Boundary Liouville theory at c=1

TL;DR

This work analyzes the boundary Liouville theory at $c=1$ by taking the $b\to i$ limit of Liouville theory and relating it to the $c=1$ limit of unitary minimal models. It identifies a 1-parameter family of boundary conditions for the Euclidean $c=1$ Liouville model, labeled by $s\in\mathbb{R}$, whose open-string spectra form momentum bands with finite gaps, and provides explicit formulas for the boundary 2-point function and the bulk-boundary OPE in this limit. The authors also establish a geometric interpretation for ZZ branes at Dirichlet points and connect the $c=1$ boundary data to the corresponding minimal-model limits and to the known FZZT/ZZ brane structure in Liouville theory. Together with these results, the paper lays out a complete set of exact conformal data for a rich, unitary, non-rational boundary CFT and offers a framework for studying tachyon condensation in this exact setting, including prospects for Lorentzian extensions. Key techniques hinge on the asymptotics of Barnes' double Gamma functions, enabling controlled $c=1$ limits of bulk and boundary couplings and revealing deep connections to the $c=1$ limit of minimal models.

Abstract

The c=1 Liouville theory has received some attention recently as the Euclidean version of an exact rolling tachyon background. In an earlier paper it was shown that the bulk theory can be identified with the interacting c=1 limit of unitary minimal models. Here we extend the analysis of the c=1-limit to the boundary problem. Most importantly, we show that the FZZT branes of Liouville theory give rise to a new 1-parameter family of boundary theories at c=1. These models share many features with the boundary Sine-Gordon theory, in particular they possess an open string spectrum with band-gaps of finite width. We propose explicit formulas for the boundary 2-point function and for the bulk-boundary operator product expansion in the c=1 boundary Liouville model. As a by-product of our analysis we also provide a nice geometric interpretation for ZZ branes and their relation with FZZT branes in the c=1 theory.

Figures (1)

• Figure 1: Boundary excitations for the boundary condition $(\rho,\sigma)$ fill a rectangle, or, after reflection, two triangles in the Kac table. In the large $m$ limit, they form bands with band-gaps around $2p=1$.