Recursion Relations for AdS/CFT Correlators

TL;DR

This paper develops recursion relations for AdS/CFT correlators, extending BCFW techniques to anti-de Sitter space, enabling more efficient computation of boundary correlation functions in holographic theories.

Contribution

It introduces new recursion relations for AdS/CFT correlators, generalizes BCFW to AdS, and extends these methods to supersymmetric theories, improving computational approaches.

Findings

Recursion relations relate vacuum correlators to lower-point transition amplitudes.

The set of polarization vectors compatible with BCFW in AdS is limited compared to flat space.

Constructing general polarization amplitudes requires combining multiple BCFW shifts.

Abstract

We expand on the results of arXiv:1011.0780 where we presented new recursion relations for correlation functions of the stress tensor and conserved currents in conformal field theories with an AdS_p dual for p > 4. These recursion relations are derived by generalizing the Britto-Cachazo-Feng-Witten (BCFW) relations to amplitudes in anti-de Sitter space (AdS) that are dual to boundary correlators, and are usually computed perturbatively by Witten diagrams. Our results relate vacuum-correlation functions to integrated products of lower-point transition amplitudes, which correspond to correlators calculated between states dual to certain normalizable modes. We show that the set of polarization vectors for which amplitudes behave well under the BCFW extension is smaller than in flat-space. We describe how transition amplitudes for more general external polarizations can be constructed by combining answers obtained by different pairs of BCFW shifts. We then generalize these recursion relations to supersymmetric theories. In AdS, unlike flat-space, even maximal supersymmetry is insufficient to permit the computation of all correlators of operators in the same multiplet as a stress-tensor or conserved current. Finally, we work out some simple examples to verify our results.