notdef]hp[angbracketright] can be written in a

basis of Jns and L(0)ns. We restrict our attention to conformal blocks where the exchanged primary is neutral, i.e. qH1 = qH2 qH; qL1 = qL2 qS. Consider the action of any of

the L(0)ns inside the three-point function [angbracketleft]hp[notdef]L1(z)L2(0)[angbracketright]:

hhp[notdef]L(0)nL1(z)L2(0)[angbracketright] = [angbracketleft]hp[notdef]

2k (J1Jn1 + J2Jn2 + : : : Jn1J1)

{ 14 {

!: (3.34)

The Sugawara generators act almost like Virasoro generators with central charge c = 1:

[Lsugn; Lsugm] = (n m)Lsugn+m +

112n(n2 1) n+m,0: (3.36)

One may think of the true stress tensor as getting a contribution from the U(1) charge through the Sugawara stress tensor, so that by subtracting out this contribution we can factor out the contributions to the extended conformal blocks from the current. More precisely, we de ne

L(0)n Ln Lsugn: (3.37)

The key point is that the L(0)n and Jn generators provide a basis that factors the algebra into separate Virasoro and U(1) sectors:

hL(0)n; L(0)mi= (n m)L(0)n+m +c 112 n(n2 1) n+m,0;

hL(0)n; Jmi= 0: (3.38)

Furthermore, it is straightforward to check that states [notdef][angbracketright] that are primary with respect

to the Lns and Jns, with weight h and charge q, are primary under L(0)n as well, with weight

h(0) = h

JHEP11(2015)200

L1(z)L2(0)[angbracketright];

(3.40)

where we have used the fact that Jn with n 0 annihilates [angbracketleft]hp[notdef]. Now, note that

hhp[notdef]L1(z)L2(0)[angbracketright] = cL1L2p

1zhL1 +hL2hp = cL1L2hp

zq2L/k

: (3.41)

Thus, we nd that L(0)n acts to the right like Ln, but with hL1; hL2 replaced with h(0)L1; h(0)L2:

hhp[notdef]L(0)nL1(z)L2(0)[angbracketright] = [angbracketleft]hp[notdef](hL1(n + 1)zn + zn+1@z

n 1

(0)

L1 +h(0)L2hp

2k q2Lzn)L1(z)L2(0)[angbracketright]

= zq2L/k(h(0)L1(n + 1)zn + zn+1@z)cL1L2hp

(0)

L1 +h(0)L2hp

: (3.42)

Descendants with multiple L(0)ns follow the same derivation; Jns commute with L(0)ns, so those of them with n 0 can still be moved to the left where they annihilate [angbracketleft]hp[notdef]. Then,

one commutes zq2L/k to the left, where it produces an extra zn q

2 L

= zq2L/k [[angbracketleft]hp[notdef]LnL1(z)L2(0)[angbracketright]]hLi !h(0)Li

k each time it commutes with a @z. Descendants with Jns can always be organized so that the Jn act to the right rst, where it just produces a h-independent power of z and some factors of qL, so the

action of the L(0)ns and Jns factors.

Thus, the conformal blocks factor into a product of Virasoro conformal blocks with c ! c 1 and hi ! hi q

2 i

JHEP11(2015)200

2k , and a U(1) block:

VT +J (c; hi; k; qi; hp; z) = VT

VJ(k; qi; z); (3.43)

where VJ is the contribution from just the U(1) descendants Jn, which can be computed

by directly summing the contribution at each level:

VJ (k; qi; z) = z

Lk (1 z)

c 1; hi q2i2k ; hp; z

qH qLk : (3.44)

We can also give a heuristic derivation of this result using background elds. Let us couple a classical background eld A[notdef](x) to a U(1) current J[notdef] = @[notdef], where is a free boson. This shifts the CFT action by

@A, leading to the equation of motion

@2(x) = @[notdef]A[notdef](x): (3.45)

We obtain a classical VEV [angbracketleft]J[notdef](x)[angbracketright] = A[notdef](x). Thus if we turn on the holomorphic Az(z) =

Q1z we can trivialize the 3-pt correlator [angbracketleft]OQ(1)OQ(1)J(z)[angbracketright] for an operator of charge

Q. This background re-absorbs the e ect of J-exchange between OQ and other operators.

By writing another operator Oq(x) of charge q as eiq(x)O0(x) and evaluating the classical part of (x), we obtain VJ.

