Moscow Mathematical Journal
Volume 20, Issue 1, January–March 2020 pp. 127–151.
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 $\mathrm{Lie}(\mathsf{G})$. We observe that
$\mathrm{Lie}(\mathsf{G})$ is isomorphic to a Lie subalgebra of the space of the vector
fields on $\mathsf{T}$. 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 $\dim (\mathfrak{G})=\dim (\mathsf{T})$. 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.
2010 Math. Subj. Class. 32M25, 37F99, 14J15, 14J32.
Authors:
Younes Nikdelan (1)
Author institution:(1) Universidade do Estado do Rio de Janeiro (UERJ), Instituto de Matemática e Estatística (IME), Departamento de Análise Matemática: Rua São Francisco Xavier, 524, Rio de Janeiro, Brazil / CEP: 20550-900
Summary:
Keywords: Complex vector fields, Gauss–Manin connection,
Dwork family, Hodge filtration, modular form.
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