FZZ formula of boundary Liouville CFT via conformal welding

Abstract

Liouville Conformal Field Theory (LCFT) on the disk describes the conformal factor of the quantum disk, which is the natural random surface in Liouville quantum gravity with disk topology. Fateev, Zamolodchikov and Zamolodchikov (2000) proposed an explicit expression, the so-called FZZ formula, for the one-point bulk structure constant for LCFT on the disk. In this paper we give a proof of the FZZ formula in the probabilistic framework of LCFT, which represents the first step towards rigorously solving boundary LCFT using conformal bootstrap. In contrast to previous works, our proof is based on conformal welding of quantum disks and the mating-of-trees theory for Liouville quantum gravity. As a byproduct of our proof, we also obtain the exact value of the variance for the Brownian motion in the mating-of-trees theory. Our paper is an essential part of an ongoing program proving integrability results for Schramm–Loewner evolutions, LCFT, and in the mating-of-trees theory.