Liouville Field, Modular Forms and Elliptic Genera

TL;DR

This paper develops a method to combine discrete and continuous representations in conformal field theories describing non-compact Calabi-Yau manifolds, simplifying their modular properties and enabling computation of elliptic genera.

Contribution

It introduces a novel approach to combine representations for better modular behavior, facilitating the calculation of elliptic genera of ALE spaces.

Findings

Elliptic genera of ALE spaces computed successfully.

Results align with decompactified K3 surface predictions.

A key theta function identity proved by Zagier supports the method.

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.