Details
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. The magnetic charges of a gauge field theory can be understood to be the group generators of the cohomology group for the fiber bundle M. The cohomology arises from the idea of classifying all possible gauge field strengths, which are manifestly exact forms, modulo all possible gauge transformations, given that the field strength F must be a closed form: . 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 everywhere except at the location of the monopole.
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