Recursion Relations for AdS/CFT Correlators

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.