Conformal Blocks Beyond the Semi-Classical Limit

Conformal Blocks Beyond the Semi-Classical Limit

A. Liam Fitzpatrick, Jared Kaplan
Abstract

Black hole microstates and their approximate thermodynamic properties can be studied using heavy-light correlation functions in AdS/CFT. Universal features of these correlators can be extracted from the Virasoro conformal blocks in CFT, which encapsulate quantum gravitational effects in AdS. At infinite central charge , the Virasoro vacuum block provides an avatar of the black hole information paradox in the form of periodic Euclidean-time singularities that must be resolved at finite .

We compute Virasoro blocks in the heavy-light, large limit, extending our previous results by determining perturbative corrections. We obtain explicit closed-form expressions for both the ‘semi-classical’ and ‘quantum’ corrections to the vacuum block, and we provide integral formulas for general Virasoro blocks. We comment on the interpretation of our results for thermodynamics, discussing how monodromies in Euclidean time can arise from AdS calculations using ‘geodesic Witten diagrams’. We expect that only non-perturbative corrections in can resolve the singularities associated with the information paradox.

\affiliation

Dept. of Physics, Boston University, Boston, MA 02215
Dept. of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218

1 Introduction and Discussion

To make predictions about the thermodynamic behavior of a system, we usually study a statistical ensemble of states codified by a partition function. In this standard ‘macroscopic’ approach, the entropy function plays a key role, counting the number of states with energy and determining the phase diagram of the theory as a function of the temperature. For example, the Cardy formula [1, 2] for predicts the asymptotic density of states in CFT, thereby counting the number of black hole states in quantum gravity theories in AdS [3, 4].

We have taken a rather different ‘microscopic’ approach to thermodynamics in AdS/CFT [5, 6, 7], studying the correlation functions of light probe operators in the background of a heavy CFT microstate. Intuitively, we expect that there should be very little difference between observables computed in a thermal density matrix and those computed in a pure state randomly chosen from the canonical ensemble. Via the operator/state correspondence, we can infer thermodynamic properties from a 4-pt correlator by comparing

(1)

where , is a heavy operator, and the last equality holds in CFT. We obtain precisely this relation [8] by approximating the left-hand side with the Virasoro vacuum conformal block, computed at large central charge in the limit . In the light-cone OPE limit [9, 10], this will be a good approximation for any CFT without additional conserved currents; more generally it provides an interesting universal contribution to the correlator capturing gravitational effects in AdS. Thus the thermodynamic properties of high energy states in CFT at large are built into the structure of the Virasoro algebra.

In this work we will study corrections to the Virasoro conformal blocks and their implications for thermodynamics. These will include both semi-classical corrections at higher orders in and genuine ‘quantum’ corrections. We use the terminology ‘semi-classical’ and ‘quantum’ because these correspond, respectively, to the gravitational backreaction of the light probe and to gravitational loop effects in AdS. In the remainder of this introduction we will discuss how our discussion relates to the black hole information paradox, and then we provide a summary of the results.

Chaos can also be studied by taking a limit of CFT 4-point correlators [11, 12, 13], with a universal bound expected for large central charge theories [14]; the implications of our results for chaos will be discussed in a forthcoming work.

1.1 The Information Paradox and the Vacuum Block

The black hole information paradox has many guises. In its most visceral and pressing form, it requires understanding the correct description of physics near black hole horizons, and in particular, the question of whether the semi-classical description can survive as a good approximation while simultaneously allowing for unitary evolution [15, 16, 17, 18]. Such problems remain extremely perplexing and important, but they are difficult (or impossible?) to formulate as a precise question about CFT observables, and progress on this front may require qualitatively new ‘observables’ [19, 20, 21].

