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.
We implement a version of conformal field theory in a doubly connected domain to connect it to the theory of annulus SLE of various types, including the standard annulus SLE, the reversible annulus SLE, and the annulus SLE with several force points. This implementation considers the statistical fields generated under the OPE multiplication by the Gaussian free field and its central/background charge modifications with a weighted combination of Dirichlet and excursion-reflected boundary conditions. We derive the Eguchi-Ooguri version of Ward's equations and Belavin-Polyakov-Zamolodchikov equati…
Irregular SLE(4) martingales and isomonodromic deformations
Mathematical Institute, University of Oxford（牛津大学数学研究所）；Institute of Mathematics, EPFL（洛桑联邦理工学院数学研究所）；Department of Mathematics and Systems Analysis, Aalto University（阿尔托大学数学与系统分析系）；Division of Mathematics, University of Cologne（科隆大学数学系）
From conformal field theory on the Riemann sphere, we implement its boundary version in a simply-connected domain using the Schottky double construction. We consider the statistical fields generated by background charge modification of the Gaussian free field with Dirichlet boundary condition under the OPE multiplications. We prove that the correlation functions of such fields with symmetric background charges form a collection of martingale-observables for (forward) chordal/radial SLE with force points and spins. We also present the connection between conformal field theory with Neumann bound…
