Modular vector fields attached to Dwork family: $\mathfrak{sl}_2(\mathbb{C})$ Lie algebra

This paper aims to show that a certain moduli space $\mathsf{T}$, which arises from the so-called Dwork family of Calabi–Yau $n$-folds, carries a special complex Lie $\{$algebra$\}$ containing a copy of $\mathfrak{sl}_2(\mathbb{C})$. In order to achieve this goal, we introduce an algebraic group $\mathsf{G}$ acting from the right on $\mathsf{T}$ and describe its Lie algebra $\mathsf{Lie(G)}$. We observe that $\mathsf{Lie(G)}$ is isomorphic to a Lie subalgebra of the space of the vector fields on $\mathsf{T}$. In this way, it turns out that $\mathsf{Lie(G)}$ and the modular vector field $\mathsf{R}$ generate another Lie algebra $\mathfrak{G}$, called AMSY-Lie algebra, satisfying $\dim (\mathfrak{G})=\dim (\mathsf{T})$. We find a copy of $\mathfrak{sl}_2(\mathbb{C})$ containing $\mathsf{R}$ as a Lie subalgebra of $\mathfrak{G}$. The proofs are based on an algebraic method calling “Gauss–Manin connection in disguise”. Some explicit examples for $n=1,2,3,4$ are stated as well.