The Spectrum of FZZT Branes Beyond the Planar Limit

TL;DR

This paper systematically examines the spectrum of FZZT branes in minimal string theory beyond the planar limit, revealing that previous identifications are incomplete and that certain branes, especially the identity brane, have unique properties.

Contribution

It demonstrates that the identification of FZZT branes as linear combinations of the identity brane with complex shifts does not hold universally beyond the planar limit, supported by Liouville and matrix model analyses.

Findings

Spoiling terms are non-universal and have universal interpretation.

Matrix model results agree with Liouville theory for these terms.

The identity brane uniquely reproduces bulk correlation numbers on the torus.

Abstract

Minimal string theory has a number of FZZT brane boundary states; one for each Cardy state of the minimal model. It was conjectured by Seiberg and Shih that all branes in a minimal string theory could be expressed as a linear combination of the brane associated to the identity operator of the minimal model with complex shifts in the boundary cosmological constant. Subsequently it was found that this identification of FZZT branes does not hold exactly for some cylinder amplitudes but was spoiled by terms that are associated with vanishing worldsheet area and are therefore non-universal. In this paper we investigate this claim systematically, using both Liouville and matrix model methods, beyond the planar limit. We find that the aforementioned identification of FZZT branes is spoiled by terms that do not admit an interpretation as non-universal terms. Furthermore, the spoiling terms as computed using the matrix model are found to be in agreement with those coming from Liouville theory, which also suggests that these terms have universal meaning. Finally, we also investigate the identification of FZZT branes by replacing the boundary state with a sum of local operators. We find in this case that the brane associated with the identity operator appears to be special as it is the only one to correctly reproduce the correlation numbers for bulk operators on the torus.