The Fermi gerbe of Weyl semimetals

In the gap topology, the unbounded self-adjoint Fredholm operators on a Hilbert space has third homotopy group the integers. We realise the generator explicitly, using a family of Dirac operators on the half-line, which arises naturally in Weyl semimetals in solid-state physics. A "Fermi gerbe" geometrically encodes how discrete spectral data of the family interpolates between essential spectral gaps. Its non-vanishing Dixmier-Douady invariant protects the integrity of the interpolation, thereby providing topological protection of the Weyl semimetal's Fermi surface.