Communications in Mathematical Physics

, Volume 339, Issue 1, pp 1–55 | Cite as

Classification of “Quaternionic" Bloch-Bundles

Topological Quantum Systems of Type AII
  • Giuseppe De Nittis
  • Kiyonori Gomi


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.


Vector Bundle Line Bundle Chern Class Topological Insulator Equivariant Cohomology 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department MathematikUniversität Erlangen-NürnbergErlangenGermany
  2. 2.Department of Mathematical SciencesShinshu UniversityNaganoJapan

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