The stringy geometry of integral cohomology in mirror symmetry

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.