Principal fiber bundle: A gauge field lives on a bundle over a base space. Parallel transport around a loop returns with a holonomy (Berry phase). For a magnetic monopole (Dirac) of charge g, the holonomy around a loop at elevation θ is γ = g·Ω, where Ω is the solid angle.

The non-triviality of the bundle (Chern number = g) is a topological obstruction — you cannot define a global gauge.