QLunch: The Mysteries of Liouville Theory
Title: The Mysteries of Liouville Theory
Abstract: A. Polyakov introduced Liouville Conformal Field theory (LCFT) in 1981 as a way to
put a natural measure on the set of Riemannian metrics over a fixed two dimensional
manifold. Ever since, the work of Polyakov has echoed in various branches of physics
and mathematics, ranging from string theory to probability theory through geometry.
In the context of 2D quantum gravity models, Polyakov’s approach is conjecturally
equivalent to the scaling limit of Random Planar Maps and through the Alday-Gaiotto-
Tachikava correspondence LCFT is conjecturally related to certain 4D Yang-Mills theories.
Through the work of Dorn,Otto, Zamolodchikov and Zamolodchikov and Teschner LCFT is
believed to be to a certain extent integrable. I will review a probabilistic approach to LCFT
based on Kahane's theory of Gaussian Multiplicative Chaos developed together with
David, Rhodes and Vargas. In particular this has recently led to proof of an integrability
conjecture on LCFT, the celebrated DOZZ formula, in a joint work with Rhodes and Vargas.