4 AdS interpretation

4.1 Matching of backgrounds

To make contact with AdS physics, it is most convenient to place the heavy operators at z = 0 and 1 in the CFT, corresponding to a heavy AdS object that exists for all time. The

{ 15 {

conformal transformation w = z from section 3.1 has a simple geometric interpretation in AdS3. A general vacuum metric can be written as

ds2 = L(z)2 dz2 +

z)2 d

z2 +

1y2 +

4 L(z)

dzd z + dy2y2 : (4.1)

A point mass source with dimension hH +

hH and spin hH

hH corresponds to

L(z) = 1

1 24hHc = 2H2z2 ;

L( z) = 1 2 z2

1 24 hHc = 2H2 z2 : (4.2)

This can be mapped to global coordinates by y = 2

2z2

zz H

H e :

JHEP11(2015)200

ds2 = d 2 + H

H cosh(2 )

dzd

z z

z +

1 4

2H dz2z2 + 2H d z2 z2

; (4.3)

which makes it manifest that the metric becomes locally pure AdS3 under z = w

1 H ;

z =

H . Furthermore, the boundary stress tensor is related to the metric in these coordinates by T (z) = 1

2z2

c12 L(z). The AdS interpretation of the coordinate transformation is therefore particularly simple: the heavy operators create a background value for T (z),

hhH[notdef]T (z)[notdef]hH[angbracketright] = hHz2 , which is removed when we transform to w coordinates. It is remark

able that this connection between the bulk and boundary descriptions is exhibited at the level of the individual conformal blocks.

To exhibit the black hole horizon, one can bring the metric into the original BTZ coordinates,

z = et+;

z = et;

2 cosh1

4r2 + 2H +

2H 2 H

; (4.4)

with metric

ds2 =

(r2 r2+)(r2 r2)

r2 dt2 +

r2(r2 r2+)(r2 r2)

dr2 + r2

d + r+r r2 dt

2: (4.5)

The inner and outer horizons r; r+ are related to H;

H by

(r+ + r)2 = 2H =

24hc 1

;

(r+ r)2 =

2H =

24 hc 1

: (4.6)

One sees by inspection that the inner and outer horizons r; r+ appear in the above metric

only in the combinations r2+ + r2 and r+r, which are real-valued for any h;

h. However,

the values r[notdef] themselves can be complex:

r[notdef] =

24h c

1/2

1/2[notdef]

24h

2 : (4.7)

Therefore, the black hole horizon exists only when both h and

h are greater than c

24 ; in

other words, to make a black hole in AdS3, states must have twist greater than c

12 .

{ 16 {

4.2 Extension to U(1)

The extension of the conformal blocks to include a U(1) conserved current in (3.43) also has a direct connection to backgrounds in AdS3. To include a U(1) current in the boundary, we need to add a gauge eld in the bulk, and in order to correctly match the boundary description it must be a Chern-Simons gauge eld without a Maxwell kinetic term F[notdef] F [notdef] .

Perhaps the most concrete way of understanding this is by considering the physical modes in the spectrum arising from the bulk gauge eld. A conserved spin-1 current on the boundary corresponds to a pure boundary mode in AdS3, much like gravitons are pure boundary modes in AdS3. This can be seen from the fact that the wavefunction [notdef] [angbracketleft]0[notdef]A[notdef][notdef]J[angbracketright] must have frequency 1 and spin 1, and be annihilated by the special

conformal generators. The former condition implies [notdef] = ei(t[notdef] )f[notdef](), and the latter

condition implies [51]

[notdef] = ( t; ; ) / iei(t[notdef] ) cos tan (1; i cot ; [notdef]1)[notdef] = @[notdef](ei(t[notdef] ) sin ): (4.8)

This is manifestly pure gauge in the bulk, and so it is a boundary mode. A Chern-Simons gauge eld in the bulk is topological and thus has only boundary modes, as required, whereas a Maxwell gauge eld has a bulk degree of freedom.

This becomes even more explicit when we consider a Chern-Simons term together with a Maxwell kinetic term in the bulk.3 In this case, the bulk photon gets a topological mass, and thus the dimension of the dual CFT mode is raised above 1, implying that it cannot be a conserved current.

The calculation of correlators arising from exchange of a bulk U(1) Chern-Simons elds was performed in [53]. Here we will discuss the generalization to include gravity as well. The action is

SE = Sbulk + Sbd; (4.9)

Sbulk =

JHEP11(2015)200

k 8

d3x [notdef] (A[notdef]@ A

A[notdef]@

A );

Sbd =

k 8

d2xp (A2 +

A2 2A [notdef]

A 2i ijAi

Aj)

k 8

d2z(AzAz +

Az 2Az

Az):

The bulk term does not depend on the metric and thus the computation of the four-point function from Chern-Simons exchange in [53] goes through almost, but not quite, unchanged. The result there exactly matches (up to a di erence in normalization of qi) the

four-point function (3.44) from summing over Jn descendants.4 However, the boundary

terms above do depend on the metric, and thus they feel the presence of gravity. Their

3The Chern-Simons term is necessarily present due to the chiral anomaly, which is given by the level k of the U(1) boundary current and thus must be a positive integer [52].

