Fiber Bundle & U(1) Gauge Field

Principal bundle: P → M
Structure group: U(1) = {e^{iθ}}
Base M: circle S¹
Fiber: U(1) phase circle

Connection 1-form A (gauge field)
Curvature F = dA (EM field tensor)
Holonomy: γ = exp(i∮A·dx)

Aharonov-Bohm: even with B=0 outside solenoid, ∮A≠0 causes observable phase shift. Physics = holonomy, not gauge field locally.

Drag slider to change flux.
Note: holonomy is gauge-invariant
despite fiber phases changing
under gauge transformation.