Kle​mm–Pandharipande genus-zero integrality conjecture for Calabi–Yau fourfolds

Let $X$ be a smooth projective Calabi–Yau $4$-fold, let $\beta$ be an effective curve class, and let $\gamma_i\in H^{m_i}(X,\mathbb{Z})$ be integral classes. The genus-zero Gopakumar–Vafa-type invariants $n_{0,\beta}(\gamma_1,\ldots,\gamma_n)$ are defined by

$$\sum_{\beta>0}\mathrm{GW}_{0,\beta}(\gamma_1,\ldots,\gamma_n)q^\beta=\sum_{\beta>0}n_{0,\beta}(\gamma_1,\ldots,\gamma_n)\sum_{d=1}^{\infty}d^{n-3}q^{d\beta}.$$

Klemm–Pandharipande's genus-zero integrality conjecture. The invariants $n_{0,\beta}(\gamma_1,\ldots,\gamma_n)$ are integers. This is the genus-zero integrality prediction for Calabi–Yau fourfold Gopakumar–Vafa-type invariants; computations by localization and mirror symmetry support it in many examples.