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

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We develop the thermal shadow formalism to study the conformal blocks decomposition in D-dimensional conformal field theory on Sβ1×SD−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.