4It is stated in [53] that the derivation is valid only at leading order in 1/k; however, in the semi-classical limit k ! 1 with qH/k and qL xed, the backreaction from the light L eld on the Chern-Simons

background can be neglected, and the dependence on k to all orders is captured.

{ 17 {

e ect is very simple: in a normalization where Jz = k

stress tensor by [52]

2k JzJz: (4.10)

Thus, the charge q of a source reduces its energy as measured at the boundary by q2=2k.

This exactly matches the shift in the conformal blocks in equation (3.43). It also implies that the matching to the spectrum in a black hole background generalizes immediately. Because the Chern-Simons eld and the metric are not coupled in the bulk, black hole solutions are just BTZ with a shifted mass parameter, M ! M q2=2k, and a background

pro le for the Chern-Simons eld [53]. For a light neutral CFT operator in the background of a heavy charged state, this shift is the only change as compared to a heavy neutral state. If the light operator is charged, the background Chern-Simons eld produces the additional dependence (3.44), which completely factorizes. The spectrum of twists (or quasi-normal modes) at large spin can then be read o from the bootstrap equation exactly as in [25]. The only two changes are that the \binding" energy depends on hi q2i=2k rather than hi,

and there is an additional contribution qBqS=k:

n = 2hB + qBqSk + 2i

4.3 Matching of correlators

It would be interesting to see how the large c Virasoro blocks arise as contributions to heavy-light 4-pt correlators obtained from Feynman diagram computations in AdS3. We

previously analyzed the case of the identity block [25], but matching with more general correlators is signi cantly more challenging. Here we will con ne our analysis to the case of 3-pt correlators. This will be su cient to show that the somewhat mysterious rescaling of H ! H= in the Virasoro conformal block is a kinematical consequence of the conformal

transformation to the w coordinates.

In a at background metric, the CFT 3-pt correlator is

[angbracketleft]OH1(1)OH2(0)OL(z)[angbracketright] = zhL+ H: (4.12) It is non-trivial to obtain this from an explicit bulk computation that includes gravitational backreaction, because the heavy operators have bulk-to-boundary propagators that are dressed by gravitons, and the back-reaction of the geometry on the bulk-to-boundary propagator is not suppressed. However, conformal symmetry guarantees the form of the nal result. The conformal transformation to w = z produces a new correlator

[angbracketleft]OH1(1)OH2(0)OL(w)[angbracketright] = whL+

: (4.13)

Let us recall how this rescaling of H from z to w coordinates can be derived from a bulk analysis.

We will write AdS in global coordinates

ds2 = 1cos2 (dt2 + d2 + sin2 d 2) (4.14)

{ 18 {

(2)1/2 Az, they shift the boundary

Sbd

Tzz ! Tzz +

zz = Tzz

r24hB q2B2kc 1

n + hS q2S 2k

: (4.11)

JHEP11(2015)200

where = =2 corresponds to the boundary of AdS. To study CFT correlators in a at metric, we approach the boundary with ( ) = =2 et and then send ! 0. This

produces a boundary metric

ds2 = e2t(dt2 + d 2) = dzd

z: (4.15)

The CFT operator OL is de ned through the boundary limit of a bulk eld L(t; ; ) as

OL(t; )

L(t; ; =2 )

: (4.16)

The CFT correlators are obtained by applying this limit to the correlators of bulk elds.

By transforming to w coordinates on the boundary, we move the CFT to a new background with metric

ds2 = dwd

z = dw

dz dzd

z: (4.17)

To obtain the new metric from AdS/CFT, we must approach the boundary via

( ) =

2 etr

dzdw (4.18)

where we take the limit ! 0. Thus we recover the conformal transformation of a primary

operator

OL(w) =

JHEP11(2015)200

OL(z(w)) (4.19)

dz dw

which gives

hw 1

ihL w

hL+ H

= whL+

[angbracketleft]OH1(1)OH2(0)OL(w)[angbracketright] /

(4.20)

as claimed. This explains the rescaling of H ! H= in the Virasoro conformal block of

equation (3.29) as the simple consequence of a conformal transformation.

5 Implications for thermodynamics

At rst glance, our results suggest that in any CFT2 at very large c, heavy operators with hH;

hH > c

24 produce a thermal background for light correlators, because in the w = 1 (1 z)2iTH coordinates we have an explicit periodicity in the Euclidean time

t log(1 z). If the full correlator had this periodicity, it would imply a version of

the eigenstate thermalization hypothesis, or alternatively a derivation of BTZ black hole thermality, as reviewed in appendix B. This interpretation is too naive, but let us rst note what it does immediately imply, and then move on to a more general discussion.

