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

Abstract

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

21 pages