Boundary Liouville Field Theory: Boundary Three Point Function

TL;DR

This work advances boundary Liouville field theory by deriving an explicit expression for the boundary three-point function $C_{\beta_3|\beta_2\beta_1}^{\sigma_3\sigma_2\sigma_1}$ through associativity constraints and fusion transformations, with normalization fixed by pole residues. The authors construct the function from a fusion-coefficient-based ansatz, solve a degenerate-field-driven finite-difference equation for the accompanying $g$-factors, and verify consistency with reflection and cyclicity properties. They further connect the result to $b$-Racah–Wigner structures and outline paths to confirm Cardy-Lewellen-type conditions in this non-rational CFT context. The paper thus provides a concrete, fully specified boundary structure constant essential for computing boundary correlators and for understanding Liouville theory on domains with boundary.

Abstract

Liouville field theory is considered on domains with conformally invariant boundary conditions. We present an explicit expression for the three point function of boundary fields in terms of the fusion coefficients which determine the monodromy properties of the conformal blocks.