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 $m\le \lfloor n/2\rfloor$, 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.

28 pages, 15 figures. arXiv admin note: substantial text overlap with arXiv:2410.21544