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

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.