Bundle gerbes for topological insulators

TL;DR

This paper reviews how bundle gerbes, a higher geometric structure, can be used to construct torsion topological invariants in topological insulators, including static and driven systems with time reversal symmetry.

Contribution

It introduces a novel geometric approach using equivariant bundle gerbes to define topological invariants in condensed matter physics.

Findings

Constructs torsion invariants for static and driven systems

Applies gerbe refinements to topological insulators

Addresses systems with time reversal invariance

Abstract

Bundle gerbes are simple examples of higher geometric structures that show their utility in dealing with topological subtleties of physical theories. I review a recent construction of torsion topological invariants for condensed matter systems via equivariant bundle gerbes. The construction covers static and periodically driven systems with time reversal invariance in 2 and 3 space dimensions. It involves refinements of geometry of gerbes that are discussed in the first lecture, the second one being devoted to the applications to topological insulators.