Derivation of all structure constants for boundary Liouville CFT

TL;DR

This paper rigorously proves the formulas for all boundary structure constants in Liouville conformal field theory, connecting probabilistic definitions with classical formulas and enabling the conformal bootstrap approach.

Contribution

It establishes the equivalence of probabilistic and classical formulas for boundary structure constants in LCFT, filling a key gap in the mathematical understanding.

Findings

Proved the boundary three-point and bulk-boundary structure constants match classical formulas.

Derived the boundary reflection coefficient formula for LCFT.

Provided exact descriptions of joint laws of area and boundary lengths in LQG surfaces.

Abstract

We prove that the probabilistic definition of the most general boundary three-point and bulk-boundary structure constants in Liouville conformal field theory (LCFT) agree respectively with the formula proposed by Ponsot-Techsner (2002) and by Hosomichi (2001). These formulas also respectively describe the fusion kernel and modular kernel of the Virasoro conformal blocks, which are important functions in various contexts of mathematical physics. As an intermediate step, we obtain the formula for the boundary reflection coefficient of LCFT proposed by Fateev-Zamolodchikov-Zamolodchikov (2000). Our proof relies on the boundary Belavin-Polyakov-Zamolodchikov differential equation recently proved by the first named author, and inputs from the coupling theory of Liouville quantum gravity (LQG) and Schramm Loewner evolution. Our results supply all the structure constants needed to perform the conformal bootstrap for boundary LCFT. They also yield exact descriptions for the joint law of the area and boundary lengths of basic LQG surfaces, including quantum triangles and two-pointed quantum disks.