## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Exploring New Structures and Natural Constructions in Mathematical Physics, K. Hasegawa, T. Hayashi, S. Hosono and Y. Yamada, eds. (Tokyo: Mathematical Society of Japan, 2011), 125 - 159

### Modular forms and elliptic genera for ALE spaces

Tohru Eguchi, Yuji Sugawara, and Anne Taormina

#### 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.

#### Article information

**Dates**

Received: 4 March 2008

Revised: 17 May 2008

First available in Project Euclid:
24 November 2018

**Permanent link to this document**

https://projecteuclid.org/
euclid.aspm/1543085346

**Digital Object Identifier**

doi:10.2969/aspm/06110125

**Mathematical Reviews number (MathSciNet)**

MR2867146

**Zentralblatt MATH identifier**

1243.81188

#### Citation

Eguchi, Tohru; Sugawara, Yuji; Taormina, Anne. Modular forms and elliptic genera for ALE spaces. Exploring New Structures and Natural Constructions in Mathematical Physics, 125--159, Mathematical Society of Japan, Tokyo, Japan, 2011. doi:10.2969/aspm/06110125. https://projecteuclid.org/euclid.aspm/1543085346