Witten diagrams revisited: the AdS geometry of conformal blocks

Abstract

We develop a new method for decomposing Witten diagrams into conformal blocks. The steps involved are elementary, requiring no explicit integration, and operate directly in position space. Central to this construction is an appealingly simple answer to the question: what object in AdS computes a conformal block? The answer is a "geodesic Witten diagram", which is essentially an ordinary exchange Witten diagram, except that the cubic vertices are not integrated over all of AdS, but only over bulk geodesics connecting the boundary operators. In particular, we consider the case of four-point functions of scalar operators, and show how to easily reproduce existing results for the relevant conformal blocks in arbitrary dimension.

Conformal blocks beyond the semi-classical limit

“…The monodromies we find in the 1/c 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 1/c corrections are likely to disappear in the full correlators.…”

On the Kerr-AdS/CFT correspondence

“…The relation between scattering amplitudes of black holes in any dimension and twodimensional conformal blocks has been alluded to by many authors and used to obtain bulk information from the boundary dynamics, see [15][16][17][18][19]. The purpose of the present work is to review the relation of semi-classical (global) conformal blocks in four dimensions to propagating (scalar) fields in five-dimensional anti-de Sitter (AdS), a fact outlined by [20] and [18,21,22] revisiting work by Ferrara and collaborators in the 70's [23][24][25].…”

“…The purpose of the present work is to review the relation of semi-classical (global) conformal blocks in four dimensions to propagating (scalar) fields in five-dimensional anti-de Sitter (AdS), a fact outlined by [20] and [18,21,22] revisiting work by Ferrara and collaborators in the 70's [23][24][25]. The construction does not rely on supersymmetry.…”

“…The picture above is well suited provided we use the reflection property (3.25) to assign the singularity to an operator -possibly heavy -with α ≈ Q. For instance, the insertion at z = ∞ can be seen to correspond to a "heavy operator" [18,19] 26) with the first solution being proper for ∆ < 2 and the second for ∆ > 2, see [41] for a detailed discussion. On the other hand, the singularities at z = 0, 1, corresponding to the inner and outer horizon have the property that 27) where δS ± is the variation of entropy of the black hole horizon at r = r ± , respectively, due to the admittance of energy ω and angular momenta m 1 and m 2 .…”

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 .…”