Multiple chordal SLE($\kappa$) and quantum Calogero-Moser system

TL;DR

This paper explores the connection between multiple chordal SLE$(ppa)$ processes in simply connected domains with boundary points and the quantum Calogero-Moser system, introducing new solutions and extending known correspondences.

Contribution

It constructs new solutions to SLE partition functions using Coulomb gas formalism and extends the correspondence to quantum Calogero-Moser eigenstates beyond standard cases.

Findings

Partition functions satisfy null vector and dilatation equations.

Two families of solutions indexed by link patterns are constructed.

Partition functions correspond to quantum Calogero-Moser eigenstates.

Abstract

We study multiple chordal SLE$(\kappa)$ systems in a simply connected domain $\Omega$, where $z_1, \ldots, z_n \in \partial \Omega$ are boundary starting points and $q \in \partial \Omega$ is an additional marked boundary point. As a consequence of the domain Markov property and conformal invariance, we show that the presence of the marked boundary point $q$ gives rise to a natural equivalence relation on partition functions. While these functions are not necessarily conformally covariant, each equivalence class contains a conformally covariant representative. Building on the framework introduced in \cite{Dub07}, we demonstrate that in the $\mathbb{H}$-uniformization with $q = \infty$, the partition functions satisfy both the null vector equations and a dilatation equation with scaling exponent $d$. Using techniques from the Coulomb gas formalism in conformal field theory, we construct two distinct families of solutions, each indexed by a topological link pattern of type $(n, m)$ with $2m \leq n$. In the special case $\Omega = \mathbb{H}$ and $q = \infty$, we further show that these partition functions correspond to eigenstates of the quantum Calogero-Moser system, thereby extending the known correspondence beyond the standard $(2n, n)$ setting.