Open/Closed BPS Correspondence and Integrality

We prove the integrality and finiteness of open BPS invariants of toric Calabi–Yau 3-folds relative to Aganagic–Vafa outer branes, defined from open Gromov–Witten invariants by the Labastida–Mariño–Ooguri–Vafa formula. Specializing to disk invariants, we extend the open/closed correspondence of Gromov–Witten invariants to BPS invariants and prove the integrality of a class of genus-zero BPS invariants of toric Calabi–Yau 4-folds, thereby providing additional examples for the conjecture of Klemm–Pandharipande.