{ 19 {

5.1 Far from the black hole: the lightcone OPE limit

As discussed in [14, 15, 25], there is a lightcone OPE limit where correlation functions are dominated by the exchange of the identity Virasoro block.5 In a CFT2 without any conserved currents besides the stress tensor and U(1)s, this means that correlators such as

[angbracketleft]OH1(0)OH2(1)OL1(1)OL2(z)[angbracketright] (5.1) are thermal in the limit where z ! 1 with

z xed, or vice versa, at large central charge. These correlators capture physical setups in AdS where the light probe is very far from the black hole. This strongly suggests that if a thermal interpretation is appropriate, the temperature must be the TH given in equation (1.6).

One can argue via the operator product expansion that more general correlators will have the same property in an appropriate limit. For example, consider a 6-pt function

[angbracketleft]OH1(0)OH2(1)OL1(z1)OL2(z2)OL3(z3)OL4(z4)[angbracketright]: (5.2) In the limit that the zi approach each other in a lightcone limit, this correlator will be dominated by identity exchange between the heavy and light operators. By expanding the light operators in the OPE we can reduce the 6-pt correlator to a 4-pt correlator, and conclude that the result must be thermal. A more physical way to obtain the same answer is to transform to w-coordinates as in section 3.1. The action of the Virasoro generators

Ln with [notdef]n[notdef] 2 are trivialized by this transformation, and so expanding in the lightcone

OPE limit we simply have

[angbracketleft]OH1(0)OH2(1)OL1(w1)OL2(w2)OL3(w3)OL4(w4)[angbracketright]

[angbracketleft]OH1(0)OH2(1)[angbracketright] [notdef] [angbracketleft]OL1(w1)OL2(w2)OL3(w3)OL4(w4)[angbracketright] (5.3) in the w-coordinate background. The heavy operator produces a thermal background for the four light operators. This can obviously be generalized to 2 + n-point correlators with two heavy operators.

5.2 Implications for entanglement entropy

CFT entanglement entropies can be computed via the replica trick

SA = lim

n!1

log (tr nA) (5.4)

where A is the reduced density matrix for a geometric region A, and we analytically continue the integer n. This trick is useful because the Renyi entropies nA for integer n can be computed by gluing together multiple copies of the system along the region A. In CFT this means that they can be obtained from correlation functions of twist operators, which admit a conformal block decomposition. See [54] for a review and [31, 37] for a relevant recent application using the Virasoro blocks. The twist operators have xed dimension

hn = c

n 1n

5More generally, the lightcone OPE limit is dominated by the exchange of twist-zero operators.

{ 20 {

JHEP11(2015)200

1n 1

(5.5)

and so at large c, for n [negationslash]= 1 their dimensions are of order c. As discussed in section 3.3,

the limit c ! 1 with h xed followed by h ! 0 should commute with the limit where

h=c is xed, followed by h=c ! 0. Thus our results should apply to the computation of

entanglement entropies using twist operators, and the recursion relations of section 3.4 could be used to obtain corrections in 1=c.

Our result for the heavy-light Virasoro blocks are more precise than those obtained in [25], but in fact we have obtained the same formula for the identity or vacuum Virasoro block

V1(z) = 1(1 z

H )2hL + O(1=c): (5.6)

Previously the uncertainty was up to an order one multiplicative function, but it has been reduced to an additive term suppressed by 1=c. This implies that the corrections to [25] are smaller than might have been imagined, particularly in theories with a gap in the spectrum above the identity operator.

One might also study the entanglement entropy of several intervals in the presence of a non-trivial background. In the limits where the analysis of [29, 37] remains valid, so that the entanglement entropy is dominated purely by exchange of operators that are Virasoro descendants of the vacuum, one could study kinematic limits where the factorization of equation (5.3) obtains. Thus we nd that the entanglement entropy for a region A composed of many disjoint intervals is [29]

SA = min c

6X(i,j) log

(wi wj)2zizj wiwj

(up to a UV-regulator-dependent normalization) where i denote the end points of various intervals in the CFT, we minimize over all possible pairings (i; j). The argument of the logarithm has the form of a distance in the w-metric multiplied by the Jacobian factor associated with the transformation rule of primary operators, as in equation (3.3).

As one would expect from the discussion of section 4.1, this is simply the minimum of the sums of geodesics connecting interval end-points in the appropriate de cit angle or BTZ background. The choice of branch cut in the logarithm determines how many times the geodesic winds [37] around the singularity.

