D-branes in Singular Calabi-Yau n-fold and N=2 Liouville Theory

TL;DR

This work develops a worldsheet CFT framework for D-branes wrapped on vanishing SUSY cycles of isolated A-D-E singular Calabi–Yau n-folds using N=2 Liouville theory. By constructing supersymmetric boundary, Ishibashi, and Cardy states and employing KPZ-type Liouville perturbations, the authors compute disc amplitudes that reproduce the holomorphic dependence and scaling of vanishing-cycle periods on the deformation parameter μ, and they relate these to geometric periods via a boundary-state realization. They also evaluate the open-string Witten index to obtain intersection numbers among SUSY cycles, finding results that agree with geometric expectations and with the SL(2,R)/U(1) approach in appropriate dimensions. The work thus provides a consistent CFT description of D-branes on singular CYs, linking boundary-state techniques to the deformation theory of singularities and to topological intersection data with potential implications for dual brane configurations and conifold physics.

Abstract

Making use of the N=2 Liouville theory and world-sheet techniques, we study the properties of D-branes wrapped around vanishing SUSY cycles of singular Calabi-Yau n-folds (n=2,3,4). After constructing boundary states describing the wrapped branes, we evaluate the disc amplitudes corresponding to the periods of SUSY cycles. We use the old technique of KPZ scaling in Liouville theory and derive holomorphicity and scaling behavior of vanishing cycles which are in agreement with geometrical considerations. We also discuss the open string Witten index using the N=2 Liouville theory and obtain the intersection numbers among SUSY cycles which also agree with geometrical expectation.