Remarks on Liouville theory with boundary

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. D-branes on compact spaces can be studied in the " stringy regime " (α ′ ∼ O(1)) by means of con-formal field theory in the presence of boundaries [1][2]. The treatment of D-branes on noncom-pact spaces requires consideration of CFT with continuous spectrum of Virasoro representations (noncompact CFT). Liouville theory may be considered as a prototypical example of such CFT. It seems to be the natural starting point for the development of techniques for the exact study of the class of CFT that describe D-branes on non-compact backgrounds. Moreover, physically interesting examples such as the SL(2)/U (1) black hole or AdS 3 are closely related to Liouville theory from the technical point of view. 2. Liouville theory w/o boundary We will very briefly assemble a few facts concerning Liouville theory with periodic boundary conditions that will be referred to later. The classical 2D field theory is defined on R × S 1 by the Lagrangian L = 1 4π (∂ a φ) 2 + µe 2bφ .