A survey on modular vector fields and CY modular forms attached to Dwork family

Abstract

This article aimes to give a survay of the works of the author on modular vector fields and Calabi-Yau (CY) modular forms attached to the Dwork family. It is mostly tried to be more objective and avoid technical details. For any positive integer $n$, it is firstly introduced an enhanced moduli space $\textsf{T}:=\textsf{T}_n$ of CY $n$-folds arising from the Dwork family. It is observed that there exists a unique vector field $\textsf{D}$ in $\textsf{T}$, known as modular vector field, whose solution components can be expressed as $q$-expansions (Fourier series) with integer coefficients. We call these $q$-expansions CY modular forms and it is verified that the space generated by them has a canonical $\mathfrak{sl}_2(\mathbb{C})$-module structure which provides it with a Rankin-Cohen algebraic structure. All these concepts are explicitly established for $n=1,2,3,4$.