Virasoro vacuum block at next-to-leading order in the heavy-light limit

Abstract

We consider the semiclassical limit of the vacuum Virasoro block describing the diagonal 4-point correlation functions on the sphere. At large central charge c, after exponentiation, it depends on two fixed ratios h H /c and h L /c, where h H,L are the conformal dimensions of the 4-point function operators. The semiclassical block may be expanded in powers of the light ratio h L /c and the leading non-trivial (linear) order is known in closed form as a function of h H /c. Recently, this contribution has been matched against AdS 3 gravity calculations where heavy operators build up a classical geometry corresponding to a BTZ black hole, while the light operators are described by a geodesic in this background. Here, we compute for the first time the next-to-leading quadratic correction O((h L /c) 2 ), again in closed form for generic heavy operator ratio h H /c. The result is a highly nontrivial extension of the leading order and may be relevant for further refined AdS 3 /CFT 2 tests. Applications to the two-interval Rényi entropy are also presented.

On information loss in AdS3/CFT2

“…Were this the case, our task would be hopeless, since we will not be able to compute the exact heavy-light correlator in any, let alone every, holographic CFT. Fortunately, in all CFT 2 there is a universal contribution to each heavy-light correlator, the Virasoro vacuum block [20][21][22][23][24][25][26][27][28], which manifests information loss in the large central charge or c → ∞ limit [29].…”

“…There is a universal contribution from the vacuum Virasoro block V 0 (1 − z) necessitated by the fact that both O L (z)O L (0) and O H (z)O H (0) contain the operator '1' in their OPE. In fact, the vacuum block can be computed directly using the Virasoro algebra at large c [20][21][22][23][24][25][26]29], and it corresponds precisely with the n = 0 term in the AdS image sum of equation (2.6). But before discussing this further, let us briefly review the physical content of the Virasoro blocks.…”

Conformal blocks beyond the semi-classical limit

“…After this work was substantially completed, the paper [30] appeared that uses a different method to compute an integral expression for the order h 2 L /c (semi-classical) result.…”

“…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 [8,28,30,[38][39][40][41][42] methods have been developed for computing the Virasoro conformal blocks in a "heavylight" 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.…”

Multi-centered AdS3 solutions from Virasoro conformal blocks

“…Recent investigations have focused on the important question of how local physics in the bulk is encoded in properties of the dual CFT. Natural localized objects to study holographically are line defects arising from adding point particles to the bulk, which have recently been shown to be intimately linked to conformal blocks in the dual CFT [2][3][4][5] (related work appears in [6][7][8][9][10][11][12][13][14][15][16][17]). In the case of AdS 3 , which we will focus on in this paper, it was shown that Virasoro conformal blocks at large central charge can be computed from the action of configurations of point particles in the bulk AdS 3 .…”