Witten diagrams for torus conformal blocks

Abstract

We give a holographic description of global conformal blocks in two dimensional conformal field theory on the sphere and on the torus. We show that the conformal blocks for one-point functions on the torus can be written as Witten diagrams in thermal AdS. This is accomplished by deriving a general conformal Casimir equation for global conformal blocks, and showing that Witten diagrams obey the same equation. We study the semiclassical limit of n-point conformal blocks, and show that these equal the action of a network of bulk world-lines obeying appropriate geodesic equations. We give an alternate description in the Chern-Simons formulation of 3D gravity, where the conformal blocks are described by networks of Wilson lines, and argue that these formulations are equivalent.

Toward the pole

“…so the behaviour of singularities in the conformal block expansion (22) changes significantly for ν ∈ Z. For instance, when ν = 1 and δ(b) = δ 0 = const, one obtains:…”

Holographic dual of the five-point conformal block

“…Such a task was recently undertaken for four-point global conformal blocks in any dimension with external scalar operators [1] (with further generalizations in Refs. [2,3,4,5,6,7,8,9,10,11,12,13]) and for Virasoro blocks in AdS 3 /CFT 2 [14,15,16,17,18,19,20,21,22,23,24].…”

Structure constants from modularity in warped CFT

“…2 As was discussed in [35], to reproduce the contributions from all the descendent states of |χ , a more careful geodesic Witten diagram [36] should be considered. In that case, φχ will be allowed to propagate in the entire BTZ spacetime, and the interaction point will be determined by minimizing the total worldline action.…”

“…In this paper, we only considered the semiclassical region and the contribution from the primary operators. It is interesting to further perform a geodesic Witten type calculation in AdS/WCFT, parallel to the calculation of [35] in AdS 3 /CFT. Such a calculation will further check the consistency of CSS boundary conditions and the proposed AdS 3 /WCFT duality at the quantum level apart from [33,40].…”

“…Here we will only consider the most general theories where such coincidence doesn't show up. See[35] for the contributions from all the descendants.…”