General Slit Stochastic L\"owner Evolution and Conformal Field Theory

TL;DR

This paper generalizes the stochastic Loewner evolution (SLE) and explores its connection with conformal field theory, providing new theoretical insights, conditions for coupling, and a unified framework for various L"owner equations.

Contribution

It introduces a generalized SLE framework, establishes conditions for coupling with Gaussian free fields, and unifies different L"owner equations within a single machinery.

Findings

Proved basic properties of the generalized SLE

Derived necessary and sufficient conditions for coupling with Gaussian free fields

Developed a unified machinery for all L"owner equations

Abstract

This monograph is dedicated to a generalization of the L\"owner equation in its stochastic form known as SLE and to its coupling with the Gaussian free field, ultimately aiming at the construction of a boundary conformal field theory with one free scalar bosonic field. This study is presented in line with a systematic, and hopefully concise, presentation and generalization of known elements of the theory of L\"owner evolution. We also study the relation to singular representations of the Virasoro algebra and methods of numerical simulation. The main results are in the proof of the basic properties of the generalized SLE and general necessary and sufficient conditions for the coupling. We also introduce a machinery that possesses to consider all L\"owner equations and known types of the coupling as different manifestations of the same thing.