Index pairings in presence of symmetries with applications to topological insulators

TL;DR

This paper studies how symmetries affect index pairings of projections and unitaries, with applications to topological insulators, revealing diverse index behaviors and proving related index theorems.

Contribution

It introduces a comprehensive framework for index pairings under various symmetries and applies these results to establish index theorems for topological insulators.

Findings

Index pairings can take arbitrary, even, or vanishing values with secondary invariants.

The framework classifies index behaviors based on symmetries in complex Hilbert spaces.

Proves index theorems for strong invariants of 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).