  In the gap topology, the unbounded self-adjoint Fredholm operators on a
Hilbert space have 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 interpolate
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.