A more straightforward manifestation of the information paradox can be formulated directly in terms of CFT correlators. In the background of a large AdS-Schwarzschild black hole, the two-point correlation function with Lorentzian time-separation decays exponentially at large time [22]. This means that information dropped into the black hole at an initial time never comes out. A CFT living on a non-compact space or a CFT at infinite central charge may also have thermal correlators that decay exponentially for all times, as can be seen explicitly by analytically continuing the right-hand side of equation (1) for the case of CFT on the thermal cylinder. However, for a CFT living on a compact space with finite central charge and at a finite temperature,111For a CFT, we can connect the non-compact and compact cases by taking the infinite temperature limit and measuring distances in units of . correlators cannot decay exponentially for all times, as this would signal loss of information concerning a perturbation to the thermal density matrix.

We add another layer to the story by studying the correlators of light operators in the background of a heavy pure state. This makes it possible to probe the pure quantum state of a one-sided BTZ black hole, instead of an ensemble of black holes. In the thermodynamic limit we expect the relation of equation (1) to hold, leading to a sharp Euclidean-time signature of information loss. Thermal 2-pt correlators are periodic under . This periodicity leads to additional singularities in equation (1) from periodic images of the OPE singularity, which occur in the Euclidean region at for any integer . Although these singularities are obligatory for thermal 2-pt correlators, they are forbidden in the 4-pt correlators of a CFT at finite central charge [23]. So these singularities are a sharp signature of information loss in the large central charge limit, analogous to the bulk point singularity [24, 25, 26], a signature of bulk locality.

In the case of either exponential decay in for thermal 2-pt correlators or periodicity in for pure-state 4-pt correlators, it would be most interesting to have a bulk computation resolving the paradox. Unfortunately, we do not have a non-perturbative definition of the bulk theory, and in fact, the bulk theory may be precisely defined only via a dual CFT.

In this paper we will focus on Euclidean time periodicity and its manifestation in the Virasoro conformal blocks. We expect that unitarity can only be restored by non-perturbative effects in , and in particular that perturbative corrections should not violate the thermal periodicity of the large heavy-light correlators. These expectations are primarily based on the expectation that corrections correspond to loop effects around the infinite gravity saddle, which is an AdS black hole background with fixed Euclidean-time periodicity, and thus such corrections should at most produce perturbative corrections to . Roughly speaking, unitarity restoration should rely on contributions from different saddles and therefore involve effects of order .222Recall that , so for this is formally . Such non-perturbative effects will be addressed more directly in future work.

We will compute corrections to the Virasoro blocks and study their behavior in Euclidean time. We find that the corrections to the vacuum block do violate periodicity, with a non-trivial monodromy under . Intriguingly, there appear to be two relevant time scales, of order and , as the correlator has non-trivial dependence on both and .

However, we do not believe that these effects have any immediate connection with the resolution of the information paradox. Conformal blocks have unphysical monodromies in the Euclidean plane that cancel when they are summed to form full CFT correlation functions. The monodromies we find in the expansion of Virasoro blocks seem to play a similar role to the more banal monodromies of global conformal blocks. In section 3 we explain how these monodromies can arise from AdS computations of the blocks in terms of ‘geodesic Witten diagrams’ [27, 28, 29]. The case of both global and Virasoro blocks can be given a parallel treatment, which suggests that the Euclidean-time monodromies of the corrections are likely to disappear in the full correlators.

1.2 Summary of Results

In previous work we showed that in the heavy-light semi-classical limit, the vacuum conformal block can be written as

(2)

where

(3)

and . Rescaling so that we measure distances in units of , and then taking , we see that the full structure of the vacuum block is preserved.

Here we show that in the large temperature limit, the first correction in a expansion of the heavy-light vacuum block is

(4)

where is the incomplete Beta function, , is the harmonic function, and .333These expressions have various branch cuts; to be precise, one should start with the conventional definition of these special functions in the region to obtain the “first sheet” behavior near the Euclidean OPE limit, and extend the function by analytic continuation.

An important point is that the methods we use in this paper can obtain terms that are not visible at any order in the “semi-classical” part of the conformal blocks. This semi-classical part is defined as

(5)

where the ratios of the external dimensions to are all held fixed. After taking the logarithm of , the correction term above can be seen to survive in this limit, but the term does not and thus goes not only beyond leading order in but beyond the semi-classical limit itself.

