Conformal field theory of Gaussian free fields in a multiply connected domain

TL;DR

This paper develops a conformal field theory framework based on Gaussian free fields for multiply connected domains, connecting it to Schramm-Loewner Evolution (SLE) and deriving explicit equations and martingale observables.

Contribution

It introduces a CFT approach for multiply connected domains using Gaussian free fields, deriving Ward's and BPZ equations, and linking to SLE with explicit drift functions.

Findings

Derived explicit Ward's equations for multiply connected domains.

Provided a conformal field theoretic realization of SLE with drift functions.

Constructed martingale observables for the SLE process.

Abstract

We implement a version of conformal field theory (CFT) that gives a connection to SLE in a multiply connected domain. Our approach is based on the Gaussian free field and applies to CFTs with central charge $c \leq 1$. In this framework we introduce the generalized Eguchi-Ooguri equations and use them to derive the explicit form of Ward's equations, which describe the insertion of a stress tensor in terms of Lie derivatives and differential operators depending on the Teicm\"{u}ller modular parameters. Furthermore, by implementing the BPZ equations, we provide a conformal field theoretic realization of an SLE in a multiply connected domain, which in particular suggests its drift function, and construct a class of martingale observables for this SLE process.