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

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.