Conformal Field Theory for Multiple SLEs

Abstract

Over the last 10-15 years Kang and Makarov have developed a version of Conformal Field Theory that studies correlation functions of the Gaussian Free Field. Their papers first lay the groundwork for several independent pieces of the theory: the Ward equations (an integration-by-parts formula), the operator product expansion (for subtracting off diverging infinities), the Girsanov theory (for representing shifts of the field), and level two degeneracy (properties of vertex exponentials). Then, in the penultimate stroke, these pieces are combined together to show that a broad class of correlation functions derived from the GFF are martingale observables for an associated SLE type process. Kang and Makarov pay special attention to doing this in a way that is coordinate free, and ultimately their technique becomes an efficient way of carrying out Ito formula computations with minimal effort.

I will explain how the Kang-Makarov framework gives another viewpoint into the SLE-GFF connection, in a way that is very natural for researchers with a primary background in complex analysis and potential theory. My main goal is to give an idea of how it can be used to explain systems of multiple SLE curves, screening techniques, deterministic limits of SLEs, rational functions, and classical integrable systems.