Pure partition functions of multiple SLEs

TL;DR

This paper introduces a quantum group method to construct a basis of solutions for multiple SLE partition functions, potentially representing extremal probability measures for these conformally invariant random curves.

Contribution

It provides a novel quantum group approach to explicitly construct pure partition functions of multiple SLEs, advancing understanding of their probabilistic structure.

Findings

Constructed a distinguished basis of solutions for multiple SLE partition functions

Conjectured correspondence between solutions and extremal probability measures

Enhanced mathematical tools for analyzing conformally invariant random processes

Abstract

Multiple Schramm-Loewner Evolutions (SLE) are conformally invariant random processes of several curves, whose construction by growth processes relies on partition functions: M\"obius covariant solutions to a system of second order partial differential equations. In this article, we use a quantum group technique to construct a distinguished basis of solutions, which conjecturally correspond to the extremal points of the convex set of probability measures of multiple SLEs.