Multiple Schramm–Loewner evolutions for conformal field theories with Lie algebra symmetries

Abstract

We provide multiple Schramm–Loewner evolutions (SLEs) to describe the scaling limit of multiple interfaces in critical lattice models possessing Lie algebra symmetries. The critical behavior of the models is described by Wess–Zumino–Witten (WZW) models. Introducing a multiple Brownian motion on a Lie group as well as that on the real line, we construct the multiple SLE with additional Lie algebra symmetries. The connection between the resultant SLE and the WZW model can be understood via SLE martingales satisfied by the correlation functions in the WZW model. Due to interactions among SLE traces, these Brownian motions have drift terms which are determined by partition functions for the corresponding WZW model. As a concrete example, we apply the formula to the suˆ(2)k-WZW model. Utilizing the fusion rules in the model, we conjecture that there exists a one-to-one correspondence between the partition functions and the topologically inequivalent configurations of the SLE traces. Furthermore, solving the Knizhnik–Zamolodchikov equation, we exactly compute the probabilities of occurrence for certain configurations (i.e. crossing probabilities) of traces for the triple SLE.