Liouville Field, Modular Forms and Elliptic Genera

Abstract

When we describe 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 whose proof, by D. Zagier, is included in an appendix.

1+19 pages, no figure; v2 A proof of identity (2.40) due to D.Zagier is included in Appendix C, minor changes, references added; v3 minor changes, references added, to be published in JHEP

References in corpus (4)

• D-branes and extended characters in SL(2,R)/U(1)
• Conifold Type Singularities, N=2 Liouville and SL(2;R)/U(1) Theories
• ZZ-Branes of N=2 Super-Liouville Theory
• N=2 Liouville Theory with Boundary

Cited by in corpus (13)

• 2d Index and Surface operators
• Non-holomorphic Modular Forms and SL(2,R)/U(1) Superconformal Field Theory
• From Liouville to Chern-Simons, Alternative Realization of Wilson Loop Operators in AGT Duality
• Modular Forms and Elliptic Genera for ALE Spaces
• N=4 Superconformal Algebra and the Entropy of HyperKahler Manifolds
• Superconformal Algebras and Mock Theta Functions
• N=2 Moonshine
• On modular properties of the AdS3 CFT
• Black Hole - String Transition and Rolling D-brane
• Liouville Theory and Elliptic Genera
• A 3D spinorial view of 4D exceptional phenomena
• Quarter-BPS states in orbifold sigma models with ADE singularities
• Topological Mathieu Moonshine