Gauss–Manin connection in disguise: Dwork family

Abstract

We study the moduli space $\mathsf{T}$ of the Calabi–Yau $n$‑folds arising from the Dwork family and enhanced with bases of the $n$‑th de Rham cohomology with constant cup product and compatible with Hodge filtration. We also describe a unique vector field $\mathsf{R}$ in $\mathsf{T}$ which contracted with the Gauss–Manin connection gives an upper triangular matrix with some non-constant entries which are natural generalizations of Yukawa couplings. For $n=1,2$ we compute explicit expressions of $\mathsf{R}$ and give a solution of $\mathsf{R}$ in terms of quasi-modular forms. The moduli space $\mathsf{T}$ is an affine variety and for $n = 4$ we give explicit coordinate system for $\mathsf{T}$ and compute the vector field $\mathsf{R}$ and the $q$-expansion of its solution.