Starting from the integral definition of gauge fields, and using a formularesembling the Gauss-Bonnet theorem - a theorem in differential geometryon compact manifolds, we derive a general conjugate relationship between the gauge charge and the dual charge. The relation between electronic charge and the magnetic monopole is an example of this conjugate relationship. For an SO (3) gauge group with a U (1) group as its invariant subgroup, we obtain the 't Hooft monopole solution as a special solution, without introducing any concept of singular strings or any mechanism such as the spontaneous breaking of symmetries.
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