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

Abstract In this paper the elliptic genus for a general Calabi-Yau 4-fold 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 K 3 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.