The Conformal Bootstrap at Finite Temperature

TL;DR

This paper develops a conformal bootstrap approach for finite-temperature CFTs, deriving a thermal inversion formula to constrain thermal data and demonstrating its effectiveness in known models.

Contribution

It introduces a thermal inversion formula and a systematic perturbation theory for thermal data, extending bootstrap methods to finite-temperature conformal field theories.

Findings

Recovered spectrum and thermal one-point functions in mean field theory

Computed thermal one-point functions for higher-spin currents in the $O(N)$ model

Developed a perturbation theory for large spin, low-twist thermal data

Abstract

We initiate an approach to constraining conformal field theory (CFT) data at finite temperature using methods inspired by the conformal bootstrap for vacuum correlation functions. We focus on thermal one- and two-point functions of local operators on the plane. The KMS condition for thermal two-point functions is cast as a crossing equation. By studying the analyticity properties of thermal two-point functions, we derive a "thermal inversion formula" whose output is the set of thermal one-point functions for all operators appearing in a given OPE. This involves identifying a kinematic regime which is the analog of the Regge regime for four-point functions. We demonstrate the effectiveness of the inversion formula by recovering the spectrum and thermal one-point functions in mean field theory, and computing thermal one-point functions for all higher-spin currents in the critical $O(N)$ model at leading order in $1/N$. Furthermore, we develop a systematic perturbation theory for thermal data in the large spin, low-twist spectrum of any CFT. We explain how the inversion formula and KMS condition may be combined to algorithmically constrain CFTs at finite temperature. Throughout, we draw analogies to the bootstrap for vacuum four-point functions. Finally, we discuss future directions for the thermal conformal bootstrap program, emphasizing applications to various types of CFTs, including those with holographic duals.