Conformal bootstrap in Liouville Theory

Colin Guillarmou, Antti Kupiainen, Rémi Rhodes, Vincent Vargas

Published 2020-05-23Version 1

Liouville conformal field theory (denoted LCFT) is a $2$-dimensional conformal field theory depending on a parameter $\gamma\in\mathbb{R}$ and studied since the eighties in theoretical physics. In the case of the theory on the $2$-sphere, physicists proposed closed formulae for the $n$-point correlation functions using symmetries and representation theory, called the DOZZ formula (for $n=3$) and the conformal bootstrap (for $n>3$). In a recent work, the three last authors introduced with F. David a probabilistic construction of LCFT for $\gamma \in (0,2]$ and proved the DOZZ formula for this construction. In this sequel work, we give the first mathematical proof that the probabilistic construction of LCFT on the $2$-sphere is equivalent to the conformal bootstrap for $\gamma\in (0,\sqrt{2})$. Our proof combines the analysis of a natural semi-group, tools from scattering theory and the use of the Virasoro algebra in the context of the probabilistic approach (the so-called conformal Ward identities).