Speaker
Description
Liouville field theory was introduced by Polyakov in the eighties in the context of string theory. Liouville theory appeared there under the form of a 2D Feynman path integral, describing fluctuating metrics over Riemann surfaces. Since then, this theory has been extensively studied in physics and this interest has more recently spread to the probabilistic community where it appears as a natural model of random Riemann surfaces. Liouville theory is a conformal field theory and, as such, the quantities of interest are the correlation functions. In this talk, I will explain some joint works with C. Guillarmou, A. Kupiainen and V. Vargas where we prove the conformal bootstrap conjecture. Conformal bootstrap can be seen as the quantum analog of the pair of pants decomposition of Riemann surfaces. It states that the correlation functions of Liouville conformal field theory on Riemann surfaces can be expressed in terms of products of 3-point correlation functions on the sphere and the conformal blocks, which are holomorphic functions on the moduli space of punctured Riemann surfaces.