5.3 General kinematics and a connection with AdS locality

To understand thermality for general kinematics we need to study the full correlator, which is an in nite sum over Virasoro blocks. The di culty is that the hypergeometric function in equation (3.29) has branch cuts in the w-coordinate, so periodicity in Euclidean time is not obvious for general blocks. Furthermore, the sum over the Virasoro blocks does not converge everywhere in the z;

z planes, so the sum does not necessarily have the same properties as its summands. Let us study these issues in more detail, and then we will formulate some general, su cient criteria for thermality, motivated in part by our present understanding [9{11, 13, 55{57] of the relationship between CFT data and AdS locality.

{ 21 {

JHEP11(2015)200

(5.7)

The full Virasoro blocks are products Vh(z)V

z), and the Euclidean time coordinate is

t = log((1 z)(1

z)) =

i2TH log((1 w)(1

w)) (5.8)

where we are considering the scalar case hH =

hH and H = 0 for simplicity. Translation

in Euclidean time corresponds to rotation about point w;

w = 1. To study the branch cut structure, it is useful to apply the hypergeometric identity6

2F1( ; ; ; 1 x) =

( ) ( )

( ) ( )2

F1( ; ; + + 1; x) (5.9)

+ ( ) ( + )

( ) ( ) x 2F1( ; ; + 1; x)

to the Virasoro blocks. The hypergeometric functions on the right hand side have a convergent Taylor expansion around x = 0, so the only branch cut emanating from x = 0 arises from the explicit power x . Applying this relation to Vh(w)V

w) produces

four terms, which can be written schematically as

Vh(w)Vh( w) = A(w)

A( w) + (1 w)2 L(1 w)2 LB(w)

B( w)

+(1 w)2 L

B(w)A(

w) + (1

w)2 LA(w)

w) (5.10)

JHEP11(2015)200

where the functions A;

A; B;

B have trivial monodromy about w;

w = 1. The quantity

L must be an integer if the light operators have half-integer or integer spin, so the terms on the rst line are automatically periodic under t ! t + 1=TH. The terms on the

second line are problematic due to the branch cut emanating from w;

w = 0. These terms must cancel in the sum over Virasoro blocks in order to give a thermal correlator.

It should be emphasized that these branch cuts are a general issue for any CFT correlator written as a sum over conformal blocks. The Virasoro blocks at large c are basically identical to global conformal blocks written in a new coordinate system. Correlators of integer spin operators must be periodic in the angle = log(1z)log(1

2 L 2

z), and the terms

on the second line of equation (5.10) do not have this property when we replace w ! z

to turn the Virasoro blocks back into global blocks. In conventional CFT correlators, the branch cuts from these problematic terms must always cancel to restore periodicity in . We would like to know when they cancel in the heavy-light correlators, since this would imply thermality.

In theories with a sparse low-dimension spectrum, one would expect that few conformal blocks contribute, and so the sum over all the blocks has simple properties. There has been major progress understanding such theories using the CFT bootstrap, and in particular, it has been shown that the o ending branch cuts in the analogue to equation (5.10) for the global conformal blocks in z coordinates are canceled by in nite sums over double-trace operators. This phenomenon can be generalized far beyond the class of theories with a sparse spectrum by using Mellin amplitudes [55, 58{61] to represent CFT correlators, as we will discuss in a forthcoming paper [62].

6There is a logarithm in the special case where is an integer.

{ 22 {

Any CFT whose correlators have exponentially bounded Mellin amplitudes will be guaranteed to have physical periodicity in . As a toy example, consider the function f(z)

f(z) = Z

ds M(s)(1 z)s (5.11)

written in terms of a Mellin transform. The function f will not have any branch cuts emanating from z = 0 if M(s) falls o exponentially fast as s ! [notdef]i1. In other words, the

asymptotic behavior of the Mellin transform M(s) determines the analyticity of the original function. This fact will be used [62] in forthcoming work to study universal properties of CFT correlators and OPE coe cients.

The asymptotic behavior of the Mellin amplitude also has a close connection with short-distance AdS locality [13, 57]. CFTs with a perturbative expansion parameter, a Fock space spectrum, and a polynomially bounded Mellin amplitude can always be described by an e ective eld theory in AdS. Exponential boundedness is a weaker condition that nevertheless implies the desired periodicity in .

