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

TL;DR

The paper extends Kawai's one-loop heterotic threshold analysis to Calabi-Yau 3- and 4-folds, producing a framework that expresses elliptic genera as product formulas tied to modular forms on Sp4-type groups. It demonstrates that the resulting products align with denominator formulas of generalized Kac-Moody algebras, hinting at a BPS algebra built from vector and hypermultiplet vertex operators. For CY4, the calculation yields a χ0-dependent automorphic product and a χ-dependent denominator product, while CY3 reduces to a simple -χ ln|F0(Ω)|^2, linking elliptic genera to automorphic structures. The work recovers known K3 results and provides a method to identify algebras associated with Calabi-Yau geometries via their elliptic genera and product expansions.

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.