Integral Cohomology and Mirror Symmetry for Calabi-Yau 3-folds

TL;DR

This paper computes the integral cohomology of Calabi-Yau 3-folds from toric hypersurfaces, revealing torsion phenomena and mirror symmetry exchanges between cohomology groups.

Contribution

It provides the first comprehensive calculation of integral cohomology, including torsion, for all toric hypersurface Calabi-Yau 3-folds, highlighting mirror symmetry effects.

Findings

32 families have non-trivial torsion in cohomology.

Torsion in H^2 and H^3 are exchanged under mirror symmetry.

Mirror symmetry relates the Picard and Brauer groups via torsion exchange.

Abstract

In this paper, we compute the integral cohomology groups for all examples of Calabi-Yau 3-folds obtained from hypersurfaces in 4-dimensional Gorenstein toric Fano varieties. Among 473 800 776 families of Calabi-Yau 3-folds $X$ corresponding to 4-dimensional reflexive polytopes there exist exactly 32 families having non-trivial torsion in $H^*(X, \Z)$. We came to an interesting observation that the torsion subgroups in $H^2$ and $H^3$ are exchanged by the mirror symmetry involution, i.e. the torsion subgroup in the Picard group of $X$ is isomorphic to the Brauer group of the mirror $X^*$