Given the simple relationship between Virasoro and global blocks, including the equivalent issues with periodicity in t and , it is natural to consider a Mellin amplitude representation directly in the w coordinates. This is also motivated by the idea that the heavy operator simply creates a new classical background in which the dynamics transpire. In analogy with the conformal cross ratios u = z

z and v = (1 z)(1

z), we write the

w)]t (5.12)

using the Mellin space representation Mw(s; t). As long as Mw(s; t) falls o exponentially as s; t ! [notdef]i1, the correlator will be analytic around w;

w]s[(1 w)(1

w = 0. Note that periodicity in is not obvious in this representation, but imposes further constraints on Mw. We

stress that Mw is not the usual Mellin amplitude M: the two di er by whether the Mellin transform is performed on the correlator in z coordinates or in w coordinates, and exponential boundedness of one does not necessarily translate into exponential boundedness of the other.

When written in w;

w coordinates as in equation (5.12), the Virasoro blocks have a simple Mellin amplitude representation Bw(s; t) [57, 58]. The existence of speci c operators in the light-light and heavy-heavy OPE limits implies the presence of speci c poles in Bw(s; t), and these poles must remain in the Mellin representation of the full CFT correlator. The individual Bw(s; t) do not fall o exponentially as s; t ! [notdef]i1, since the Virasoro

blocks are not analytic near w = 1. But if the sum of the Bw(s; t) has a good asymptotic behavior, then the full CFT correlator will be thermal. One might also investigate factorizations as in equation (5.3) in this Mellin space language.

6 Future directions

Intuition suggests that CFT operators carrying a large conserved charge may generate a classical background eld a ecting correlation functions in a universal way. This idea

{ 23 {

JHEP11(2015)200

heavy-light correlator

[angbracketleft]OH1(0)OH2(1)OL1(1)OL2(w;

w)[angbracketright] =

dsdt Mw(s; t)[w

has obvious validity in the case of AdS/CFT, where heavy or charged states generate gravitational or gauge elds in AdS, but one can also look for manifestations of this e ect in general CFTs. We have shown that heavy operators in 2d CFTs have exactly this e ect, and that it can be used to compute Virasoro conformal blocks at large central charge. We also used this idea to sum up contributions from U(1) currents in the CFT. Classical gravitational and gauge elds in AdS3 have a universal manifestation in all CFT2 at large central charge.

To leading order at large central charge c, the heavy-light Virasoro blocks bear a striking imprint of thermality, and it would be interesting to explore this property in more detail. In particular, one should expect that the probability of a transition from one heavy state to another should be weighted by a Boltzmann factor. The rescaled H = 2iEH1EH2TH appearing in the conformal blocks of equation (3.29) seems tantalizing in this regard. In fact, in the limit of small h; L the conformal blocks approach a universal form proportional to (1 w) H/ . This result might be used to argue that the rates for transitions among

the heavy states take a universal thermal form.

Unitary CFT correlators should not be exactly thermal. One can use the recursion relations of section 3.4 to compute corrections to our c = 1 result, but at present this

is only practical if we work to low order in the coordinate w, or perhaps to rst order in 1=c. However, corrections to thermality are most likely non-perturbative in c, showing up in the Virasoro blocks at nite z or w. It would be very exciting to understand these non-perturbative corrections. CFT correlators are made from an in nite sum of Virasoro blocks, so it would also be interesting to understand whether corrections to thermality arise primarily from non-perturbative corrections to the individual Virasoro blocks, or from patterns in the in nite sum.

An immediate question is whether one can obtain similar results using classical background elds for CFTs in d > 2 dimensions. We were able to obtain very precise results for CFT2 by making use of the in nite dimensional Virasoro algebra, so one might not expect higher dimensional generalizations to be possible. However, we believe that the results of [47, 48] on the expectation value of the stress-tensor in conformally at backgrounds can be used to study heavy operators in large central charge CFTs in d = 4 and 6, perhaps after imposing some physical constraints [63] on the CFT data. It would be very interesting to generalize the analysis of [47, 48, 64], and to understand which CFT data a ect the result.

Our analysis motivates studying CFTs in a non-trivial background metric chosen to trivialize stress tensor correlators. We have worked with a pure CFT, but one might also consider a CFT with a UV cuto , or a general local QFT. By recapitulating the computations of section 3.1 in a CFT with a UV cuto , one might attempt to derive the bulk Einstein equations. The idea would be to identify the cuto with the position of a UV brane, and determine an appropriate metric as a function of the cuto . Perhaps one could obtain a more precise understanding of the relationship between UV cuto schemes and bulk locality.

