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

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.