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

Abstract

In this paper we obtain an ordinary differential equation $\mathsf{H}$ from a Picard-Fuchs equation associated with a nowhere vanishing holomorphic $n$-form. We work on a moduli space $\mathsf{T}$ 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 $\mathsf{H}$ on $\mathsf{T}$ such that its composition with the Gauss-Manin connection satisfies certain properties. The ordinary differential equation given by $\mathsf{H}$ is a generalization of differential equations introduced by Darboux, Halphen and Ramanujan.

27 pages

References in corpus (1)

• Eisenstein type series for Calabi-Yau varieties