A Natural Language for AdS/CFT Correlators

TL;DR

This paper demonstrates that Mellin space provides a natural framework for AdS/CFT correlators, revealing their pole structure, recursive construction rules, and connection to flat space scattering amplitudes, thus offering a new holographic perspective.

Contribution

It introduces Mellin space as the natural setting for AdS/CFT correlators, deriving explicit factorization formulas, recursive diagrammatic rules, and connecting these to flat space S-matrix properties.

Findings

Mellin amplitudes have poles corresponding to OPE decompositions.

Derived explicit residues for Mellin amplitudes at factorization poles.

Established a connection between Mellin amplitudes and flat space S-matrix.

Abstract

We provide dramatic evidence that `Mellin space' is the natural home for correlation functions in CFTs with weakly coupled bulk duals. In Mellin space, CFT correlators have poles corresponding to an OPE decomposition into `left' and `right' sub-correlators, in direct analogy with the factorization channels of scattering amplitudes. In the regime where these correlators can be computed by tree level Witten diagrams in AdS, we derive an explicit formula for the residues of Mellin amplitudes at the corresponding factorization poles, and we use the conformal Casimir to show that these amplitudes obey algebraic finite difference equations. By analyzing the recursive structure of our factorization formula we obtain simple diagrammatic rules for the construction of Mellin amplitudes corresponding to tree-level Witten diagrams in any bulk scalar theory. We prove the diagrammatic rules using our finite difference equations. Finally, we show that our factorization formula and our diagrammatic rules morph into the flat space S-Matrix of the bulk theory, reproducing the usual Feynman rules, when we take the flat space limit of AdS/CFT. Throughout we emphasize a deep analogy with the properties of flat space scattering amplitudes in momentum space, which suggests that the Mellin amplitude may provide a holographic definition of the flat space S-Matrix.