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

TL;DR

This paper demonstrates that a moduli space from the Dwork family of Calabi-Yau n-folds possesses a Lie algebra structure containing an $rak{sl}_2(b C)$ subalgebra, revealing deep algebraic symmetries in the geometric setting.

Contribution

It introduces an algebraic group acting on the moduli space and constructs a Lie algebra containing $rak{sl}_2(b C)$, connecting geometric moduli with Lie algebra structures.

Findings

Identification of a Lie algebra containing $rak{sl}_2(b C)$ in the moduli space.

Construction of the AMSY-Lie algebra generated by modular vector fields.

Explicit examples for dimensions n=1,2,3,4 illustrating the theory.

Abstract

This paper aims to show that a certain moduli space $\textsf{T}$, 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 $\sf G$ acting from the right on $\textsf{T}$ and describe its Lie algebra ${\rm Lie}({\sf G})$. We observe that ${\rm Lie}({\sf G})$ is isomorphic to a Lie subalgebra of the space of the vector fields on $\textsf{T}$. In this way, it turns out that ${\rm Lie}({\sf G})$ and the modular vector field ${\sf R}$ generate another Lie algebra $\mathfrak{G}$, called AMSY-Lie algebra, satisfying $\dim (\mathfrak{G})=\dim (\textsf{T})$. We find a copy of $\mathfrak{sl}_2(\mathbb{C})$ containing ${\sf R}$ 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.