  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 2D conformal field theory (CFT) and are closely
related to four 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 $(25,\infty)$ 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.
