The elliptic genus of Calabi-Yau 3- and 4-folds, product formulae and generalized Kac-Moody algebras

TL;DR

This paper derives the elliptic genus for Calabi-Yau fourfolds and extends previous work on heterotic string threshold corrections, revealing a connection to generalized Kac-Moody algebras through denominator formulas.

Contribution

It provides a general formula for the elliptic genus of Calabi-Yau fourfolds and links string threshold corrections to generalized Kac-Moody algebras.

Findings

Elliptic genus for Calabi-Yau fourfolds is explicitly derived.

Threshold corrections involve a generalized Kac-Moody algebra denominator formula.

The work extends previous Kawai results to higher-dimensional Calabi-Yau manifolds.

Abstract

In this paper the elliptic genus for a general Calabi-Yau fourfold is derived. The recent work of Kawai calculating N=2 heterotic string one-loop threshold corrections with a Wilson line turned on is extended to a similar computation where K3 is replaced by a general Calabi-Yau 3- or 4-fold. In all cases there seems to be a generalized Kac-Moody algebra involved, whose denominator formula appears in the result.