  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.
