Eisenstein type series for Calabi-Yau varieties

TL;DR

This paper introduces a differential equation linked to Calabi-Yau varieties, generalizing Ramanujan's Eisenstein series, with implications for mirror symmetry and string theory.

Contribution

It presents a novel differential equation for Calabi-Yau families, extending classical Eisenstein series concepts to complex algebraic geometry.

Findings

The differential equation is satisfied by seven specific functions in q-expansion form.

The Yukawa coupling is shown to be rational in these functions.

The work generalizes Ramanujan's differential equation to Calabi-Yau contexts.

Abstract

In this article we introduce an ordinary differential equation associated to the one parameter family of Calabi-Yau varieties which is mirror dual to the universal family of smooth quintic three folds. It is satisfied by seven functions written in the $q$-expansion form and the Yukawa coupling turns out to be rational in these functions. This is a generalization of the Ramanujan differential equation satisfied by three Eisenstein series.