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Let $A$ be a Yang-Mills field with $A_0 = 0$ and we also have time independent scalar field $\phi$ in the adjoint representation of our gauge group with zero potential (no mass too). I have to show that $$F=*D \phi$$ satisfies equations of motion (Hodge star is taken in Euclidean space).

So EOM for Y-M fields that I'm interested in is $$D*F=0$$ Covariant derivative acts like $$D\phi = d\phi + [A,\phi]$$ Two Hodge stars are proportional to identity so my equation is $$DD\phi=0=dd\phi + d[A,\phi]+[A,d\phi +[A,\phi]]=0+[dA,\phi]+2[A, d\phi] + [A,[A,\phi]] $$ I don't really see how that can be zero. Any tips?

Edit: Also: is $DF = 0$?

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  • $\begingroup$ I guess I am trying to understand what $F$ is. Usually for $D = d + A$, a 1-form, we have $D\wedge D\phi = F\phi$. In this case $F$ is a 2-form. Your $F$ looks different from this for $F = *D\phi$ means that $F$ is a 1-form, where as a scalar $\phi$ is a 0-form. $\endgroup$ – Lawrence B. Crowell Jun 19 '16 at 2:03
  • $\begingroup$ It's a 2 form. $D\phi$ is a 1 form and you dualize over Euclidean space so you get a 2-form. And yes, connection curvature is defined in this way in this problem. $\endgroup$ – Caims Jun 19 '16 at 22:37

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