Integrability of SLE via conformal welding of random surfaces

TL;DR

This paper establishes integrable results for SLE by connecting Liouville conformal field theory and mating-of-trees frameworks, using conformal welding of random surfaces to derive exact laws for SLE derivatives.

Contribution

It introduces a novel approach linking SLE, LCFT, and LQG through conformal welding and uniform embedding, extending previous equivalence results with fewer marked points.

Findings

Derived an exact law for a conformal derivative of SLE.

Extended equivalence results between LQG descriptions.

Connected SLE interfaces with random surface welding.

Abstract

We demonstrate how to obtain integrable results for the Schramm-Loewner evolution (SLE) from Liouville conformal field theory (LCFT) and the mating-of-trees framework for Liouville quantum gravity (LQG). In particular, we prove an exact formula for the law of a conformal derivative of a classical variant of SLE called $\mathrm{SLE}_\kappa(\rho_-;\rho_+)$. Our proof is built on two connections between SLE, LCFT, and mating-of-trees. Firstly, LCFT and mating-of-trees provide equivalent but complementary methods to describe natural random surfaces in LQG. Using a novel tool that we call the uniform embedding of an LQG surface, we extend earlier equivalence results by allowing fewer marked points and more generic singularities. Secondly, the conformal welding of these random surfaces produces SLE curves as their interfaces. In particular, we rely on the conformal welding results proved in our companion paper [AHS20]. Our paper is an essential part of a program proving integrability results for SLE, LCFT, and mating-of-trees based on these two connections.