Liouville conformal field theory was introduced by Polyakov in 1981 as an essential ingredient in his path integral construction of string theory. Since then Liouville theory has appeared in a wide variety of contexts ranging from random conformal geometry to 4d Yang-Mills theory with supersymmetry.
Recently, a probabilistic (or constructive) construction of Liouville theory was provided using the 2d Gaussian Free Field. This construction can be seen as a rigrous construction of the 2d path integral introduced in Polyakov's 1981 work. In contrast to this construction, modern conformal field theory is based on representation theory and the so-called bootstrap procedure (based on recursive techniques) introduced in 1984 by Belavin-Polyakov-Zamolodchikov. In particular, a bootstrap contruction for Liouville theory has been proposed in the mid 90's by Dorn-Otto-Zamolodchikov-Zamolodchikov (DOZZ). The aim of this talk is to review ongoing work which aims at showing the equivalence between the probabilistic (or path integral) construction and the bootstrap construction of Liouville theory.
Based on numerous joint works with F. David, C. Guillarmou, A. Kupiainen, R. Rhodes.