Integral affine structures on spheres and torus fibrations of Calabi-Yau toric hypersurfaces I

TL;DR

This paper explores combinatorial models of integral affine structures on spheres related to Calabi-Yau toric hypersurfaces and their connection to mirror symmetry and the Strominger-Yau-Zaslow conjecture.

Contribution

It introduces a purely combinatorial description of dual integral affine structures on spheres arising from Calabi-Yau toric hypersurfaces and their relation to special Lagrangian torus fibrations.

Findings

Combinatorial models match the topological torus fibration in the large complex structure limit.

Dual pairs of integral affine structures are described on spheres.

The structures relate to the metric collapse in mirror symmetry.

Abstract

We describe in purely combinatorial terms dual pairs of integral affine structures on spheres which come from the conjectural metric collapse of mirror families of Calabi-Yau toric hypersurfaces. The same structures arise on the base of a special Lagrangian torus fibration in the Strominger-Yau-Zaslow conjecture. We study the topological torus fibration in the large complex structure limit and show that it coincides with our combinatorial model.