Topological insulators and K-theory. (English) Zbl 1537.81187
Summary: We analyze topological invariants, in particular \(\mathbb{Z}_2\) invariants, which characterize time reversal invariant topological insulators, in the framework of index theory and K-theory. After giving a careful study of the underlying geometry and K-theory, we formalize topological invariants as elements of KR theory. To be precise, the strong topological invariants lie in the higher KR groups of spheres; \(\widetilde{KR}^{- j - 1}(\mathbb{S}^{D + 1, d})\). Here \(j\) is a \(KR\)-cycle index, as well as an index counting off the Altland-Zirnbauer classification of Time Reversal Symmetry (TRS) and Particle Hole Symmetry (PHS) – as we show. In this setting, the computation of the invariants can be seen as the evaluation of the natural pairing between \(KR\)-cycles and \(KR\)-classes. This fits with topological and analytical index computations as well as with Poincaré Duality and the Baum-Connes isomorphism for free Abelian groups. We provide an introduction starting from the basic objects of real, complex and quaternionic structures which are the mathematical objects corresponding to TRS and PHS. We furthermore detail the relevant bundles and \(K\)-theories (Real and Quaternionic) that lead to the classification as well as the topological setting for the base spaces.
©2024 American Institute of Physics
©2024 American Institute of Physics
MSC:
| 81T45 | Topological field theories in quantum mechanics |
| 55N15 | Topological \(K\)-theory |
| 19M05 | Miscellaneous applications of \(K\)-theory |
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