Index Pairings in Presence of Symmetries with Applications to Topological Insulators

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).