Conformal welding is an operation that encodes a large class of Jordan curves on the Riemann sphere, including the loop version of Schramm–Loewner evolution (SLE), in terms of circle homeomorphisms. 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 circle homeomorphisms on the welding homeomorphism. While this result was hinted at by Carfagnini and Wang's identification of the Onsager–Machlup action functional of the SLE loop measure with the Kähler potential of the unique right-invariant Kähler metric on the group of Weil–Petersson circle homeomorphisms (Loewner energy), 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).