I'm confused about application of the Hodge star operation within Yang Mills theory. Using differential forms, and given the connection A, the curvature in Yang-Mills theory is F=dA+A∧A. The Bianchi identities and the field equations are then dAF=0 and ∗dA∗F=0, where dA is the covariant derivative dAF=dF+[A,F]. Now the commutator of a P-form and Q-form is [P,Q]=P∧Q−Q∧P if either or both of P, Q are even, and [P,Q]=P∧Q+Q∧P if both P and Q are odd. I want to express the Bianci identity and the YM equations in full in terms of A and the wedge product. In full, the Bianchi identity in terms of connections and wedges is dAF=dA∧A−A∧dA+A∧dA+A∧A∧A−dA∧A−A∧A∧A=0.
Saturday, June 17, 2017
homework and exercises - Hodge star operator in Yang-Mills theory and derivation of the YM equations
But what is the corresponding full expression in terms of A for the YM equations ∗dA∗F? And/or how is it derived directly from the action ∫∗F∧F? It is the correct use of the Hodge star operation that I'm messing up on. (Can't find any notes/texts that answers this in detail.)
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