SLE-type growth processes and the Yang-Lee singularity

TL;DR

This paper generalizes Schramm-Loewner Evolution (SLE) processes to a hierarchy of stochastic conformal maps, revealing a connection to the Yang-Lee singularity and conformal field theory with central charge -22/5.

Contribution

It introduces a hierarchy of SLE-type growth processes and establishes a link to conformal field theory related to the Yang-Lee singularity, including explicit null vector construction.

Findings

Hierarchy of stochastic conformal maps developed

Connection established between SLE hierarchy and Yang-Lee CFT

Explicit null vector constructed in Virasoro algebra at level four

Abstract

The recently introduced SLE growth processes are based on conformal maps from an open and simply-connected subset of the upper half-plane to the half-plane itself. We generalize this by considering a hierarchy of stochastic evolutions mapping open and simply-connected subsets of smaller and smaller fractions of the upper half-plane to these fractions themselves. The evolutions are all driven by one-dimensional Brownian motion. Ordinary SLE appears at grade one in the hierarchy. At grade two we find a direct correspondence to conformal field theory through the explicit construction of a level-four null vector in a highest-weight module of the Virasoro algebra. This conformal field theory has central charge c=-22/5 and is associated to the Yang-Lee singularity. Our construction may thus offer a novel description of this statistical model.