Genus zero Gopakumar–Vafa invariants of multi-banana configurations

The multi-Banana configuration $\widehat{F}_\mathrm{\small{MB}}$ is a local Calabi–Yau threefold of Schoen type. Namely, $\widehat{F}_\mathrm{\small{MB}}$ is a conifold resolution of $\widehat{I}_v \times_D I_w$, where $\widehat{I}_v \to \mathbf{D}$ is an elliptic surface over a formal disc $\mathbf{D}$ with an $I_v$ singularity 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 $\mathbf{C}^2$, or classical theta functions, respectively.

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Published 5 January 2024