After this work was substantially completed, the paper [30] appeared that uses a different method to compute an integral expression for the order (semi-classical) result.

2 Corrections to the Vacuum Conformal Block

2.1 Review

Conformal blocks in 2d CFTs are contributions to four-point correlation functions from irreducible representations of the full Virasoro algebra, and as such resum contributions from all powers of the stress tensor. These contributions are dual to those of all multi-graviton contributions in AdS, and thus automatically encode an enormous amount of information about gravity in AdS. To distinguish these conformal blocks from simpler expression that contain irreducible representations of the global subgroup , we refer to the former as Virasoro conformal blocks and the latter as global conformal blocks. The explicit form for global conformal blocks in 2d has been known for some time and is just a hypergeometric function [31]; this is in contrast with Virasoro blocks, where, despite various systematic expansions [31, 32, 33, 34, 35, 36, 37], no closed form expression is known. In [8, 38, 39, 40, 30, 41, 28, 42] methods have been developed for computing the Virasoro conformal blocks in a “heavy-light” limit, where the central charge as well as the conformal weight of two “heavy” external operators are taken to be large, while the conformal weight of two “light” external operators is held fixed. The most efficient technique [38] works by using the conformal anomaly to absorb the leading order contribution of the stress tensor in this limit into a deformation of the metric.

To be more precise, recall that the Laurent coefficients of the stress tensor depend on the coordinates being used:

(6)

The usual Virasoro generators are the Laurent coefficients in the flat coordinate , where the CFT lives in the metric . The subset of with are raising operators which, when acting on a primary state, provide a natural basis for all states in a conformal block. So, one can work out the conformal blocks for a four-point function by expanding the state created by in this natural basis

(7)

where is the primary state of the conformal block and are coefficients that are fixed by conformal symmetry. Recall that primary states are defined as those annihilated by the lowering operators with . One way to compute the Virasoro block is to construct a projector :

(8)

Acting with to make , one automatically obtains the sum over the basis in (7) with coefficients given by evaluating . The conformal block itself is just given by , the four point correlator projectioned onto the irreducible representation of Virasoro built from the primary state .

However, in the heavy-light limit, this is not a very efficient basis to use. Although the normalization factors grow with for most contributions and thus produce a large suppression, these can be compensated in the Virasoro block by factors of the heavy operator dimension coming from the numerator . Fortunately, there exists another natural basis that avoids this difficulty. It is easy to see that any other set of coordinates which begins linearly in Euclidean coordinates at small will again have the property that with , and thus also provides a natural basis. In [38] it was noted that the choice of coordinates

(9)

leads to remarkable simplifications in the basis generated by ; in particular, at leading order in , the only basis elements that contribute are those of the form . The reason is that when one forms the projector in this basis, there is no longer any enhancement from the conformal weight of the heavy operator in . The simplest way to see this is to note that due to the conformal anomaly,

(10)

This does not grow with , and therefore factor of cannot compensate for the suppression by factors of from the norms .

To go to subleading orders in , we have to include some of these suppressed terms. Clearly, we have to include terms where the suppression from the norm involves only one factor of , but there are also some contributions that must be included where the norm produces two factors of . The reason is that in the sum over modes, factors of the form with two ’s can produce a factor of upstairs. This is again easiest to understand by looking at correlators with the stress tensor, where this positive factor of arises from the limit when two ’s are brought together. In general, a correlator with insertions of can have at most upstairs from such OPE singularities, and there will be a suppression by coming from the norm of the physical modes. Thus, to compute to order we will have to consider factors of ’s.

2.2 Computation

The projector for the is similar to the original Euclidean basis projector . Inside a four-point function, it takes the form

(11)

As shown in [38], this correctly acts as a projector onto the modes when the overlap factor is just given by the analogous Euclidean overlap factor after a conformal transformation on the ’s:

(12)

The norm factors are unchanged from the Euclidean basis. The only piece that changes substantially is the overlap with the heavy operators:

(13)

Our strategy for computing these will be to compute the corresponding correlators and read off the Laurent coefficients. In the following, we will focus on the vacuum block with for simplicity, and relegate the calculation of the general case to appendix 4.

