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New Recursion Relations and a Flat Space Limit for AdS/CFT Correlators

Suvrat Raju

TL;DR

The paper develops new recursion relations for AdS/CFT correlators in momentum space that are valid in all dimensions, including d=3, enabling efficient computation of stress-tensor and conserved-current correlators. It then introduces a novel flat-space limit in which the (d+1)-dimensional graviton or gluon amplitude emerges from singular structures in the d-dimensional boundary correlators, valid at tree and loop levels with dimensional regularization. The recursion relations are shown to be consistent with this flat-space limit, providing a powerful cross-check and a bridge between holographic boundary data and bulk scattering. The work lays groundwork for incorporating higher-derivative corrections and exploring extensions to higher-spin theories, while offering practical tools for explicit AdS/CFT calculations and for connecting holographic results to flat-space physics.

Abstract

We consider correlation functions of the stress-tensor or a conserved current in AdS_{d+1}/CFT_d computed using the Hilbert or the Yang-Mills action in the bulk. We introduce new recursion relations to compute these correlators at tree level. These relations have an advantage over the BCFW-like relations described in arXiv:1102.4724 and arXiv:1011.0780 because they can be used in all dimensions including d=3. We also introduce a new method of extracting flat-space S-matrix elements from AdS/CFT correlators in momentum space. We show that the (d+1)-dimensional flat-space amplitude of gravitons or gluons can be obtained as the coefficient of a particular singularity of the d-dimensional correlator of the stress-tensor or a conserved current; this technique is valid even at loop-level in the bulk. Finally, we show that our recursion relations automatically generate correlators that are consistent with this observation: they have the expected singularity and the flat-space gluon or graviton amplitude appears as its coefficient.

New Recursion Relations and a Flat Space Limit for AdS/CFT Correlators

TL;DR

The paper develops new recursion relations for AdS/CFT correlators in momentum space that are valid in all dimensions, including d=3, enabling efficient computation of stress-tensor and conserved-current correlators. It then introduces a novel flat-space limit in which the (d+1)-dimensional graviton or gluon amplitude emerges from singular structures in the d-dimensional boundary correlators, valid at tree and loop levels with dimensional regularization. The recursion relations are shown to be consistent with this flat-space limit, providing a powerful cross-check and a bridge between holographic boundary data and bulk scattering. The work lays groundwork for incorporating higher-derivative corrections and exploring extensions to higher-spin theories, while offering practical tools for explicit AdS/CFT calculations and for connecting holographic results to flat-space physics.

Abstract

We consider correlation functions of the stress-tensor or a conserved current in AdS_{d+1}/CFT_d computed using the Hilbert or the Yang-Mills action in the bulk. We introduce new recursion relations to compute these correlators at tree level. These relations have an advantage over the BCFW-like relations described in arXiv:1102.4724 and arXiv:1011.0780 because they can be used in all dimensions including d=3. We also introduce a new method of extracting flat-space S-matrix elements from AdS/CFT correlators in momentum space. We show that the (d+1)-dimensional flat-space amplitude of gravitons or gluons can be obtained as the coefficient of a particular singularity of the d-dimensional correlator of the stress-tensor or a conserved current; this technique is valid even at loop-level in the bulk. Finally, we show that our recursion relations automatically generate correlators that are consistent with this observation: they have the expected singularity and the flat-space gluon or graviton amplitude appears as its coefficient.

Paper Structure

This paper contains 21 sections, 80 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Setting
  3. Notation:
  4. New Flat Space Recursion Relations
  5. Recursion in Yang-Mills
  6. Recursion in Gravity
  7. Recursion Relations in AdS
  8. Large $w$ Behaviour for Current Correlators
  9. Large $w$ Behaviour for Stress-Tensor Correlators
  10. Recursion Relations for Currents
  11. Recursion Relations for Stress Tensors
  12. Another form of the relations
  13. A New Flat Space Limit
  14. Flat Space Limit for Tree Amplitudes
  15. Contact Interactions
  16. ...and 6 more sections

Figures (2)

  • Figure 1: Gravity Feynman diagram, with 4 momenta of $O(w)$, that scales like $w^{n-2}$
  • Figure 2: One loop AdS diagram