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

TL;DR

This paper explores the relationship between SL(2,R)/U(1) supercoset theory and N=2 Liouville theory, analyzing their representations and computing elliptic genera for non-compact Calabi-Yau manifolds, revealing complex modular properties.

Contribution

It establishes a precise correspondence between supercoset and Liouville representations and computes elliptic genera for various non-compact Calabi-Yau spaces, highlighting their unique modular features.

Findings

Discrete representations correspond to massless N=2 Liouville representations.

Elliptic genera of non-compact Calabi-Yau manifolds have complex modular properties.

Continuous representations contribute volume-divergent parts to the partition function.

Abstract

We first discuss the relationship between the SL(2;R)/U(1) supercoset and N=2 Liouville theory and make a precise correspondence between their representations. We shall show that the discrete unitary representations of SL(2;R)/U(1) theory correspond exactly to those massless representations of N=2 Liouville theory which are closed under modular transformations and studied in our previous work hep-th/0311141. It is known that toroidal partition functions of SL(2;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;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.