'Real' gerbes and Dirac cones of topological insulators

TL;DR

This paper demonstrates that 'Real' gerbes provide a geometric framework to understand the spectral and topological properties of time-reversal invariant topological insulators, especially their edge states and Dirac cones.

Contribution

It introduces the use of 'Real' gerbes to encode the discrete spectrum data of topological insulators, offering a new geometric perspective beyond counting Dirac points.

Findings

Gerbe invariant captures edge state filling of the bulk gap

Discrete spectrum data is geometrically encoded in 'Real' gerbes

Provides a topological classification of edge states in insulators

Abstract

A time-reversal invariant topological insulator occupying a Euclidean half-space determines a 'Quaternionic' self-adjoint Fredholm family. We show that the discrete spectrum data for such a family is geometrically encoded in a non-trivial 'Real' gerbe. The gerbe invariant, rather than a na\"ive counting of Dirac points, precisely captures how edge states completely fill up the bulk spectral gap in a topologically protected manner.