Loewner Dynamics of 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.