Modular Forms and Elliptic Genera for ALE Spaces

Abstract

When we describe string propagation on non-compact or singular Calabi-Yau manifolds by CFT, continuous as well as discrete representations appear in the theory. These representations mix in an intricate way under the modular transformations. In this article, we propose a method of combining discrete and continuous representations so that the resulting combinations have a simpler modular behavior and can be used as conformal blocks of the theory. We compute elliptic genera of ALE spaces and obtain results which agree with those suggested from the decompactification of K3 surface. Consistency of our approach is assured by some remarkable identity of theta functions. We include in the appendix some new materials on the representation theory of $\mathcal{N}=4$ superconformal algebra.

28+1 pages, no figure, Contribution to the Proceedings of the workshop in honour of Professor Tsuchiya's retirement, Nagoya University, March 2007

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Cited by in corpus (6)

• N=4 Superconformal Bootstrap of the K3 CFT
• From Liouville to Chern-Simons, Alternative Realization of Wilson Loop Operators in AGT Duality
• Compact Formulas for the Completed Mock Modular Forms
• Geodesics on Calabi-Yau manifolds and winding states in nonlinear sigma models
• Superconformal Algebras and Mock Theta Functions 2. Rademacher Expansion for K3 Surface
• Liouville Theory and Elliptic Genera