AdS Field Theory from Conformal Field Theory

TL;DR

This paper establishes criteria for when a Conformal Field Theory can be described by a perturbative Effective Field Theory in Anti-de Sitter space, linking Mellin amplitudes, unitarity, and higher spin considerations.

Contribution

It introduces a new third condition involving Mellin amplitudes for CFTs to have an AdS EFT description and explores the implications for higher spin fields and scattering analogies.

Findings

Mellin amplitudes must be polynomial-bounded for EFT description

Connection between conformal blocks and AdS diagrams established

Constraints on higher spin interactions in AdS/CFT

Abstract

We provide necessary and sufficient conditions for a Conformal Field Theory to have a description in terms of a perturbative Effective Field Theory in AdS. The first two conditions are well-known: the existence of a perturbative `1/N' expansion and an approximate Fock space of states generated by a finite number of low-dimension operators. We add a third condition, that the Mellin amplitudes of the CFT correlators must be well-approximated by functions that are bounded by a polynomial at infinity in Mellin space, or in other words, that the Mellin amplitudes have an effective theory-type expansion. We explain the relationship between our conditions and unitarity, and provide an analogy with scattering amplitudes that becomes exact in the flat space limit of AdS. The analysis also yields a simple connection between conformal blocks and AdS diagrams, providing a new calculational tool very much in the spirit of the S-Matrix program. We also begin to explore the potential pathologies associated with higher spin fields in AdS by generalizing Weinberg's soft theorems to AdS/CFT. The AdS analog of Weinberg's argument constrains the interactions of conserved currents in CFTs, but there are potential loopholes that are unavailable to theories of massless higher spin particles in flat spacetime.