Calabi-Yau manifolds and sporadic groups

TL;DR

This paper investigates the potential links between the elliptic genera of higher-dimensional Calabi-Yau manifolds and sporadic groups, extending previous work on K3 surfaces and Mathieu groups.

Contribution

It explores possible connections between elliptic genus expansions of Calabi-Yau manifolds of dimension greater than three and sporadic symmetry groups, including calculations for various models.

Findings

Elliptic genera of certain Calabi-Yau d-folds may relate to sporadic groups.

Computed twined elliptic genera for multiple Calabi-Yau 5-folds.

Potential underlying sporadic symmetries suggested by genus expansions.

Abstract

A few years ago a connection between the elliptic genus of the K3 manifold and the largest Mathieu group M$_{24}$ was proposed. We study the elliptic genera for Calabi-Yau manifolds of larger dimensions and discuss potential connections between the expansion coefficients of these elliptic genera and sporadic groups. While the Calabi-Yau 3-fold case is rather uninteresting, the elliptic genera of certain Calabi-Yau $d$-folds for $d>3$ have expansions that could potentially arise from underlying sporadic symmetry groups. We explore such potential connections by calculating twined elliptic genera for a large number of Calabi-Yau 5-folds that are hypersurfaces in weighted projected spaces, for a toroidal orbifold and two Gepner models.