There is also a fascinating connection between conformal blocks and entanglement entropy at large central charge [28] which has recently led to increasingly rich matching of such e ects between gravity and eld theory computations [29, 31{33, 35, 37]. We

{ 24 {

JHEP11(2015)200

have discussed how an extension to entanglement entropies of multiple intervals in the background of an excited state follows naturally from the results here, and it would be interesting to consider connections to other deformations as well. For instance, the addition of light states in the gravity theory produces quantum corrections to the Ryu-Takayanagi formula [65{67], and in terms of the eld theory these should involve conformal blocks for states with corresponding conformal weights; it would be interesting to investigate the precise decomposition. Finally, even within the pure gravitational theory, loop corrections to Renyi entropies on the gravity side are reproduced by conformal blocks including not just Virasoro descendants of the vacuum but also additional descendants that arise from acting with operators from the di erent \replicas" [32, 33]. It seems natural to think of these contributions as descendants of the vacuum under an enhanced symmetry, and to try to simplify the eld theory computation using some of the methods employed here.

It would be interesting to study more general background eld con gurations, and couplings that source more general CFT operators. One can also use the methods of section 3.1 to nd more complicated background metrics that simplify stress tensor correlators in the presence of more than two heavy operators, for instance by combining global conformal transformations with repeated application of z ! z . Such conformal mappings

can be used to study states created by several heavy operators. Supersymmetric theories and higher spin conserved currents would also be of interest, and our methods could provide interesting corrections to the results of [68] on conformal blocks in theories with

WN symmetry.

Acknowledgments

We would like to thank Chris Brust, Liang Dai, Tom Faulkner, Shamit Kachru, Ami Katz, Zuhair Khandker, Logan Maingi, Gustavo Marques Tavares, Junpu Wang, and Xi Dong for valuable discussions. JK is supported in part by NSF grant PHY-1316665 and by a Sloan Foundation fellowship. ALF was partially supported by ERC grant BSMOXFORD no. 228169. MTW was supported by NSF grant PHY-1214000 and DOE grant DE-SC0010025.

A Details of recursion formula

In this appendix we will present explicit formulas necessary for the evaluation of the recursion formula, which were rst derived in [45, 46]. Instead of using the central charge c, it is often more convenient to work with the parameter T :

c = 13 6(T + 1=T ): (A.1) For each c, there are clearly two solutions for T , and which one we use will be de ned in context momentarily.

The poles in c arise from degeneracy conditions at special values determined by Kac. For any positive integers m and n, there is a degenerate state when

hp = c 124 +

( +m + n)2

4 ; (A.2)

{ 25 {

JHEP11(2015)200

where

[notdef] =

1 p24

p1 c [notdef] p25 c

: (A.3)

Note that + = 1. Now, we will de ne T to be the solution such that + = pT ; = 1=pT : (A.4)

Thus, the degeneracy condition can be written

hp =

(1 T )2

4T +

(mpT n=pT )2

4 : (A.5)

This has the solution

Tm,n(hp) = 2hp + mn 1 +

p(2hp + mn 1)2 (m2 1)(n2 1) n2 1

: (A.6)

The poles values cm,n(hp) are then just given by

cm,n = 13 6(Tm,n(hp) + 1=Tm,n(hp)): (A.7)

There is another branch of solutions to (A.5) corresponding to a plus sign in front of the square root. However, it is straightforward to verify that exchanging branches is equivalent to interchanging m and n. Thus, if we consider all pairs of integers m; n such m 1 and

n 2, then we cover all the degeneracy values cm,n(hp) using just the branch in (A.6).

The dependence of the coe cient factors Rm,n on the external weights is completely xed by the fact that they must vanish at speci c values allowed in theories with c = cm,n(hp),

together with the fact that according to the OPE, the coe cient of each power of z must be a polynomial of known order in the his. The remaining dependence on T was determined in [45, 46] to be

Rm,n(c; hi) = Jacm,n

Yp,q

JHEP11(2015)200

1 + 2 pq2 1 2 pq2 [notdef]

[notdef]

3 + 4 pq2 3 + 4 pq2 [notdef] 12[prime]

Yk,[lscript] 1k[lscript]: (A.8)

The product over p; q in the rst line is from p = (m 1); (m 3); (m 5); : : : ; m

3; m1, and q = (n1); (n3); (n5); : : : ; n3; n1. The primed product over k;

in the second line is from k = (m1); (m2); : : : ; m and = (n1); (n2); : : : ; n,

but omitting the pairs (k; ) = (0; 0) and (k; ) = (m; n). The i and p,qs denote

i =

rhi + (1 T )2 4T ;

p,q = qpT p=pT : (A.9)

The Jacobian factor Jacm,n is

dcm,n(hp) dhp

Jacm,n = dhpdc =

dhp=dT dc=dT =

24(1 T 2m,n)

1 + m2 T 2m,n(1 + n2)

: (A.10)

{ 26 {

B Periodicity in Euclidean time and thermodynamics

The simple thermodynamic relation

dS = dE

T (B.1)

implies that if we know the e ective temperature as a function of energy, T (E), then we can relate it to S(E), giving the density of states.

