A Higgs mechanism for Abelian theories over non-trivial background flat connections is proposed. It is found that the mass generated for the spin 1 excitation is the same as the one obtained from the standard Higgs mechanism over trivial backgrounds, however, the dynamical structure of the action for the Higgs scalar is completely different from the usual approach. There is a topological contribution to the mass term of the Higgs field. After functional integration over all backgrounds, it is shown that the action for the massive spin 1 excitation is dual to the Topologically Massive Models in any dimension.
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