Wu-yang Ambiguity in Connection Space

A good understanding of the Wu-Yang ambiguity [1], with all its aspects and consequences, does not seem to have been as yet achieved. Activity on the subject has been intensive in the first years after its discovery [2, 3, 4, 5, 6, 7] and declined afterwards [8]. Progress has been made step by step, sometimes through the discovery of general properties of formal character [9], most of times by unearthing particular cases which elucidate special points [10]. This note is devoted to the presentation of one more formal property, which shows up in the space A of gauge potentials, or connections (“A-space”). We shall mostly use invariant notation, such as A = JaA a μdx μ for gauge potentials (connections), F = 1 2 JaF a μνdx μ ∧ dx for the field strength (curvatures), K = JaK a μdx μ for covectors in the adjoint representation of the gauge algebra, etc. Wedge products will be implicit. Consider two connection forms A and A, with curvatures F = dA+ AA and F ♯ = dA + AA.