Integrality of Calabi–Yau four-fold Gopakumar–Vafa type invariants

Prove that the genus-zero and genus-one Gopakumar–Vafa type invariants defined from Gromov–Witten invariants of compact Calabi–Yau 4-folds are integers.

Background

Klemm–Pandharipande define genus-zero and genus-one Gopakumar–Vafa type invariants for compact Calabi–Yau 4-folds through generating-series identities involving Gromov–Witten invariants. Their proposed integrality statement concerns both types of invariants.

The paper records this integrality assertion as a conjecture and does not establish it in general; it only discusses verification in examples and uses the corresponding invariants in its local calculations.