Witten Diagrams Revisited: The AdS Geometry of Conformal Blocks

TL;DR

This paper introduces a new, elementary method for decomposing Witten diagrams into conformal blocks using geodesic Witten diagrams, simplifying calculations in AdS/CFT without explicit integrations.

Contribution

It presents a novel approach to compute conformal blocks via geodesic Witten diagrams, avoiding explicit integrations and working directly in position space.

Findings

Reproduces known conformal blocks in various dimensions

Simplifies the decomposition process of Witten diagrams

Provides a new geometric perspective on 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.