Virasoro conformal blocks in closed form

TL;DR

This work advances the analytic control of Virasoro conformal blocks by delivering three closed-form expansions valid for generic dimensions and central charge $c$. It develops a global block decomposition, an elliptic $q$-series representation, and a heavy-light semiclassical framework, each enabling a tractable $1/c$ expansion and practical calculations. The results illuminate connections to entanglement entropy, thermality, and three-dimensional gravity, and offer new avenues for exploring exponentiation, bulk scattering, and potential resummations. Overall, the paper provides versatile tools for probing the structure and applications of Virasoro blocks beyond standard recursion methods.

Abstract

Virasoro conformal blocks are fixed in principle by symmetry, but a closed-form expression is unknown in the general case. In this work, we provide three closed-form expansions for the four-point Virasoro blocks on the sphere, for arbitrary operator dimensions and central charge $c$. We do so by solving known recursion relations. One representation is a sum over hypergeometric global blocks, whose coefficients we provide at arbitrary level. Another is a sum over semiclassical Virasoro blocks obtained in the limit in which two external operator dimensions scale linearly with large $c$. In both cases, the $1/c$ expansion of the Virasoro blocks is easily extracted. We discuss applications of these expansions to entanglement and thermality in conformal field theories and particle scattering in three-dimensional quantum gravity.