Modular Bootstrap for Boundary N=2 Liouville Theory

TL;DR

This work develops a modular bootstrap framework for boundary ${ m N}=2$ Liouville theory at rational central charges by defining extended characters as sums over spectral flows, ensuring integral ${U(1)}$ charges and closed modular behavior. It classifies Cardy states into three classes, yielding ZZ-like localized branes and FZZT-like extended branes, and demonstrates how these states reproduce open-string spectra and intersection numbers when coupled to ${ m N}=2$ minimal models, thereby modeling D-branes on Calabi–Yau vanishing cycles. The analysis also extends to 2d type 0 vacua with ${ m hat c}=5$, discussing stability, tachyons, and decompactification limits, and sketches connections to non-perturbative CY dynamics and potential matrix-model duals. Overall, the paper provides a concrete, modular-data-driven construction of D-branes in ${ m N}=2$ Liouville backgrounds and links them to CY singularities and string vacua, with open avenues for disk amplitudes and dual descriptions.

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.