Conformal transformations and the SLE partition function martingale

TL;DR

This paper develops a conformal field theory framework for analyzing SLE martingales, explicitly constructing conformal transformations and partition functions, and connecting them to Virasoro algebra representations.

Contribution

It provides explicit constructions of conformal transformations fixing a point and links these to SLE martingales through Virasoro algebra representations.

Findings

Derived equations connecting SLE to Virasoro algebra

Proved all polynomial SLE martingales are enumerated by this construction

Identified the SLE partition function as a known local martingale

Abstract

We present an implementation in conformal field theory (CFT) of local finite conformal transformations fixing a point. We give explicit constructions when the fixed point is either the origin or the point at infinity. Both cases involve the exponentiation of a Borel subalgebra of the Virasoro algebra. We use this to build coherent state representations and to derive a close analog of Wick's theorem for the Virasoro algebra. This allows to compute the conformal partition function in non trivial geometries obtained by removal of hulls from the upper half plane. This is then applied to stochastic Loewner evolutions (SLE). We give a rigorous derivation of the equations, obtained previously by the authors, that connect the stochastic Loewner equation to the representation theory of the Virasoro algebra. We give a new proof that this construction enumerates all polynomial SLE martingales. When one of the hulls removed from the upper half plane is the SLE hull, we show that the partition function is a famous local martingale known to probabilists, thereby unravelling its CFT origin.