Elliptic genus and modular differential equations

TL;DR

This paper investigates modular differential equations satisfied by elliptic genera of Calabi--Yau varieties, revealing degrees of these equations vary with the dimension and providing explicit examples and analogies to known modular form equations.

Contribution

It establishes specific degrees of modular differential equations for elliptic genera of various Calabi--Yau dimensions and introduces new examples of such equations, extending known modular form results.

Findings

Elliptic genus of CY_3 satisfies degree one differential equation.

Elliptic genus of CY_5 satisfies degree three differential equation.

CY_4 elliptic genus satisfies degree five differential equation.

Abstract

We study modular differential equations for the basic weak Jacobi forms in one abelian variable with applications to the elliptic genus of Calabi--Yau varieties. We show that the elliptic genus of any $CY_3$ satisfies a differential equation of degree one with respect to the heat operator. For a $K3$ surface or any $CY_5$ the degree of the differential equation is $3$. We prove that for a general $CY_4$ its elliptic genus satisfies a modular differential equation of degree $5$. We give examples of differential equations of degree two with respect to the heat operator similar to the Kaneko--Zagier equation for modular forms in one variable. We find modular differential equations of Kaneko--Zagier type of degree $2$ or $3$ for the second, third and fourth powers of the Jacobi theta-series.