Remi Rhodes: Liouville conformal field theory and the DOZZ formula

Liouville conformal field theory (LCFT hereafter), introduced by Polyakov in his 1981 seminal work « Quantum geometry of bosonic strings », can be seen as a probabilistic version of Riemann surfaces. 

LCFT appears in Polyakov’s work as a 2D version of the Feynman path integral with an exponential interaction term. Since then, LCFT has emerged in a wide variety of contexts in the physics literature.

A major issue in theoretical physics was to solve the theory, namely compute the correlation functions. In this direction, an intriguing formula for the three point correlations of LCFT was proposed in the middle of the 90’s by Dorn-Otto and Zamolodchikov- Zamolodchikov, the celebrated DOZZ formula.

The purpose of this talk is twofold. First I will present a rigorous probabilistic construction of Polyakov’s path integral formulation of LCFT based on the Gaussian Free Field. Second, I will present the DOZZ formula and show that our probabilistic construction indeed satisfies the DOZZ formula. Time permitting, I will explain the main lines of our proof, which can be seen as a quantum analog of Poincaré’s proof of the uniformization theorem though the stress-tensor and the Fuchsian equation.