Rankin-Cohen brackets for Calabi-Yau modular forms

For any positive integer n, we introduce a modular vector field R on a moduli space T of enhanced Calabi-Yau n-folds arising from the Dwork family. By Calabi-Yau quasi-modular forms associated to R we mean the elements of the graded C-algebra (M) over tilde generated by solutions of R, which are provided with natural weights. The modular vector field R induces the derivation R and the Ramanujan-Serre type derivation partial derivative on (M) over tilde. We show that they are degree 2 differential operators and there exists a proper subspace M subset of (M) over tilde, called the space of Calabi-Yau modular forms associated to R , which is closed under partial derivative. Using the derivation R , we define the RankinCohen brackets for (M) over tilde and prove that the subspace generated by the positive weight elements of M is closed under the RankinCohen brackets. We find the mirror map of the Dwork family in terms of the Calabi-Yau modular forms.