Calabi-Yau modular forms in limit: Elliptic Fibrations

TL;DR

This work shows that in the limit of elliptically fibred Calabi–Yau manifolds, Calabi–Yau modular forms reduce to classical modular objects. By formulating a two-parameter Picard–Fuchs framework and constructing the Calabi–Yau modular forms field ${\sf M}_n$, the authors prove that q-expansion coefficients organize into (quasi-)modular forms for specific groups determined by $(a_0,a_1,a_2)$. The approach is applied to CY3 and CY4, yielding modular expressions for genus-zero Gromov–Witten data and delivering new modular identities for CY4 Yukawa couplings, including an anomaly-type structure. The main explicit example—an elliptic fibration over $\mathbb{P}^3$—demonstrates the mechanism with explicit PF operators, period expansions, and modular decompositions in terms of $E_4,E_6$ and $\eta$, illustrating the practical impact on mirror symmetry and enumerative geometry.

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.

Theorems & Definitions (16)

• Theorem 1
• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Proposition 4
• Proposition 5
• proof