Saturday, June 17, 2017

homework and exercises - Hodge star operator in Yang-Mills theory and derivation of the YM equations


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+AA. The Bianchi identities and the field equations are then dAF=0 and dAF=0, where dA is the covariant derivative dAF=dF+[A,F]. Now the commutator of a P-form and Q-form is [P,Q]=PQQP if either or both of P, Q are even, and [P,Q]=PQ+QP 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=dAAAdA+AdA+AAAdAAAAA=0.

But what is the corresponding full expression in terms of A for the YM equations dAF? And/or how is it derived directly from the action FF? 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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