‘Real’ Gerbes and Dirac Cones of Topological 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.