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

Tom Alberts,
Sung-Soo Byun,
Nam-Gyu Kang

July 2024
SLE

### 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≤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 Teicmüller 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.