The monodromy of the flat connection only depends on the topological type of the loop in the field free region (in fact on the loops homology class). The holonomy description is general, however, and works inside as well as outside the superconductor. Outside of the conducting tube containing the magnetic field, the field strength F = 0 . In other words, outside the tube the connection is flat, and the monodromy of the loop contained in the field-free region depends only on the winding number around the tube. The monodromy of the connection for a loop going round once (winding number 1) is the phase difference of a particle interfering by propagating left and right of the superconducting tube containing the magnetic field. If one wants to ignore the physics inside the superconductor and only describe the physics in the outside region, it becomes natural and mathematically convenient to describe the quantum electron by a section in a complex line bundle with an "external" flat connection ∇ ∇ with monodromy