Remarks on Liouville theory with boundary

TL;DR

This paper discusses the bootstrap approach for Liouville theory with boundary, exploring boundary conditions, spectrum determination, and connections to quantum groups, highlighting the complexity of noncompact conformal field theories with boundaries.

Contribution

It provides new insights into the boundary bootstrap for Liouville theory, including spectrum analysis and the role of quantum group connections.

Findings

Determined the spectrum of Liouville theory on the strip with boundary conditions.

Established an analogue of the Cardy condition for boundary two-point functions.

Highlighted the richer structure of bootstrap in noncompact conformal field theories with boundary.

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.