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

We develop the thermal shadow formalism to study the conformal blocks decomposition in D-dimensional conformal field theory on Sβ1×SD−1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathbbm{S}}_{\beta}^1\times {\mathbbm{S}}^{D-1} $$\end{document}, where the temperature is T = β−1. It is demonstrated that both the 1-point thermal (T ≠ 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.