Abstract
Several criteria are known for determining which connectionsA are determined uniquely by their curvatureF, or byF and its covariant derivatives. On a principal bundle with semi-simple gauge groupG over a 4-manifoldM, a sufficient condition forF to determineA uniquely is that the linear mapB → [F ∧B] from Lie algebra-valued 1-forms to 3-forms (pulled back toM via a local gauge) be invertible on an open dense set inM. Recently F. A. Doria has claimed that this condition is also necessary. We present counterexamples to this claim, and also to his assertion thatF determinesA uniquely if the restriction of the bundle to every open subset ofM has holonomy group equal toG andF is “not degenerate as a 2-form over spacetime.”
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