Virasoro Module Structure of Local Martingales of SLE Variants

TL;DR

This paper explores the Virasoro module structure of local martingales in SLE variants, revealing a rich algebraic framework and methods to generate multiple SLE geometries using Coulomb gas formalism.

Contribution

It introduces a natural submodule of local martingales for SLE variants and demonstrates how Coulomb gas integrals produce candidates for multiple SLE geometries.

Findings

The module of local martingales has a rich structure with a natural submodule.

Coulomb gas formalism can generate martingale functions and multiple SLE geometries.

Polynomial local martingales form a Virasoro module in chordal SLE.

Abstract

Martingales often play an important role in computations with Schramm-Loewner evolutions (SLEs). The purpose of this article is to provide a straightforward approach to the Virasoro module structure of the space of local martingales for variants of SLEs. In the case of ordinary chordal SLE, it has been shown in Bauer & Bernard: Phys.Lett.B 557 that polynomial local martingales form a Virasoro module. We will show for more general variants that the module of local martingales has a natural submodule M that has the same interpretation as the module of polynomial local martingales of chordal SLE, but it is in many cases easy to find more local martingales than that. We discuss the surprisingly rich structure of the Virasoro module M and construction of the ``SLE state'' or ``martingale generating function'' by Coulomb gas formalism. In addition, Coulomb gas or Feigin-Fuchs integrals will be shown to transparently produce candidates for multiple SLE pure geometries.