SLE martingales and the Virasoro algebra

TL;DR

This paper establishes a direct link between Virasoro algebra representations and polynomial martingales in SLE, revealing the algebra as the spectrum generator for SLE martingales through a novel Borel-Weil inspired representation.

Contribution

It introduces a new Virasoro algebra representation acting on conformal map parameters, connecting algebraic structures with stochastic processes in SLE.

Findings

Virasoro algebra acts as spectrum generator for SLE martingales

New representation inspired by Borel-Weil construction

Explicit relation between algebra representations and polynomial martingales

Abstract

We present an explicit relation between representations of the Virasoro algebra and polynomial martingales in stochastic Loewner evolutions (SLE). We show that the Virasoro algebra is the spectrum generating algebra of SLE martingales. This is based on a new representation of the Virasoro algebra, inspired by the Borel-Weil construction, acting on functions depending on coordinates parametrizing conformal maps.