Quasi-invariance for SLE welding measures

TL;DR

This paper proves that SLE welding measures are quasi-invariant under Weil-Petersson circle homeomorphisms, revealing a deep connection between random conformal weldings, Gaussian multiplicative chaos, and the Weil-Petersson group structure.

Contribution

It establishes a Cameron-Martin type quasi-invariance result for SLE welding measures under Weil-Petersson homeomorphisms, linking group actions with Gaussian chaos and quantum gravity.

Findings

Welding measures are quasi-invariant under Weil-Petersson transformations.

Characterization of composition action via Hilbert-Schmidt operators.

Representation of SLE welding as quantum gravity disk weldings.

Abstract

A large class of Jordan curves on the Riemann sphere can be encoded by circle homeomorphisms via conformal welding, among which we consider the welding homeomorphism of the random SLE loops and the Weil-Petersson class of quasicircles. It is known from the work of Carfagnini and Wang (arXiv:2311.00209) that the Onsager-Machlup action functional of SLE loop measures - the Loewner energy - coincides with the K\"ahler potential of the unique right-invariant K\"ahler metric on the group of Weil-Petersson circle homeomorphisms. This identity suggests that the group structure given by the composition shall play a prominent role in the law of SLE welding, which is so far little understood. In this paper, we show a Cameron-Martin type result for random weldings arising from Gaussian multiplicative chaos, especially the SLE welding measures, with respect to the natural group action by Weil-Petersson circle homeomorphisms. More precisely, we show that these welding measures are quasi-invariant when pre- or post-composing the random welding by a fixed Weil-Petersson circle homeomorphism. Our proof is based on the characterization of the composition action in terms of Hilbert-Schmidt operators on the Cameron-Martin space of the log-correlated Gaussian field and the description of the SLE welding as the welding of two independent Liouville quantum gravity (LQG) disks.