Conformal welding is an operation that encodes Jordan curves on the Riemann sphere in terms of circle homeomorphisms. Thus, composition defines a natural group action of circle homeomorphisms on Jordan curves. In this talk, I will discuss a Cameron–Martin type quasi-invariance result for the SLE loop measure under the right group action by Weil–Petersson homeomorphisms. While this result was hinted by Carfagnini and Wang's identification of Loewner energy as the Onsager–Machlup action functional of the SLE loop measure, the group structure of SLE welding has been little understood previously.
Our proof is based on the characterization of the composition operator associated with Weil–Petersson circle homeomorphisms using Hilbert–Schmidt operators and the description of the SLE loop measure in terms of the welding of two independent quantum disks by Ang, Holden, and Sun. This is joint work with Shuo Fan (Tsinghua University and IHES).