Rankin–Cohen brackets for Calabi–Yau modular forms

Abstract

$\def\M{\mathscr{M}}\def\Rscr{\mathscr{R}}\def\Rsf{\mathsf{R}}\def\Tsf{\mathsf{T}}\def\tildeM{\widetilde{\M}}$For any positive integer $n$, we introduce a modular vector field $\Rsf$ on a moduli space $\Tsf$ of enhanced Calabi–Yau $n$-folds arising from the Dwork family. By Calabi–Yau quasi-modular forms associated to $\Rsf$ we mean the elements of the graded $\mathbb{C}$-algebra $\tildeM$ generated by solutions of $\Rsf$, which are provided with natural weights. The modular vector field $\Rsf$ induces the derivation $\Rscr$ and the Ramanujan–Serre type derivation $\partial$ on $\tildeM$. We show that they are degree $2$ differential operators and there exists a proper subspace $\M \subset \tildeM$, called the space of Calabi–Yau modular forms associated to $\Rsf$, which is closed under $\partial$. Using the derivation $\Rscr$, we define the Rankin–Cohen brackets for $\tildeM$ and prove that the subspace generated by the positive weight elements of $\M$ is closed under the Rankin–Cohen brackets. We find the mirror map of the Dwork family in terms of the Calabi–Yau modular forms.