Conformal welding and the matter--Liouville--ghost factorisation

TL;DR

This paper develops a Cameron--Martin-type formula for SLE loop measure welding homeomorphisms, relating conformal transformations of Gaussian free fields to string theory ghost systems, and introduces new methods for quantum surface welding.

Contribution

It provides a novel Cameron--Martin formula for SLE loop measure weldings and links conformal welding with ghost systems from string theory, offering new insights into quantum surface welding.

Findings

Derived a Cameron--Martin-type formula for SLE loop measure

Connected conformal welding to the bc-ghost system from string theory

Introduced an intrinsic approach to quantum surface welding

Abstract

We study the action of local conformal transformations on several measures related to the Gaussian free field and Schramm--Loewner evolutions. The main novelty of our work is a Cameron--Martin-type formula for the welding homeomorphism of the SLE loop measure ($\kappa\in(0,4]$); its proof relies on a rigorous interpretation (and computation) of the "Jacobian" of the conformal welding map, which we relate to the "$bc$-ghost system" from bosonic string theory. We also give an intrinsic definition of the trace of the GFF on SLE, and prove a characterisation of the free boundary GFF in $\mathbb{D}$. As an application, we introduce a new and intrinsic approach to the conformal welding of quantum surfaces.