Recursion Relations for AdS/CFT Correlators
Suvrat Raju
TL;DR
The paper develops a framework to compute CFT correlators with AdS duals by extending BCFW recursion to AdS, linking higher‑point boundary correlators to integrals of lower‑point transition amplitudes. By deforming external momenta along a null vector and analyzing large‑$w$ behavior, the authors derive recursion relations for Yang–Mills and gravity in AdS, and extend them to supersymmetric theories; a key feature is that the recursion factorizes through normalizable bulk modes, with a boundary term vanishing under suitable polarizations. They verify the approach with explicit Ward identities and four‑point tests, and discuss polarization decompositions across dimensions $d=4,5,6$, highlighting polarization constraints and the need to combine different shifts to access all tensor structures. The results offer a practical, tree‑level reduction of AdS/Witten diagrams to lower‑point data, with potential to illuminate loops, higher‑derivative corrections, and the space of CFTs with gravity duals, while also suggesting connections to generating function formalisms and broader amplitude program insights.
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.