SLE partition functions via conformal welding of random surfaces

TL;DR

This paper explores how conformal welding of Liouville quantum gravity surfaces can be used to derive SLE partition functions across various settings, providing an alternative to traditional stochastic calculus methods.

Contribution

It demonstrates a novel approach to obtaining SLE partition functions through conformal welding of LQG surfaces, linking LCFT and topological configurations.

Findings

SLE partition functions arise from conformal welding of LQG surfaces.

The approach applies to multiple SLE, imaginary geometry flow lines, and boundary Green functions.

Provides an alternative to stochastic calculus for studying SLE partition functions.

Abstract

SLE curves describe the scaling limit of interfaces from many 2D lattice models. Heuristically speaking, the SLE partition function is the continuum counterpart of the partition function of the corresponding discrete model. It is well known that conformally welding of Liouville quantum gravity (LQG) surfaces gives SLE curves as the interfaces. In this paper, we demonstrate in several settings how the SLE partition function arises from conformal welding of LQG surfaces. The common theme is that we conformally weld a collection of canonical LQG surfaces which produces a topological configuration with more than one conformal structure. Conditioning on the conformal moduli, the surface after welding is described by Liouville conformal field theory (LCFT), and the density of the random moduli contains the SLE partition function for the interfaces as a multiplicative factor. The settings we treat includes the multiple SLE for $\kappa\in (0,4)$, the flow lines of imaginary geometry on the disk with boundary marked points, and the boundary Green function. These results demonstrate an alternative approach to construct and study the SLE partition function, which complements the traditional method based on stochastic calculus and differential equation.