QLunch: The Mysteries of Liouville Theory

Speaker: Antti Kupiainen, University of Helsinki

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.