Conformal Bootstrap in Liouville theory

Abstract: Liouville conformal field theory (denoted LCFT) is a 2-dimensional conformal field theory depending on a parameter $\gamma\in\R$ and studied since the eighties in theoretical physics. In the case of the theory on the Riemann sphere, physicists proposed closed formulae for the n-point correlation functions using symmetries and representation theory, called the DOZZ formula (when n=3) and the conformal bootstrap (for n>3). A probabilistic construction of LCFT was recently proposed by David-Kupiainen-Rhodes-Vargas for $\gamma \in (0,2]$ and the last three authors later proved the DOZZ formula. In this talk I will present a proof of equivalence between the probabilistic and the bootstrap construction (proposed in physics) for the n point correlation functions with n greater or equal to 4, valid for $\gamma\in (0,1)$. Our proof combines the analysis of a natural semi-group, tools from scattering theory and the use of Virasoro algebra in the context of the probabilistic approach (the so-called conformal Ward identities).

Based on joint work with C. Guillarmou, A. Kupiainen and V. Vargas.

mathematical physicsprobability

Audience: researchers in the topic

Probability and the City Seminar

Series comments: The Probability and the City Seminar is organized jointly by the probability groups of Columbia University and New York University.

Video recordings of talks are posted online at www.youtube.com/channel/UC0CXjG-ZSIZHy0S40Px2FEQ .