Abstract
We show that a gauge field uniquely determines its potential if and only if its holonomy group coincides with the gauge group on every open set in spacetime, provided that the field is not degenerate as a 2-form over spacetime. In other words, there is no potential ambiguity whenever such a field is irreducible everywhere in spacetime. We then show that the ambiguous potentials for those gauge fields are partitioned into gauge-equivalence classes (modulo certain homotopy classes) as a consequence of the nontrivial connectivity of spacetime. These homotopy classes depend on the gauge group, on the holonomy group and on this last group's centralizer in the gauge group.
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We exclude the cases of connections which are kept invariant when restricted to a bundle over a lower-dimensional submanifold ofM. In the 3-dimensional case the connection form when extended to a 4-dimensionalM may lead to the degeneracy of *ℱ without however implying the existence of copies over the 3-manifold (see [37]). The 2-dimensional case is unique in that every 2-curvature is copied. See Deser and Wilczek in [1]
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Communicated by R. Stora
To the Memory of Jorge André Swieca
Research supported by C.N.Pq. and M.E.C. (Brazil)
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Doria, F.A. The geometry of gauge field copies. Commun.Math. Phys. 79, 435–456 (1981). https://doi.org/10.1007/BF01208502
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DOI: https://doi.org/10.1007/BF01208502