Rankin-Cohen brackets for Calabi-Yau modular forms

Abstract

For any positive integer $n$, we introduce a quasi-homogeneous vector field $\textsf{D}$ of degree $2$ on a moduli space $\textsf{T}$ 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 $\widetilde{\mathcal{M}}$ generated by the components of a particular solution of $\textsf{D}$, which are provided with natural weight. Using $\textsf{D}$ we introduce the derivation $\mathcal{D}$ and the Ramanujan-Serre type derivation $\partial$ on $\widetilde{\mathcal{M}}$. 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 $\partial$. 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.