Fiber Bundles are the geometric language of gauge theory. A principal G-bundle P → M has a base manifold M (e.g. spacetime), fibers G (e.g. U(1), SU(2), SU(3)), and a connection A (gauge field).
Dψ = dψ + A ∧ ψ (covariant derivative)
F = dA + A ∧ A (curvature = field strength)
Holonomy is the transformation a fiber vector acquires after parallel transport around a closed loop γ: hol(γ) = P exp(∮_γ A). On S², the holonomy of a loop enclosing solid angle Ω is a rotation by Ω — the Berry phase for a spin in a magnetic field. Non-Abelian gauge holonomy = Wilson loops (QCD: quark confinement). The visualization shows parallel transport of a tangent vector around a spherical triangle — the vector rotates by the enclosed solid angle.