Darboux-Halphen-Ramanujan Vector Field on a Moduli of Calabi-Yau Manifolds

TL;DR

This paper derives a generalized Darboux-Halphen-Ramanujan differential equation from the Picard-Fuchs equation of Calabi-Yau manifolds, establishing a unique vector field on the moduli space that satisfies specific geometric properties.

Contribution

It introduces a new differential equation on the moduli space of Calabi-Yau manifolds, extending classical equations by Darboux, Halphen, and Ramanujan, and proves the existence of a unique associated vector field.

Findings

Derived a differential equation from Picard-Fuchs equations of Calabi-Yau manifolds.

Established the existence of a unique vector field satisfying certain properties.

Generalized classical differential equations to higher-dimensional Calabi-Yau moduli spaces.

Abstract

In this paper we obtain an ordinary differential equation ${\sf H}$ from a Picard-Fuchs equation associated with a nowhere vanishing holomorphic $n$-form. We work on a moduli space ${\sf T }$ constructed from a Calabi-Yau $n$-fold $W$ together with a basis of the middle complex de Rham cohomology of $W$. We verify the existence of a unique vector field ${\sf H}$ on ${\sf T }$ such that its composition with the Gauss-Manin connection satisfies certain properties. The ordinary differential equation given by ${\sf H}$ is a generalization of differential equations introduced by Darboux, Halphen and Ramanujan.