Genus zero Gopakumar-Vafa invariants of Multi-Banana configurations

TL;DR

This paper computes genus zero Gopakumar-Vafa invariants for multi-Banana configurations, a class of local Calabi-Yau threefolds, using generalized techniques and expressing results via elliptic genera and theta functions.

Contribution

It extends previous methods to compute invariants for complex fiber curve classes in multi-Banana configurations, providing explicit formulas for specific cases.

Findings

Explicit computation for v=1 and v=w=2 cases

Partition functions expressed via elliptic genera and theta functions

Generalization of earlier techniques to new geometric configurations

Abstract

The multi-Banana configuration $\widehat{F}_{mb}$ is a local Calabi-Yau threefold of Schoen type. Namely, $\widehat{F}_{mb}$ is a conifold resolution of $\widehat{I}_v \times_{\bf{D}} \widehat{I}_w$, where $\widehat{I}_v \to {\bf{D}}$ is an elliptic surface over a formal disc ${\bf{D}}$ with an $I_v$ singulararity on the central fiber. We generalize the technique developed in our earlier paper to compute genus 0 Gopakumar-Vafa invariants of certain fiber curve classes. We illustrate the computation explicitly for $v=1$ and $v=w=2$. The resulting partition function can be expressed in terms of elliptic genera of $\bf{C}^2$, or classical theta functions, respectively.