| 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 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). |
