Note on Schramm-Loewner evolution for superconformal algebras

TL;DR

This paper extends Schramm-Loewner evolution (SLE) to superconformal algebras by formulating stochastic processes on superconformal transformation groups, enabling new insights into their algebraic and probabilistic structures.

Contribution

It introduces variants of SLE related to superconformal algebras using a group-theoretical approach and develops methods to derive local martingales from superconformal representations.

Findings

Formulation of superconformal SLE variants.

Derivation of stochastic differential equations from superconformal groups.

Method to obtain local martingales via Grassmann integration.

Abstract

We propose variants of Schramm-Loewner evolution (SLE) that are related to superconformal algebras following the group theoretical formulation of SLE, in which the relevant stochastic differential equation is derived from a random process on an infinite dimensional Lie group. In this paper, we consider random processes on certain kind of groups of superconformal transformations generated by exponentiated elements of the Grassmann envelop of superconformal algebras. We also provide a prescription of obtaining local martingales from a representation of the superconformal algebra after integration by Grassmann variables.