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

TL;DR

This work establishes a precise correspondence between the $SL(2;\mathbf{R})/U(1)$ supercoset and ${\cal N}=2$ Liouville theory, identifying discrete massless representations that are closed under modular transformations and correspond to localized states on non-compact Calabi–Yau geometries. By decomposing toroidal partition functions into continuous (volume-divergent) and discrete (localized) parts, the authors demonstrate how elliptic genera isolate the discrete content and reveal intricate modular properties described by Appell functions rather than Jacobi forms. Coupling to rational ${\cal N}=2$ RCFTs and enforcing integral $U(1)$-charges yields modular-invariant partition functions per unit volume, enabling universal expressions for elliptic genera across a range of singular geometries, including conifolds, ALE spaces, and non-compact CY$_3$'s. The results support a deep, perhaps exact, T-duality between the KS coset and ${\cal N}=2$ Liouville theory and provide a framework to interpret geometric data of non-compact Calabi–Yau spaces through modular-invariant spectra and Appell-function structures.

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.