We will now review a more formal argument for this result based entirely on the periodicity of correlation functions in Euclidean time. Let us assume that a correlator

[angbracketleft]OH[notdef]O(t)O(0)[notdef]OH[angbracketright] (B.2) is periodic in imaginary t with period . We will demonstrate the relation between and the density of states in equation (B.1).

We can write the 4-pt correlation function as

[angbracketleft]OH[notdef]O(t)O(0)[notdef]OH[angbracketright] =

= dE[prime]eS(E[prime])ei(EE[prime])t[notdef][angbracketleft]E[notdef]O(0)[notdef]E[prime][angbracketright][notdef]2 (B.4)

where we wrote OH as the state E. If we assume that the original correlator is order one,

while S(E) is a large number, then we expect

hE[notdef]O[notdef]E[prime][angbracketright] e(S(E[prime])+S(E))/4 or eS[parenleftBig]

dE[prime]eS(E[prime])ei(EE[prime])t+ (EE[prime])[notdef][angbracketleft]E[notdef]O(0)[notdef]E[prime][angbracketright][notdef]2: (B.7)

Now let us inverse Fourier transform from t ! !. We obtaineS(E!)[notdef][angbracketleft]E[notdef]O(0)[notdef]E ![angbracketright][notdef]2 = eS(E+!)e ![notdef][angbracketleft]E[notdef]O(0)[notdef]E + ![angbracketright][notdef]2: (B.8)

Clearly when ! = 0 this equation becomes trivial, so we do not learn anything about the overall magnitude of [angbracketleft]E[notdef]O[notdef]E[angbracketright] from it. This tells us that

eS(E!)S(E+!)+ ! = [notdef][angbracketleft]E[notdef]O(0)[notdef]E + ![angbracketright][notdef]2

[notdef][angbracketleft]E[notdef]O(0)[notdef]E ![angbracketright][notdef]2

{ 27 {

JHEP11(2015)200

dE[prime]eS(E[prime])[angbracketleft]E[notdef]O(t)[notdef]E[prime][angbracketright][angbracketleft]E[prime][notdef]O(0)[notdef]E[angbracketright] (B.3)

E+E[prime]

2 /2 (B.5)

where we imposed symmetry under E $ E[prime], but we cannot be more precise without more

information about the theory. Of course it remains possible that most of these matrix elements are identically zero, while others are much larger than this estimate, but here we are assuming continuity in E and E[prime].

The periodicity in Euclidean time, or more precisely the KMS relation, implies that

hE[notdef]O(t i )O(0)[notdef]E[angbracketright] = [angbracketleft]E[notdef]O(t)O(0)[notdef]E[angbracketright]: (B.6) This means that we can write

dE[prime]eS(E[prime])ei(EE[prime])t[notdef][angbracketleft]E[notdef]O(0)[notdef]E[prime][angbracketright][notdef]2 =Z

: (B.9)

Now if we assume that the [angbracketleft]E[notdef]O(0)[notdef]E[prime][angbracketright] correlators take either of the forms from equa

tion (B.5) and expand in !, we get the same result. Using the rst form, we obtain

eS(E!)S(E+!)+ ! = e

so that we nd

@E : (B.11)

This result relates the periodicity in imaginary time to the density of states, identifying = 1=T . The additional assumptions we needed were that we can view the correlator as the integral of a smooth function of energy E[prime], with a smooth entropy function S(E), and also the corresponding rough estimate of OPE coe cients from equation (B.5).

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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Abstract

We show that in 2d CFTs at large central charge, the coupling of the stress tensor to heavy operators can be re-absorbed by placing the CFT in a non-trivial background metric. This leads to a more precise computation of the Virasoro conformal blocks between heavy and light operators, which are shown to be equivalent to global conformal blocks evaluated in the new background. We also generalize to the case where the operators carry U(1) charges. The refined Virasoro blocks can be used as the seed for a new Virasoro block recursion relation expanded in the heavy-light limit. We comment on the implications of our results for the universality of black hole thermality in AdS^sub 3^, or equivalently, the eigenstate thermalization hypothesis for CFT^sub 2^ at large central charge.

Details

Title

Virasoro conformal blocks and thermality from classical background fields

Author

Fitzpatrick, A Liam; Kaplan, Jared; Walters, Matthew T

Pages

1-32

Publication year

2015

Publication date

Nov 2015

Publisher

Springer Nature B.V.

e-ISSN

10298479

Source type

Scholarly Journal

Language of publication

English

DOI

https://doi.org/10.1007/JHEP11(2015)200

ProQuest document ID

1749595938

SISSA, Trieste, Italy 2015

Virasoro conformal blocks and thermality from classical background fields

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