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

This paper aims to show that a certain moduli space $\mathsf{\text{T}}$, which arises from the so-called Dwork family of Calabi-Yau $n$-folds, carries a special complex Lie algebra containing a copy of ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{C})$. In order to achieve this goal, we introduce an algebraic group $\mathsf{G}$ acting from the right on $\mathsf{\text{T}}$ and describe its Lie algebra ${\rm Lie}({\sf G})$\mathrm{.}Weobservethat${\rm Lie}({\sf G})$ is isomorphic to a Lie subalgebra of the space of the vector fields on $\mathsf{\text{T}}$. In this way, it turns out that $\mathrm{L}\mathrm{i}\mathrm{e}(\mathsf{G})$ and the modular vector field $\mathsf{R}$ generate another Lie algebra $\mathfrak{G}$, called AMSY-Lie algebra, satisfying $\dim (\mathfrak{G})=\dim ($\mathsf{\text{T}}$)$\mathrm{.}Wefindacopyof$\mathfrak{sl}_2($\mathbb{C}$)$containing${\sf R}$asaLiesubalgebraof$\mathfrak{G}$. 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.