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

Abstract

We develop the thermal shadow formalism to study the conformal blocks decomposition in $D$-dimensional conformal field theory on $\mathbb{S}_β^{1} \times \mathbb{S}^{D-1}$, where the temperature is $T = β^{-1}$. It is demonstrated that both the 1-point thermal ($T\ne 0$) 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.

18 pp, v2: misprints corrected, extended discussion in Section 2, more clarifying comments, references added; v3: extended discussion of the thermal conformal block in the high-temperature regime as well as at self-dual conformal dimensions; more detail on non-parametric conformal and thermal integrals and their relation to the Bloch-Wigner function in a new appendix A; refs added, journal version