Slit holomorphic stochastic flows and Gaussian free field

TL;DR

This paper characterizes slit holomorphic stochastic flows that generate Gaussian free field level lines, linking SLE variants with conformal field theory modifications and providing martingale-observables.

Contribution

It identifies conditions for slit holomorphic stochastic flows to produce GFF level lines and develops a conformal field theory framework with new martingale-observables.

Findings

Characterization of slit holomorphic stochastic flows generating GFF level lines

Descriptions of GFF modifications corresponding to SLE with drifts

Development of martingale-observables for these SLE types

Abstract

It was realized recently that the chordal, radial and dipolar SLEs are special cases of a general slit holomorphic stochastic flow. We characterize those slit holomorphic stochastic flows which generate level lines of the Gaussian free field. In particular, we describe the modifications of the Gaussian free field (GFF) corresponding to the chordal and dipolar SLE with drifts. Finally, we develop a version of conformal field theory based on the background charge and Dirichlet boundary condition modifications of GFF and present martingale-observables for these types of SLEs.