Virasoro conformal blocks in closed form

TL;DR

This paper derives three explicit closed-form expansions for four-point Virasoro conformal blocks on the sphere, enabling easier analysis of their properties and applications in conformal field theory and quantum gravity.

Contribution

It provides the first explicit closed-form expansions of Virasoro blocks for arbitrary operator dimensions and central charge, solving known recursion relations.

Findings

Three closed-form expansions derived for Virasoro blocks

Explicit coefficients for hypergeometric global blocks provided

Facilitates analysis of entanglement, thermality, and quantum gravity scattering

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.