Probabilistic conformal blocks for Liouville CFT on the torus

Abstract

Virasoro conformal blocks are a family of important functions defined as power series via the Virasoro algebra. They are a fundamental input to the conformal bootstrap program for 2-dimensional (2D) conformal field theory (CFT) and are closely related to 4-dimensional supersymmetric gauge theory through the Alday–Gaiotto–Tachikawa correspondence. The present work provides a probabilistic construction of the 1-point toric Virasoro conformal block for central change greater than 25. More precisely, we construct an analytic function using a probabilistic tool called Gaussian multiplicative chaos (GMC) and prove that its power series expansion coincides with the 1-point toric Virasoro conformal block. The range $(25,\mathrm{\infty })$ of central charges corresponds to Liouville CFT, an important CFT originating from 2D quantum gravity and bosonic string theory. Our work reveals a new integrable structure underlying GMC and opens the door to the study of non-perturbative properties of Virasoro conformal blocks such as their analytic continuation and modular symmetry. Our proof combines an analysis of GMC with tools from CFT such as Belavin–Polyakov–Zamolodchikov differential equations, operator product expansions, and Dotsenko–Fateev type integrals.