Given a trivialization of the line-bundle, a non-vanishing section, the U(1)-connection is given by the 1- form corresponding to the electromagnetic four-potential A as ∇ ∇ = d + i A where d means exterior derivation on the Minkowski space. The monodromy is the holonomy of the flat connection. The holonomy of a connection, flat or non flat, around a closed loop γ γ is e i ∫ ∫ γ γ A (one can show this does not depend on the trivialization but only on the connection). For a flat connection one can find a gauge transformation in any simply connected field free region(acting on wave functions and connections) that gauges away the vector potential. However, if the monodromy is nontrivial, there is no such gauge transformation for the whole outside region. In fact as a consequence of Stokes' theorem, the holonomy is determined by the magnetic flux through a surface σ σ bounding the loop γ γ , but such a surface may exist only if σ σ passes through a region of non trivial field: