Liouville conformal field theory and the quantum zipper

Abstract

Sheffield showed that conformally welding a ${\hat{I}}^3$-Liouville quantum gravity (LQG) surface to itself gives a Schramm-Loewner evolution (SLE) curve with parameter $κ= γ^2$ as the interface, and Duplantier-Miller-Sheffield proved similar results for $κ= \frac{16}{γ^2}$ for ${\hat{I}}^3$-LQG surfaces with boundaries decorated by looptrees of disks or by continuum random trees. We study these dynamics for LQG surfaces coming from Liouville conformal field theory (LCFT). At stopping times depending only on the curve, we give an explicit description of the surface and curve in terms of LCFT and SLE. This has applications to both LCFT and SLE. We prove the boundary BPZ equations for LCFT, a crucial input for subsequent work with Remy, Sun and Zhu deriving the structure constants of boundary LCFT. With Yu we prove the reversibility of whole-plane SLE${}_{\hat{I}}º$ for $\hat{I}º>8$ via a novel radial mating-of-trees, and will show the space of LCFT surfaces is closed under conformal welding.

47 pages, 12 figures. Final version, to appear in the Annals of Probability