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For time-independent fields the Aharonov-Bohm effect has been obtained by idealizing the coordinate space as multiply-connected and using representations of its fundamental homotopy group to provide information on what is physically identified as the magnetic flux. With a time-dependent field, multiple-connectedness introduces the same degree of ambiguity; by taking into account electromagnetic fields induced by the time dependence, full physical behavior is again recovered once a representation is selected. The selection depends on a single arbitrary time (hence the so-called holonomies are not unique), although no physical effects depend on the value of that particular time. These features can also be phrased in terms of the selection of self-adjoint extensions, thereby involving yet another question that has come up in this context, namely, boundary conditions for the wave function.

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I argue that the Holonomy Interpretation, at least as it has been presented in Richard Healey's Gauging What's Real, faces serious problems. These problems are revealed when certain approximations and idealizations that are innate in the original formulation of the Aharonov-Bohm effect are thrust aside; in particular, when the temporal dimension is taken into account. There are two ways in which time re-appears in the picture: by considering complete solutions to the original problem, where the magnetic flux is static, and by examining the effects of time dependent magnetic fluxes. Both cases expose explanatory gaps in the Interpretation, as well as conflicts between it and customary ideas about relativistic locality and local action on which the Interpretation depends.

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