Nonperturbative Mellin Amplitudes: Existence, Properties, Applications

TL;DR

This paper establishes that nonperturbative conformal field theory correlation functions can be represented by Mellin amplitudes, exploring their properties, applications, and implications for bootstrap bounds and holographic theories.

Contribution

It introduces a nonperturbative Mellin amplitude framework for CFTs, analyzing their properties and applying them to bootstrap bounds and holography.

Findings

Mellin amplitudes exist nonperturbatively for CFTs

Derived bootstrap bounds using dispersion relations and Polyakov conditions

Validated functionals with 3d Ising model data and holographic theories

Abstract

We argue that nonperturbative CFT correlation functions admit a Mellin amplitude representation. Perturbative Mellin representation readily follows. We discuss the main properties of nonperturbative CFT Mellin amplitudes: subtractions, analyticity, unitarity, Polyakov conditions and polynomial boundedness at infinity. Mellin amplitudes are particularly simple for large N CFTs and 2D rational CFTs. We discuss these examples to illustrate our general discussion. We consider subtracted dispersion relations for Mellin amplitudes and use them to derive bootstrap bounds on CFTs. We combine crossing, dispersion relations and Polyakov conditions to write down a set of extremal functionals that act on the OPE data. We check these functionals using the known 3d Ising model OPE data and other known bootstrap constraints. We then apply them to holographic theories.