Aharonov-Bohm Effect, Dirac Monopole, and Bundle Theory

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We discuss the Aharonov-Bohm (A-B) effect and the Dirac (D) monopole of magnetic charge g=12 in the context of bundle theory, exhibiting a purely geometric relation between them. If ξₐ₋B and ξD are the respective U(1)-bundles, we show that ξₐ₋B is isomorphic to the pull-back of ξD induced by the inclusion of the corresponding base spaces ι:(D₀²)*→ S². The fact that the A-B effect disappears when the magnetic flux in the solenoid equals an integer times the quantum of flux Φ₀=2π e associated with the electric charge e, reflects here as a consequence of the pull-back by ι of the Dirac connection in ξD to ξₐ₋B, and the Dirac quantization condition. We also show the necessary vanishing in ξₐ₋B of the pull-back of the Chern class c₁ in ξD.