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 ${\mathbb{S}}_{\beta }^1\times {\mathbb{S}}^{D-1}$ \( {\mathbbm{S}}_{\beta}^1\times {\mathbbm{S}}^{D-1} \) , 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.