How to Succeed at Witten Diagram Recursions without Really Trying

TL;DR

This paper introduces simple methods to derive recursion relations for exchange Witten diagrams in AdS space, revealing new dimensional reduction formulas and extending to boundary CFTs, thus simplifying their evaluation.

Contribution

The paper presents novel recursion relations for exchange Witten diagrams derived from conformal block recursions, including dimensional reduction formulas and boundary CFT extensions.

Findings

Discovered five-term recursion relation relating diagrams in d and d-2 dimensions.

Derived dimensional reduction formulas for exchange Witten diagrams.

Extended recursion techniques to boundary conformal field theories.

Abstract

Witten diagrams are basic objects for studying dynamics in AdS space, and also play key roles in the analytic functional bootstrap. However, these diagrams are notoriously hard to evaluate, making it extremely difficult to search for recursion relations among them. In this note, we present simple methods to obtain recursion relations for exchange Witten diagrams from conformal block recursion relations. We discover a variety of new relations, including the dimensional reduction formulae for exchange Witten diagrams. In particular, we find a five-term recursion relation relating exchange Witten diagrams in $d$ and $d-2$ dimensions. This gives the holographic analogue of a similar formula for conformal blocks due to Parisi-Sourlas supersymmetry. We also extend the analysis to two-point functions in CFTs with conformal boundaries, and obtain similar results.