N=2 Liouville Theory with Boundary

TL;DR

The paper analyzes boundary phenomena in $N=2$ Liouville theory by establishing two complementary constructions of boundary states: a modular bootstrap for A-branes and a bootstrap of disc correlators involving degenerate bulk operators, with a careful treatment of the $ heta$-momentum/winding quantization. It derives explicit boundary wavefunctions, boundary interactions, and reflection coefficients, then demonstrates consistency between modular data and reflection data by comparing open-string spectra from annulus amplitudes and boundary reflection phases. Boundary interactions, including novel A-type boundary terms and Chan-Paton factors for B-branes, are shown to map precisely to boundary states, and disc correlators yield exact structure constants that tie boundary couplings to brane labels. The results illuminate D-brane physics in non-compact $N=2$ CFTs and offer a framework extendable to related dual models such as sine-Liouville and the $SL(2,f R)/U(1)$ coset, highlighting the role of boundary fermions and degenerate representations in shaping open-string spectra.

Abstract

We study N=2 Liouville theory with arbitrary central charge in the presence of boundaries. After reviewing the theory on the sphere and deriving some important structure constants, we investigate the boundary states of the theory from two approaches, one using the modular transformation property of annulus amplitudes and the other using the bootstrap of disc two-point functions containing degenerate bulk operators. The boundary interactions describing the boundary states are also proposed, based on which the precise correspondence between boundary states and boundary interactions is obtained. The open string spectrum between D-branes is studied from the modular bootstrap approach and also from the reflection relation of boundary operators, providing a consistency check for the proposal.