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 ${\mathbb{S}}_{\beta }^1\times {\mathbb{S}}^{D-1}$, where the temperature is $T={\beta }^{-1}$. It is demonstrated that both the 1-point thermal ($T\mathrm{\ne }0$) 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.