  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 k=1, there are several situations
when boundary one-point correlators are SLE(kappa)-martingales. If the boundary
field is labelled by the defining n-dimensional representation of SU(n), we
obtain kappa=2. For n even, by choosing the boundary field labelled by the
(unique) self-adjoint fundamental representation, we get kappa=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 SLE(kappa) evolution is replaced by
SLE(kappa,rho) with rho=kappa-6.
