# Thermal CFTs in momentum space

## TL;DR

This paper explores the properties of thermal conformal field theories in momentum space, deriving formulas for thermal conformal blocks, analyzing their analytic structure, and establishing sum rules and spectral properties at finite temperature.

## Contribution

It introduces a Fourier transform formula for thermal conformal blocks, studies their analytic behavior, and connects the thermal block expansion with momentum space Green's functions.

## Key findings

- Fourier transform of thermal conformal blocks vanishes for double twist operators.
- Spectral density at high momenta supports the spectrum condition || > |k|.
- Explicit matching of thermal block expansion with Green's functions at finite temperature.

## Abstract

We study some aspects of conformal field theories at finite temperature in momentum space. We provide a formula for the Fourier transform of a thermal conformal block and study its analytic properties. In particular we show that the Fourier transform vanishes when the conformal dimension and spin are those of a "double twist" operator $\Delta = 2\Delta_\phi + \ell + 2n$. By analytically continuing to Lorentzian signature we show that the spectral density at high spatial momenta has support on the spectrum condition $|\omega| > |k|$. This leads to a series of sum rules. Finally, we explicitly match the thermal block expansion with the momentum space Green's function at finite temperature in several examples.