It will be convenient to choose the insertions of the heavy operators to be at 0 and rather than at 1 and ; this corresponds to and compared to above. Correlators can be computed in most easily by using the OPE:

(14)

where and . The notation “” here means “equal up to regular terms.” Since is holomorphic in , these OPEs determine the singularities and therefore the complete functional dependence of correlators in terms of correlators without insertions. Since the transformation from to is regular except at , the last OPE above, is just a rewriting of the standard OPE .

Applying the OPE to one or two insertions of we find

(15)

Expanding the above correlator at , one can see that there is only a fourth-order pole at and the higher order poles cancel, as is enforced by the OPE. To compute the correction to the leading order heavy-light Virasoro blocks, the only modes we need to sum are single- and double- modes. Calculating the overlap factors with the light operators and the inner product factors that enter is a straightforward application of the Virasoro algebra. It will be convenient to use a basis of double- modes that are symmetric in the indices, i.e. of the form . One finds

(16)

and

(17)

Inverting and expanding to ,

(18)

Now, these factors can be substituted into the sum that defines the projector. We can take advantage of the fact that is a generating function for in order to write these terms as contour integrals in the following form:

(19)

where

(20)

The sum on in can be done in closed form, and we get a combination of powers, logs, and hypergeometrics of the form

(21)

The integration contour in (19) must have , since the sum over powers of converges in when , and the sum over powers of in converges when . Starting with the contour integral over , we can shrink it down as far as possible. However, the sum over produces branch cuts that prevent one from shrinking the contour all the way down to the origin. These branch cuts in are along the real axis between 0 and ; the discontinuities across this branch cut can be read off from the coefficients of the logarithms in , together with the following expressions for the discontinuities of the hypergeometric functions:

(22)

We therefore reduce to

(23)

We interpret the term as a true ‘quantum’ correction while is ‘semi-classical’. The former would correspond to a loop effect in AdS, while the latter captures effects from classical gravitational backreaction from the light probe object.444Note that there is no piece. In fact, this is true at all orders in , since such a term would have to survive in the limit that . But in that case, would have to be the identity operator, so the vacuum “block” would be the two-point function, which is just constant normalized to 1. Finally, the remaining sum on converges in the region that , and gives

(24)

Thus, we can shrink the contour onto the branch cut from 0 to . However, note that after we do this, the branch cut from is crossed when , but not when . One also crosses a pole at in . However, as explained below equation (15), the only such singularity is . This does not contribute to any overlap term with , since in a small expansion it does not have any terms with non-negative powers of both and , so we can just subtract it out. Taking this into account, we finally obtain

(25)

The primes on the correlators indicate that we are to subtract out their singularities.

The function contributes to the conformal block at times a function of , and consequently it is part of the “semi-classical” piece. The semi-classical part is defined as the piece of that is formally of in the limit where is large and are held fixed. However, the function contributes only to at in this limit and therefore goes beyond the semi-classical part of the block.

We were able to evaluate both the semi-classical and quantum corrections, which are written closed form in equation 4. In what follows we will examine some interesting limits of the general result.

2.3 Small limit

The main reason that the integrals in (25) are difficult is that written as a function of contains non-integer powers of arising from . In the limit that is small, we can expand the correlator around , and these become integer powers and logarithms. At , one has

(26)

The resulting integrals in (25) can be easily evaluated. The result is

(27)

Since , there is no difference between using vs in the expression at leading order in above. We have checked these expressions against a direct small expansion up to using the methods of [31].

2.4 Large limit

It is more interesting to consider limits that allow to be imaginary, since that is the regime where the heavy state develops a horizon in AdS and a temperature. The limit that is most likely to be generic is that where is taken to . In particular, as mentioned in the introduction, in this limit one can rescale distance as to obtain the infinite radius limit of the circle. While the two-point function on the circle at finite radius and finite temperature is equivalent to a two-point function on the torus and is thus not a universal quantity, the two-point function on the plane at finite temperature <