Genus zero Gopakumar-Vafa invariants of the Banana manifold

TL;DR

This paper computes genus zero Gopakumar-Vafa invariants for the Banana manifold, a specific Calabi-Yau threefold, revealing their generating function as a weak Jacobi form and interpreting invariants as counts of certain genus 0 curves.

Contribution

It provides the first explicit calculation of genus zero Gopakumar-Vafa invariants for the Banana manifold and links these invariants to a weak Jacobi form and geometric curve counts.

Findings

Invariants form a weak Jacobi form of weight -2 and index 1.

Invariants count structure sheaves of possibly nonreduced genus 0 curves.

Explicit formula for invariants related to the universal cover of singular fibers.

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 $X_{\text{Ban}}\to \mathbf{P}^1$.