Holonomy angle: 0.00 rad
Loop area: 0.00
Curvature F: 0.00
Fiber bundles formalize the notion of fields defined over a base space. A principal G-bundle P→M carries a group G as fiber. A connection A defines "horizontal" lifts of paths in M to P — the way to transport fiber elements consistently.
Parallel transport along a closed loop γ in M returns the fiber element rotated by the holonomy Hol(γ) ∈ G. For a U(1) bundle over S² (like a magnetic monopole), the holonomy equals exp(i·Ω) where Ω is the solid angle enclosed — directly giving Berry's phase.
The curvature 2-form F = dA + A∧A measures the infinitesimal holonomy. By the non-abelian Stokes theorem, Hol(γ) = exp(∫∫ F). Move the loop to change the enclosed solid angle and watch the fiber phase accumulate.