A survey on modular vector fields and CY modular forms attached to Dwork family

Abstract

This article aims to give a survay of the works of the author on modular vector fields and Calabi-Yau (CY) modular forms attached to the Dwork family and avoid technical details. For any positive integer n, it is introduced a moduli space T := Tn of enhanced CY n-folds arising from the Dwork family. It is observed that there exists a unique vector field D in T, known as modular vector field, whose solution components can be expressed as q-expansions Max Planck Institute for Mthematics (MPIM), Vivatsgasse 7, 53111, Bonn, Germany; Departamento de Análise Matemática, Instituto de Matemática e Estat́ıstica (IME), Universidade do Estado do Rio de Janeiro (UERJ), Rua São Francisco Xavier, 524, Rio de Janeiro, Brazil; ORCID: https://orcid.org/0000-0002-2479-7697 E-mail: younes.nikdelan@ime.uerj.br Younes Nikdelan Modular vector fields and CY modular forms 101 (Fourier series) with integer coefficients. We call these q-expansions CY modular forms and it is verified that the space generated by them has a canonical sl2(C)-module structure which provides it with a Rankin-Cohen algebraic structure. All these concepts are explicitly established for n = 1, 2, 3, 4.