Local martingales associated with Schramm-Loewner evolutions with internal symmetry

TL;DR

This paper develops a framework for constructing local martingales related to Schramm-Loewner evolutions with internal symmetries using affine Lie algebra representations, extending previous formulations and revealing new symmetries.

Contribution

It introduces explicit stochastic differential equations for internal degrees of freedom and constructs new local martingales associated with affine Lie algebra symmetries in SLEs.

Findings

Constructed local martingales for SLEs with internal degrees of freedom.

Formulated stochastic differential equations for Heisenberg and affine sl_2 algebras.

Discovered affine sl_2 symmetry in the space of SLE local martingales.

Abstract

We consider Schramm-Loewner evolutions (SLEs) with internal degrees of freedom that are associated with representations of affine Lie algebras, following group theoretical formulation of SLEs. We reconstruct the SLEs considered by Bettelheim {\it et al.} [Phys. Rev. Lett. {\bf 95}, 251601 (2005)] and Alekseev {\it et al.} [Lett. Math. Phys. {\bf 97}, 243-261 (2011)] in correlation function formulation. We also explicitly formulate stochastic differential equations on internal degrees of freedom for Heisenberg algebras and the affine $\mathfrak{sl}_{2}$. Our formulation enables us to find several local martingales associated with SLEs with internal degrees of freedom from computation on a representation of an affine Lie algebra. Indeed, we formulate local martingales associated with SLEs with internal degrees of freedom described by Heisenberg algebras and the affine $\mathfrak{sl}_{2}$. We also find an affine $\mathfrak{sl}_{2}$ symmetry of a space of SLE local martingales for the affine $\mathfrak{sl}_{2}$.