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 socalled the FZZ formula, for the onepoint 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 matingoftrees 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 matingoftrees theory. Our paper is an essential part of an ongoing program proving integrability results for SchrammLoewner evolutions, LCFT, and in the matingoftrees theory.
 Publication:

arXiv eprints
 Pub Date:
 April 2021
 DOI:
 10.48550/arXiv.2104.09478
 arXiv:
 arXiv:2104.09478
 Bibcode:
 2021arXiv210409478A
 Keywords:

 Mathematics  Probability;
 Mathematical Physics
 EPrint:
 49 pages, 4 figures