Local martingales associated with Schramm-Loewner evolutions with internal symmetry

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 $\mathfrak{sl}_{2}$.