• Corpus ID: 233296120

Integrability of SLE via conformal welding of random surfaces

@inproceedings{Ang2021IntegrabilityOS,
  title={Integrability of SLE via conformal welding of random surfaces},
  author={Morris Ang and Nina Holden and Xin Sun},
  year={2021}
}

• M. Ang, N. Holden, Xin Sun
• Published 19 April 2021
• Mathematics

We demonstrate how to obtain integrability results for the Schramm-Loewner evolution (SLE) from Liouville conformal field theory (LCFT) and the mating-of-trees framework for Liouville quantum gravity (LQG). In particular, we prove an exact formula for the law of a conformal derivative of a classical variant of SLE called SLEκ(ρ−; ρ+). Moreover, we prove that the SLE loop measure constructed by Zhan (2020) arises naturally from the conformal welding of two quantum disks in LQG. Our proofs are built on two connections between SLE, LCFT, and mating-of-trees. Firstly, LCFT and mating-of-trees provide equivalent but complementary methods to describe natural random surfaces in LQG. We extend earlier equivalence results in two directions, namely by allowing fewer marked points and more generic singularities. Secondly, the conformal welding of these random surfaces produces SLE curves as their interfaces. In particular, we rely on the conformal welding results proved in our companion paper [AHS20b]. Our paper is an essential part of a program proving integrability results for SLE, LCFT, and mating-of-trees based on these two connections.