Conformal weldings of random surfaces: SLE and the quantum gravity zipper

TL;DR

This paper constructs a conformal welding of Liouville quantum gravity surfaces, revealing that the interface is an SLE curve, and explores symmetries and conjectures related to random planar maps and quantum gravity.

Contribution

It provides a rigorous construction linking Liouville quantum gravity surfaces with SLE curves, advancing understanding of their geometric and probabilistic relationship.

Findings

Interface between quantum surfaces is an SLE curve

Identifies symmetries consistent with quantum gravity conjectures

Proposes conjectures and open questions in the field

Abstract

We construct a conformal welding of two Liouville quantum gravity random surfaces and show that the interface between them is a random fractal curve called the Schramm-Loewner evolution (SLE), thereby resolving a variant of a conjecture of Peter Jones. We also demonstrate some surprising symmetries of this construction, which are consistent with the belief that (path decorated) random planar maps have (SLE-decorated) Liouville quantum gravity as a scaling limit. We present several precise conjectures and open questions.