Multiple chordal SLE(0) and classical Calogero-Moser system

TL;DR

This paper develops a comprehensive theory of multiple chordal SLE(0) systems of type (n,m), linking their traces to rational functions with specific critical points and poles, and shows their evolution follows the Calogero-Moser Hamiltonian.

Contribution

It extends the construction of multiple chordal SLE(0) systems beyond previous cases and connects their dynamics to classical integrable Hamiltonian systems.

Findings

Traces correspond to real rational functions with specified critical points and poles.

Loewner dynamics evolve according to the Calogero-Moser Hamiltonian.

Generalization of multiple SLE(0) systems to broader parameter ranges.

Abstract

We develop a general theory of multiple chordal $\mathrm{SLE}(0)$ systems of type $(n, m)$ for positive integers $n$ and $m$ with $m \leq \lfloor n/2 \rfloor$, extending the construction of~\cite{ABKM20} beyond the previously studied case $n = 2m$. By applying integrals of motion associated with the Loewner evolution, we show that, in the $\mathbb{H}$-uniformization with the marked point $q = \infty$, the traces of type $(n, m)$ multiple chordal $\mathrm{SLE}(0)$ systems correspond to the real locus of real rational functions with $n$ real simple critical points, $m$ simple poles, and a pole of order $n - 2m + 1$ at infinity. Furthermore, we demonstrate that, under a common capacity parametrization, the Loewner dynamics evolve according to the classical Calogero-Moser Hamiltonian.