Elliptic genus and modular differential equations

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 CY3 satisfies a differential equation of degree one with respect to the heat operator. For a K3 surface or any CY5 the degree of the differential equation is 3. We prove that for a general CY4 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.