Review on the probabilistic construction and Conformal bootstrap in Liouville Theory

TL;DR

This paper reviews recent probabilistic methods and the conformal bootstrap approach in Liouville CFT, highlighting their role in unifying various theoretical frameworks and advancing understanding of non-compact conformal field theories.

Contribution

It synthesizes recent developments in probabilistic construction and bootstrap formalization of Liouville theory, bridging geometry, probability, and representation theory.

Findings

Probabilistic construction of Liouville CFT clarified

Conformal bootstrap formalized for non-compact CFTs

Unified framework connecting geometry, probability, and algebra

Abstract

In the paper, we review the recent construction of the Liouville conformal field theory (CFT) from probabilistic methods, and the formalization of the conformal bootstrap. This model has offered a fruitful playground to unify the probabilistic construction of the path integral, the geometric axiomatics of CFT by Segal and the representation theoretical content of the conformal bootstrap. We explain and extract the main steps and ideas behind the construction and resolution of this non-compact CFT.