The parafermionic observable in SLE

TL;DR

This paper introduces a continuum version of the parafermionic observable for SLE, proves its existence, computes its value, and shows it becomes holomorphic under certain conditions, linking discrete models to conformal invariance.

Contribution

It defines a continuum parafermionic observable for SLE, proves its existence, computes its value, and demonstrates holomorphicity for specific parameters, advancing understanding of conformal invariance in models.

Findings

The continuum parafermionic observable exists and is well-defined.

The observable's value is computed up to a non-zero constant.

Holomorphicity is achieved for a specific parameter choice.

Abstract

The parafermionic observable has recently been used by number of authors to study discrete models, believed to be conformally invariant and to prove convergence results for these processes to SLE. We provide a definition for a one parameter family of continuum versions of the paraferminonic observable for SLE, which takes the form of a normalized limit of expressions identical to the discrete definition. We then show the limit defining the observable exists, compute the value of the observable up to a finite multiplicative constant, and prove this constant is non-zero for a wide range of kappa. Finally, we show our observable for SLE becomes a holomorphic function for a particular choice of the parameter, which helps illuminate a fundamental property of the discrete observable.