On SLE Martingales in Boundary WZW Models

Abstract

Following Bettelheim et al. (Phys Rev Lett 95:251601, 2005), we consider the boundary WZW model on a half-plane with a cut growing according to the Schramm–Loewner stochastic evolution and the boundary fields inserted at the tip of the cut and at infinity. We study necessary and sufficient conditions for boundary correlation functions to be SLE martingales. Necessary conditions come from the requirement for the boundary field at the tip of the cut to have a depth two null vector. Sufficient conditions are established using Knizhnik–Zamolodchikov equations for boundary correlators. Combining these two approaches, we show that in the case of G = SU(2) the boundary correlator is an SLE martingale if and only if the boundary field carries spin 1/2. In the case of G = SU(n) and the level k = 1, there are several situations when boundary one-point correlators are SLE _κ -martingales. If the boundary field is labelled by the defining n-dimensional representation of SU(n), we obtain ${\varkappa=2}$ . For n even, by choosing the boundary field labelled by the (unique) self-adjoint fundamental representation, we get ${\varkappa=8/(n {+} 2)}$ . We also study the situation when the distance between the two boundary fields is finite, and we show that in this case the ${{\rm SLE}_\varkappa}$ evolution is replaced by ${{\rm SLE}_{\varkappa,\rho}}$ with ${\rho=\varkappa -6}$ .