# Simplify Yang-Mills Equation of Motion in the 1-form gauge field $A$

We know the Yang-Mills theory Equation of Motion (eom) without source

$$* D * F = * (d (* F ) + [A, (* F )])= 0.$$

My question is that what are the most simple form we can boil down this eom to its minimal?

$$* (d (* (d A + A \wedge A) ) + [A, (* (d A + A \wedge A) )])$$

$$=* d * d A + * d * (A \wedge A) +A( * (d a + A \wedge A) ) - (* (dA + A \wedge A) ) A=0$$

This is what I get. How can we massage it further in order to make it as simple as possible but similar to the Maxwell's ---

$$* d * d A + ... =0?$$ What is the simplest form of $$...$$ term?

p.s. What I got so far is that $$...$$ term is $$C=* d * (A \wedge A) +A( * (d a + A \wedge A) ) - (* (dA + A \wedge A) ) A.$$ Do we have better way to simplify this complicated $$C$$ in terms of $$A$$?

• $\star d \star dA$ is not gauge invariant, so why do you want to isolate it? – Ryan Thorngren May 20 at 21:07
• $F$ is also not gauge invariant, but only covariant. I want to see $C just for comparing the higher order terms. – annie heart May 20 at 21:17 •$F$transforms homogeneously so it's much easier to understand than$\star d \star A\$. – Ryan Thorngren May 20 at 21:33