Genus zero Gopakumar-Vafa invariants of the Banana manifold

Abstract

The Banana manifold $X_{\text{Ban}}$ is a compact Calabi-Yau threefold constructed as the conifold resolution of the fiber product of a generic rational elliptic surface with itself, first studied by Bryan. We compute Katz's genus 0 Gopakumar-Vafa invariants of fiber curve classes on the Banana manifold $X_{\text{Ban}}\to \mathbf{P}^1$. The weak Jacobi form of weight -2 and index 1 is the associated generating function for these genus 0 Gopakumar-Vafa invariants. The invariants are shown to be an actual count of structure sheaves of certain possibly nonreduced genus 0 curves on the universal cover of the singular fibers of $X_{\text{Ban}}\to \mathbf{P}^1$.