Remarks on Liouville theory with boundary

TL;DR

This work analyzes Liouville theory on a strip with conformal boundary conditions, focusing on the boundary bootstrap for noncompact CFTs. It lays out the data needed to bootstrap boundary Liouville theory, derives two core consistency conditions connecting bulk and boundary data, and presents explicit expressions for the one-point and boundary two-point functions along with their unitarity properties. It also discusses the spectrum of the theory on the strip via Cardy-like constructions and canonical quantization, and outlines a path toward a quantum-group formulation using continuous Racah–Wigner data. The results highlight the richer structure of noncompact boundary CFTs, including reflection amplitudes and a potential link to ${\mathcal U}_q(sl(2,\mathbb{R}))$.

Abstract

The bootstrap for Liouville theory with conformally invariant boundary conditions will be discussed. After reviewing some results on one- and boundary two-point functions we discuss some analogue of the Cardy condition linking these data. This allows to determine the spectrum of the theory on the strip, and illustrates in what respects the bootstrap for noncompact conformal field theories with boundary is richer than in RCFT. We briefly indicate some connections with $U_q(sl(2,R))$ that should help completing the bootstrap.