A tauberian theorem for the conformal bootstrap

TL;DR

This paper establishes a rigorous mathematical connection between the asymptotic behavior of spectral densities and singularities in conformal bootstrap expansions, with applications to 1D CFTs, higher-dimensional theories, and large N gauge theories.

Contribution

It provides a new tauberian theorem linking spectral density asymptotics to crossing channel singularities in conformal bootstrap analysis.

Findings

Rigorous link between spectral density and singularities in 1D conformal blocks.

Application to SL(2,R)-invariant correlators and 1D CFTs.

Control of spectral density asymptotics in large N gauge theories.

Abstract

For expansions in one-dimensional conformal blocks, we provide a rigorous link between the asymptotics of the spectral density of exchanged primaries and the leading singularity in the crossed channel. Our result has a direct application to systems of SL(2,R)-invariant correlators (also known as 1d CFTs). It also puts on solid ground a part of the lightcone bootstrap analysis of the spectrum of operators of high spin and bounded twist in CFTs in d>2. In addition, a similar argument controls the spectral density asymptotics in large N gauge theories.