Modular Bootstrap for Boundary N=2 Liouville Theory

TL;DR

This paper develops a modular bootstrap approach for boundary N=2 Liouville theory, introducing extended characters, classifying D-branes, and constructing analogues of ZZ- and FZZT-branes with applications to Calabi-Yau singularities and string vacua.

Contribution

It introduces extended characters with modular properties, classifies boundary states, and constructs D-brane analogues in N=2 Liouville theory, advancing understanding of boundary conformal field theories.

Findings

Constructed localized and extended D-branes in N=2 Liouville theory.

Reproduced intersection numbers of vanishing cycles in Calabi-Yau singularities.

Identified stable and unstable D-branes in 2D string vacua.

Abstract

We study the boundary N=2 Liouville theory based on the ``modular bootstrap'' approach. As fundamental conformal blocks we introduce the ``extended characters'' that are defined as the proper sums over spectral flows of irreducible characters of the N=2 superconformal algebra (SCA) and clarify their modular transformation properties in models with rational central charges. We then try to classify the Cardy states describing consistent D-branes based on the modular data. We construct the analogues of ZZ-branes (hep-th/0101152), localized at the strong coupling region, and the FZZT-branes (hep-th/0001012, hep-th/0009138), which extend along the Liouville direction. The former is shown to play important roles to describe the BPS D-branes wrapped around vanishing cycles in deformed Calabi-Yau singularities, reproducing the correct intersection numbers of vanishing cycles. We also discuss the non-BPS D-branes in 2d type 0 (and type II) string vacua composed of the N=2 Liouville with $\hat{c}(\equiv c/3)=5$. Unstable D0-branes are found as the ZZ-brane analogues mentioned above, and the FZZT-brane analogues are stable due to the existence of mass gap despite the lack of GSO projection.