Meromorphic modular differential equations for the elliptic genus of Calabi--Yau varieties

Source abstract

We study differential equations -- not necessarily monic -- for Jacobi modular forms. As an application, we derive a second-order differential equation for the elliptic genus of a $K3$ surface involving the logarithmic derivative of the modular invariant $j(τ)$, and we explicitly find a third-order equation with a leading coefficient of weight 12 for the elliptic genus of any four-dimensional Calabi--Yau variety. Furthermore, we propose a simple recursive algorithm for investigating second-order differential equations with leading coefficient $E_4(τ)$ or $E_6(τ)$.