Rankin-Cohen brackets for Calabi-Yau modular forms

For any positive integer $n$, we introduce a quasi-homogeneous vector field $\mathsf{\text{D}}$ of degree $2$ on a moduli space $\mathsf{\text{T}}$ of enhanced Calabi-Yau $n$-folds arising from the Dwork family. By Calabi-Yau quasi-modular forms for Dwork family we mean the elements of the graded $\mathbb{C}$-algebra $\tilde{\mathcal{M}}$ generated by the components of a particular solution of $\mathsf{\text{D}}$, which are provided with natural weight. Using $\mathsf{\text{D}}$ we introduce the derivation $\mathcal{D}$ and the Ramanujan-Serre type derivation $\mathrm{\partial }$ on $\tilde{\mathcal{M}}$. We show that they are degree $2$ differential operators and there exists a proper subspace $$\mathcal{M}$\subset \widetilde{$\mathcal{M}$}$, called the space of Calabi-Yau modular forms, which is closed under $\mathrm{\partial }$. Using the derivation $\mathcal{D}$, we define the Rankin-Cohen brackets for Calabi-Yau quasi-modular forms and prove that the subspace generated by the positive weight elements of $\mathcal{M}$ is closed under the Rankin-Cohen brackets.