Modular Bootstrap for Boundary N=2 Liouville Theory

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.

52 pages, no figure; v2 a reference added, minor corrections, a footnote added in page 8; v3 comments added for clarifications, summary of proposed boundary states added, version to appear in JHEP; v4 a reference added

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• Two-dimensional superstrings and the supersymmetric matrix model
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• Non-compact String Backgrounds and Non-rational CFT
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• Remarks on free field realization of SL(2,R)/U(1) x U(1) WZNW model
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• Notes on D-branes in 2D Type 0 String Theory
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• Falling D0-Branes in 2D Superstring Theory
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• Three-Charge Black Holes and Quarter BPS States in Little String Theory
• Comments on geometric and universal open string tachyons near fivebranes
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• Notes on S-Matrix of Non-critical N=2 String
• Duality in N=4 Liouville Theory and Moonshine Phenomena
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• Orientifolds in N=2 Liouville Theory and its Mirror
• `Analytic Continuation' of N=2 Minimal Model
• Non-Compact SCFT and Mock Modular Forms
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• Deformed Type 0A Matrix Model and Super-Liouville Theory for Fermionic Black Holes