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Classification of “Quaternionic" Bloch-Bundles

Topological Quantum Systems of Type AII

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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 \({d\leqslant 3}\). 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.

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Correspondence to Giuseppe De Nittis.

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Communicated by N. A. Nekrasov

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De Nittis, G., Gomi, K. Classification of “Quaternionic" Bloch-Bundles. Commun. Math. Phys. 339, 1–55 (2015).

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