Integrality of mirror maps for the Batyrev mirror pairs studied

Prove the integrality of mirror maps for the Batyrev mirror pairs considered, namely demonstrate that the mirror map b() for these pairs has integral (ℤ) coefficients when the constructions are carried out over ℤ rather than over the coefficient field 𝔟𝑘, in analogy with the Greene–Plesser case.

Background

The mirror map b() identifies deformation parameters on the B-side with symplectic/Kähler parameters on the A-side. Beyond proving HMS, the paper discusses a refined version where b() is specified by explicit formulae (Cox–Katz §6.3.4). Prior work established integrality of mirror maps in the Greene–Plesser setting, and the authors expect an analogous result for the Batyrev mirror pairs studied here when working over ℤ.

However, the paper does not undertake this integrality proof. The statement thus points to an open arithmetic aspect of mirror symmetry for these toric Calabi–Yau hypersurfaces, namely that the mirror map’s coefficients should be integral once the entire construction is performed over ℤ.