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

Abstract: In this second of two articles, we study the multiple radial $\mathrm{SLE}(0)$ systems as the deterministic limits of the random multiple radial $\mathrm{SLE}(\kappa)$ systems. We construct the multiple radial SLE(0) systems from stationary relations by heuristically taking the limit of partition functions as $\kappa \rightarrow 0$. 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 of $\kappa=0$, 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.