Calabi-Yau modular forms in limit: Elliptic Fibrations

TL;DR

This paper investigates the behavior of Calabi-Yau modular forms in certain limits, explores their connection to classical modular forms, and uncovers new identities in the context of elliptic fibrations of Calabi-Yau manifolds.

Contribution

It provides a mathematical framework linking Calabi-Yau modular forms to classical modular forms and derives new identities for fourfolds, extending previous observations on threefolds.

Findings

Proof of modularity properties of topological string amplitudes for Calabi-Yau threefolds.

Description of modular forms from degeneracy loci in elliptic fibrations.

New identities for Calabi-Yau fourfolds not previously computed.

Abstract

We study the limit of Calabi-Yau modular forms, and in particular, those resulting in classical modular forms. We then study two parameter families of elliptically fibred Calabi-Yau fourfolds and describe the modular forms arising from the degeneracy loci. In the case of elliptically fibred Calabi-Yau threefolds our approach gives a mathematical proof of many observations about modularity properties of topological string amplitudes starting with the work of Candelas, Font, Katz and Morrison. In the case of Calabi-Yau fourfolds we derive new identities not computed before.