Integrability in random conformal geometry

Abstract

Liouville quantum gravity (LQG) is a random surface arising as the scaling limit of random planar maps. Schramm-Loewner evolution (SLE) is a random planar curve describing the scaling limits of interfaces in many statistical physics models. Liouville conformal field theory (LCFT) is the quantum field theory underlying LQG. Each of these satisfies conformal invariance or covariance. This thesis proves exact formulas in random conformal geometry; we highlight a few here.

The Brownian annulus describes the scaling limit of uniform random planar maps with the annulus topology, and is the canonical annular 𝛾-LQG surface with 𝛾 = √︀8/3. We obtain the law of its modulus, which is as predicted from the ghost partition function in bosonic string theory.

The conformal loop ensemble (CLE) is a random collection of loops in the plane which locally look like SLE, corresponding to the scaling limit of all interfaces in several important statistical mechanics models. We derive the three-point nesting statistic of simple CLE on the sphere. It agrees with the imaginary DOZZ formula of Zamolodchikov (2005) and Kostov-Petkova (2007), which is the three-point structure constant of the generalized minimal model conformal field theories.

We compute the one-point bulk structure constant for LCFT on the disk, thereby proving the formula proposed by Fateev, Zamolodchikov and Zamolodchikov (2000). This is a disk analog of the DOZZ constant for the sphere. Our result represents the first step towards solving LCFT on surfaces with boundary via the conformal bootstrap.

Our arguments depend on the interplay between LQG, SLE and LCFT. Firstly, LQG behaves well under conformal welding with SLE curves as the interfaces. Secondly, LCFT and LQG give complementary descriptions of the same geometry.