Large deviations of multichordal SLE0+, real rational functions, and zeta-regularized determinants of Laplacians
We introduce a Loewner potential, associated to a collection of n disjoint simple chords (multichords) joining 2n boundary points of a simply connected domain in the complex plane. It is first motivated by the large deviations of multiple SLE, a probabilistic model of interfaces in 2D statistical mechanics configurations: We prove a strong large deviation principle (LDP) for multiple chordal SLE0+ curves with respect to the Hausdorff metric. The rate function differs from the potential by an additive constant depending only on the boundary data, that satisfies PDEs arising as a semiclassical limit of the Belavin-Polyakov-Zamolodchikov equations in conformal field theory. Moreover, we show that the potential can be expressed intrinsically in terms of determinants of Laplacians.
Furthermore, we prove that multiple SLE_0 can be defined as the unique multichord minimizing the potential in the upper half-plane for a given boundary data and show it to be the real locus of a rational function. As a by-product, we obtain an analytic proof of the Shapiro conjecture in real enumerative geometry, first proved by Eremenko and Gabrielov: if all critical points of a rational function are real, then the function is real up to post-composition by a Mobius map.