Schramm-Loewner evolution with Lie superalgebra symmetry

TL;DR

This paper extends Schramm-Loewner evolution (SLE) to include internal degrees of freedom modeled by affine Lie superalgebras, providing a new framework for analyzing conformal invariance with supersymmetry.

Contribution

It introduces a generalized SLE framework incorporating affine Lie superalgebra symmetries, including stochastic equations for internal degrees of freedom and martingale computations.

Findings

Formulation of SLE with affine Lie superalgebra symmetry

Derivation of stochastic differential equations for internal degrees

Calculation of local martingales from superalgebra representations

Abstract

We propose a generalization of Schramm-Loewner evolution (SLE) that has internal degrees of freedom described by an affine Lie superalgebra. We give a general formulation of SLE corresponding to representation theory of an affine Lie superalgebra whose underlying finite dimensional Lie superalgebra is basic classical type, and write down stochastic differential equations on internal degrees of freedom in case that the corresponding affine Lie superalgebra is $\widehat{\mathfrak{osp}(1|2)}$. We also demonstrate computation of local martingales associated with the solution from a representation of $\widehat{\mathfrak{osp}(1|2)}$.