Multiple radial SLE(0) and classical Calogero-Sutherland System

Abstract

We develop a theory of multiple radial SLE(0) -- a smooth system of curves in a simply connected domain $\Omega$ with marked boundary points $z_1, \ldots, z_n \in \partial \Omega$ and a marked interior point $q$ -- arising as the deterministic limit of random multiple radial SLE($\kappa$) systems. We construct multiple radial SLE(0) systems by starting from the stationary relations, which arise heuristically as the $\kappa \to 0$ limit of partition functions. By constructing the field integrals of motion for the Loewner dynamics, we show that the traces of multiple radial SLE(0) systems are the horizontal trajectories of an equivalence class of quadratic differentials. These trajectories have limiting ends at the boundary points $\{z_1, z_2, \ldots, z_n\}$. The stationary relations connect the classification of multiple radial SLE(0) systems to the enumeration of critical points of the master function of trigonometric Knizhnik--Zamolodchikov (KZ) equations. In the deterministic case $\kappa = 0$, we show that the Loewner dynamics with a common parametrization of capacity form a special class of classical Calogero--Sutherland systems, restricted to a submanifold of phase space defined by the Lax matrix.