Mirror symmetry for tropical hypersurfaces and patchworking

TL;DR

This paper establishes a mirror symmetry isomorphism between tropical homology groups of mirror Calabi-Yau hypersurfaces and applies it to connect patchworking methods with the topology of real Calabi-Yau hypersurfaces.

Contribution

It proves a new mirror symmetry isomorphism for tropical Calabi-Yau hypersurfaces and links patchworking constructions to the topology of real Calabi-Yau hypersurfaces.

Findings

Mirror symmetry isomorphism between tropical homology groups

Patchworking yields connected real Calabi-Yau hypersurfaces under certain conditions

Divisor class on the mirror influences hypersurface connectivity

Abstract

In the first part of the paper, we prove a mirror symmetry isomorphism between integral tropical homology groups of a pair of mirror tropical Calabi-Yau hypersurfaces. We then apply this isomorphism to prove that a primitive patchworking of a central triangulation of a reflexive polytope gives a connected real Calabi-Yau hypersurface if and only if the corresponding divisor class on the mirror is not zero.