The stringy geometry of integral cohomology in mirror symmetry

TL;DR

This paper explores how torsion cohomology in Calabi-Yau three-folds influences mirror symmetry, revealing new topological structures like flat gerbes that refine the duality beyond traditional geometric considerations.

Contribution

It identifies two torsion subgroups in cohomology and shows how they affect mirror symmetry through orbifold constructions and flat gerbes, extending the understanding of dualities.

Findings

Torsion cohomology encodes orbifold and gerbe data.

Inclusion of flat gerbes refines mirror symmetry.

Topology of gerbes affects duality structure.

Abstract

We examine the physical significance of torsion co-cycles in the cohomology of a projective Calabi-Yau three-fold for the (2,2) superconformal field theory (SCFT) associated to the non-linear sigma model with such a manifold as a target space. There are two independent torsion subgroups in the cohomology. While one is associated to an orbifold construction of the SCFT, the other encodes the possibility of turning on a topologically non-trivial flat gerbe for the NS-NS B-field. Inclusion of these data enriches mirror symmetry by providing a refinement of the familiar structures and points to a generalization of the duality symmetry, where the topology of the flat gerbe enters on the same footing as the topology of the underlying manifold.