Winter School

in Mathematical Physics - 2020


Antti Kupiainen, "An Introduction to Probabilistic Liouville Theory"

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 I will discuss a proof in a joint work with Rhodes and Vargas of an integrability conjecture on LCFT, the celebrated DOZZ formula.

Lecture notes
Exercises: sheet 1, solutions

Nicolai Reshetikhin, "Integrable Systems on moduli spaces of flat connections"

Moduli spaces of flat connections over surfaces as symplectic manifolds were introduced by Atyiah and Bott in the context of geometry of gauge theories. Integrable systems on a closely related spaces, moduli space of Higgs bundles, were introduced by Hitchin. The goal of these lectures is to review a natural construction of superintegrable systems on moduli spaces of flat connections where Hamiltonians are given by invariant function on monodromies along simple curves on a surface. It turns out that such systems and their deformations include various versions of spin Calogero-Moser systems and, in particular, an integrable system that can be naturally regarded as a classical counterpart of the Harish-Chandra theory. The lectures are based on joint work with S. Artamonov and J. Stokman.

Lecture notes: lecture 1, lecture 2, lecture 3, lecture 4
Exercises: sheet 1

Jörg Teschner, "The free field approach to Liouville CFT"

The goal of my lectures will be to describe an older approach to the solution to Liouville theory which can supplement the powerful probabilistic approach of Kupiainen, Rhodes and Vargas by providing detailed information on the chiral factorisation and on the spectrum of Liouville conformal field theory. The main idea of this approach is to quantise the well-known representation of the solution to the Liouville equation of motion in terms of a free bosonic field. We will present the crucial ingredient in the verification that the quantised Liouville field is local, the braid relations of the chiral vertex operators representing the main building blocks of the construction. If time permits we will explain how the calculation of braid relations gives hints on connections to quantum Teichmueller theory and the harmonic analysis of Diff(S^1).

Exercises: sheet 1