Classification of "Quaternionic" Bloch-Bundles Topological Quantum Systems of Type AII

We provide a classification of type AII topological quantum systems in dimension d = 1, 2, 3, 4. Our analysis is based on the construction of a topological invariant, the FKMM-invariant, which completely classifies "Quaternionic" vector bundles (a.k.a. "symplectic" vector bundles) in dimension . This invariant takes value in a proper equivariant cohomology theory and, in the case of examples of physical interest, it reproduces the familiar Fu-Kane-Mele index. In the case d = 4 the classification requires a combined use of the FKMM-invariant and the second Chern class. Among the other things, we prove that the FKMM-invariant is a bona fide characteristic class for the category of "Quaternionic" vector bundles in the sense that it can be realized as the pullback of a universal topological invariant.