Loewner Dynamics for Real Rational Functions and the Multiple SLE(0) Process

Abstract

Recently Peltola and Wang (see Peltola talk of last week) 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($\kappa$) - 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 (with commutation holding in a very strong sense), and use this to directly show that the curves satisfy a geodesic multichord property. We also show that our SLE(0, $\rho$) processes lead to relatively simple solutions for the degenerate versions of the BPZ equations in terms of the poles and critical points of the rational function, and that the dynamics of these poles and critical points come from the Calogero-Moser integrable system. Although our results are purely deterministic they are again motivated by taking limits of probabilistic constructions in conformal field theory.