Fiber Bundle — Connection, Curvature & Gauge Theory
Principal U(1) bundle over S² · Parallel transport · Holonomy = Berry phase
Monopole charge n1
Loop latitude θ45°
Loop speed1.0
A U(1) principal bundle over S² with connection A = n(1−cosθ)dφ/2 has curvature F = (n/2)sinθ dθ∧dφ. Parallel transport around a loop at latitude θ₀ gives holonomy γ = exp(i·n·Ω/2) where Ω = 2π(1−cosθ₀) is the solid angle — this is the Dirac monopole and Aharonov-Bohm phase.