One-point thermal conformal blocks from four-point conformal integrals

TL;DR

This paper introduces a formalism to analyze conformal blocks in thermal conformal field theories, revealing that thermal and zero-temperature blocks are connected through a common integral and expressed via special functions.

Contribution

The authors develop the thermal shadow formalism and demonstrate the equivalence of thermal and zero-temperature conformal blocks through a unified integral representation.

Findings

Thermal 1-point conformal blocks are expressed by the fourth Appell function.

Both thermal and zero-temperature conformal blocks are derived from the same 4-point conformal integral.

The formalism provides a new perspective on the structure of conformal blocks at finite temperature.

Abstract

We develop the thermal shadow formalism to study the conformal blocks decomposition in $D$-dimensional conformal field theory on $\mathbb{S}_{\beta}^{1} \times \mathbb{S}^{D-1}$, where the temperature is $T = \beta^{-1}$. It is demonstrated that both the 1-point thermal ($T\neq 0$) conformal blocks and the 4-point plane ($T=0$) conformal blocks are defined by the same 4-point conformal integral. It is shown that up to power prefactors the 1-point thermal conformal block is given by the fourth Appell function.