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arXiv · 1503.04834

Index pairings in presence of symmetries with applications to topological insulators

Julian Grossmann·Hermann Schulz-Baldes
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Abstract

In a basic framework of a complex Hilbert space equipped with a complex conjugation and an involution, linear operators can be real, quaternionic, symmetric or anti-symmetric, and orthogonal projections can furthermore be symplectic. This paper investigates index pairings of projections and unitaries submitted to such symmetries. Various scenarios emerge: Noether indices can take either arbitrary integer values or only even integer values or they can vanish and then possibly have secondary $\mathbb{Z}_2$-invariants. These general results are applied to prove index theorems for the strong invariants of topological insulators. The symmetries come from the Fermi projection ($K$-theoretic part of the pairing) and the Dirac operator ($K$-homological part of the pairing depending on the dimension of physical space).

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BibTeXRIS

Julian Grossmann, Hermann Schulz-Baldes. 2015-12-12. Index pairings in presence of symmetries with applications to topological insulators. https://doi.org/10.1007/s00220-015-2530-6

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