Liouville conformal field theory: equivalence between the path integral and the bootstrap construction

Liouville conformal field theory (LCFT) is a family of Conformal field theories which arise in a wide variety of contexts in the physics and the probabilistic literature: SUSY Yang-Mills, the scaling limit of large planar maps, etc...

There are two main and seemingly unrelated approaches to LCFT in the physics literature: one in the Feynman path integral formulation and one in the conformal bootstrap approach. Recently, we constructed rigorously LCFT in the Feynman path integral formulation via probability theory (and more specifically the Gaussian Free Field). In this talk, I will present recent work which shows that both approaches (probabilistic construction of the Feynman path integral and conformal bootstrap) are in fact identical. A key ingredient in our work is the analysis of an infinite dimensional semigroup, the so-called Liouville semigroup.

Based on a series of joint works with C. Guillarmou, F. David, A. Kupiainen and R. Rhodes.