Conformal welding of quantum disks and multiple SLE: the non-simple case

TL;DR

This paper extends the conformal welding of quantum disks to the non-simple case and constructs multiple SLEs for , linking Liouville quantum gravity with SLE and LCFT, and proving existence of multiple SLE partition functions.

Contribution

It generalizes conformal welding to non-simple quantum disks and constructs multiple SLEs for , establishing connections with LCFT and proving partition function existence.

Findings

Extended conformal welding to non-simple quantum disks.

Constructed multiple SLEs for  with link patterns.

Proved existence and smoothness of multiple SLE partition functions.

Abstract

Two-pointed quantum disks with a weight parameter $W>0$ is a canonical family of finite-volume random surfaces in Liouville quantum gravity. We extend the conformal welding of quantum disks in [AHS23] to the non-simple regime, and give a construction of the multiple SLE associated with any given link pattern for $\kappa\in(4,8)$. Our proof is based on connections between SLE and Liouville conformal field theory (LCFT), where we show that in the conformal welding of multiple forested quantum disks, the surface after welding can be described in terms of LCFT, and the random conformal moduli contains the SLE partition function for the interfaces as a multiplicative factor. As a corollary, for $\kappa\in(4,8)$, we prove the existence of the multiple SLE partition functions, which are smooth functions satisfying a system of PDEs and conformal covariance.