Loewner Dynamics for the Multiple SLE(0) Process

Abstract

Recently Peltola and Wang introduced the multiple SLE(0) process as the deterministic limit of the random multiple SLE($\kappa$) curves as $\kappa$ goes to zero. They also showed that the limiting curves have important geometric characterizations that are independent of their relation to SLE - they are the real locus of real rational functions, and they can be generated by a deterministic Loewner evolution driven by multiple points. We prove that the Loewner evolution is a very special family of commuting SLE(0, $\rho$) processes. We also show that our SLE(0,$\rho$) processes lead to relatively simple solutions to a particular high-dimensional system of quadratic equations called the degenerate BPZ equations. In addition, the dynamics of these poles and critical points come from the Calogero-Moser integrable system. Although our results are purely deterministic they are motivated by taking limits of probabilistic constructions in conformal field theory.