Gauss-Manin Connection in Disguise: Dwork Family

Abstract: We study the moduli space $\textsf{T}$ 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 $\textsf{R}$ in $\textsf{T}$ 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 $\textsf{R}$ and give a solution of $\textsf{R}$ in terms of quasi-modular forms. The moduli space $\textsf{T}$ is an affine variety and for $n=4$ we give explicit coordinate system for $\textsf{T}$ and compute the vector field $\textsf{R}$ and the $q$-expnasion of its solution.