Modular bootstrap, Segal's axioms and resolution of Liouville conformal field theory
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, which can be thought of as a measure (or expectation value) over the space of configurations of the system. 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 (averages of relevant observables). In this talk, we will explain some joint works with G. Baverez, C. Guillarmou and A. Kupiainen where we show that the correlation functions of Liouville conformal field theory on Riemann surfaces can be expressed purely 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. The proof is strongly based on the decomposition of the path integral (written as an average of some random variables) into path integrals over surfaces with boundaries (building blocks/pairs of pants), which is a verification of Segal's axioms in our setting, and on the spectral analysis of a certain self-adjoint Hamiltonian.