Also, a change of position of a Dirac string corresponds to a gauge transformation.
This shows that Dirac strings are not gauge invariant, which is consistent with the fact that they are not observable.
The quantization forced by the Dirac string can be understood in terms of the cohomology of the fibre bundle representing the gauge fields over the base manifold of space-time.
for the fiber bundle M. The cohomology arises from the idea of classifying all possible gauge field strengths
Here, A is the vector potential and d represents the gauge-covariant derivative, and F the field strength or curvature form on the fiber bundle.
Informally, one might say that the Dirac string carries away the "excess curvature" that would otherwise prevent F from being a closed form, as one has that