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 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 and 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 construction of LCFT developed together with David,
Rhodes and Vargas and recent proofs concerning the integrability of LCFT:
-The proof in a joint work with Rhodes and Vargas of the DOZZ formula
(Annals of Mathematics, 81-166,191 (2020)
-The proof in a joint work with Guillarmou, Rhodes and Vargas of the
bootstrap conjecture for LCFT (arXiv:2005.11530).