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I understand the global gauge symmetry results conserved charges associated with the symmetry. My question is, why don't we have conserved charges associated with local gauge symmetry of the gauge fields $A$ itself? Why don't we have a conserved charge for the Lagrangian $\int \mathrm{Tr}\left( F\wedge\star F\right)$? Please elaborate in both abelian case and non-abelian case.

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