Communications in Mathematical Physics

, Volume 343, Issue 2, pp 477–513 | Cite as

Index Pairings in Presence of Symmetries with Applications to Topological Insulators

  • Julian Großmann
  • Hermann Schulz-Baldes


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 Lagrangian. 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 disordered 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).


Dirac Operator Chiral Symmetry Clifford Algebra Fredholm Operator Topological Insulator 
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© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department MathematikFriedrich-Alexander-Universität Erlangen-NürnbergErlangenGermany

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