SL(2;R)/U(1) Supercoset and Elliptic Genera of Non-compact Calabi-Yau Manifolds

We first discuss the relationship between the SL(2;Bbb R)/U(1) supercoset and Script N = 2 Liouville theory and make a precise correspondence between their representations. We shall show that the discrete unitary representations of SL(2;Bbb R)/U(1) theory correspond exactly to those massless representations of Script N = 2 Liouville theory which are closed under modular transformations and studied in our previous work [18]. It is known that toroidal partition functions of SL(2;Bbb R)/U(1) theory (2D Black Hole) contain two parts, continuous and discrete representations. The contribution of continuous representations is proportional to the space-time volume and is divergent in the infinite-volume limit while the part of discrete representations is volume-independent. In order to see clearly the contribution of discrete representations we consider elliptic genus which projects out the contributions of continuous representations: making use of the SL(2;Bbb R)/U(1), we compute elliptic genera for various non-compact space-times such as the conifold, ALE spaces, Calabi-Yau 3-folds with A_n singularities etc. We find that these elliptic genera in general have a complex modular property and are not Jacobi forms as opposed to the cases of compact Calabi-Yau manifolds.