Conformal welding of random surfaces from Liouville theory

Abstract

Liouville quantum gravity (LQG) is a natural model describing random surfaces, which arises as the scaling limit for random planar maps. Liouville conformal field theory (LCFT) is the underlying 2D CFT that governs LQG. Schramm-Loewner evolution (SLE) is a random planar curve, which describes the scaling limits of interfaces in many statistical physics models. As discovered by Sheffield (2010), one of the deepest results in random geometry is that SLE curves arises as the interfaces under conformal welding of LQG surfaces. In this thesis, we present some new results on conformal welding of LQG surfaces as well as their applications towards the theory of SLE. We first define a three-parameter family of random surfaces in LQG which can be viewed as the quantum version of triangles. Then we prove the conformal welding result of a quantum triangle and a two-pointed quantum disk, and deduce integrability results for chordal SLE with three force points. The second main result is regarding the conformal welding of a multiple number of LQG surfaces, where under several scenarios, we prove that the output surfaces can be described in terms of LCFT, and the random moduli of the surface is encoded in terms of the partition functions for the SLE curves. The third part is about the conformal welding of the quantum disks with forested boundary, where we prove that this conformal welding gives a two-pointed quantum disk with an independent SLEκ for κ ∈ (4,8). We further extend to the conformal welding of a multiple number of forested quantum disks, where as an application, for κ ∈ (4,8), we prove the existence of the multiple SLE partition functions, which are smooth functions satisfying a system of PDEs and conformal covariance. This was open for κ ∈ (6,8) and N ≥ 3 prior to our work. The conformal loop ensemble (CLE) is a random collection of planar loops which locally look like SLE. For κ ∈ (4,8), the loops are non-simple and may touch each other and the boundary. As a second application, we derive the probability that the loop surrounding a given point in the non-simple conformal loop ensemble touches the domain boundary.