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I'm studying $SU(N)$ gauge theory, but I'm confused.

Here(Gauge fields -- why are they traceless hermitian?), the reason why gauge field is in Lie albgebra of gauge group $G$ is that we have to cancel out the term which comes from Kinetic term by acting gauge transformation.

For simplification, I want to use the $F_{\mu \nu}$. I know this transforms as $$ F_{\mu \nu} \to gF_{\mu\nu}g^{-1} $$ and it's called adjoint representation ($g\in G$). However, is it true that $F_{\mu \nu}$ belongs to the Lie algebra, just because it undergoes a change as an adjoint representation? In my understanding, adjoint representation means that an element of a certain set $x\in X$ changes $$ Ad(g)x=gxg^{-1} $$ and then $x$ does not have to be in $G$. For example, we can show this act becomes representation; $Ad(g_1)Ad(g_2)=Ad(g_1g_2)$, even if $x\notin G$.

Anyway, My question is:

Why adjoint representation have the important role for $A_{\mu}, F_{\mu \nu}\in$ (Lie algebra of $G$)?

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  • $\begingroup$ The important thing is that $A_\mu$ is a Lie-algebra valued one-form $\endgroup$ – Mathphys meister Dec 4 '20 at 7:38
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The field strength $F_{\mu \nu}$ is a (local representation of) a Lie-algebra valued two-form. In components, it is often written as $F_{\mu \nu}^a$, where the $\mu \nu$ are the space-time indices (making it a two-form) and a is the Lie-algebra index. As such, it is indeed in the Lie-algebra.

However, it is not any Lie-algebra valued two form. Instead, it is the (local representation of) the covariant derivative of a special one-form, the connection one-form (whose local representation is usually denoted by $A_{\mu}$). This necessarily transforms in the adjoint representation. An intuitive reasoning for this can be found in this question (1).

This is the mathematical viewpoint. The physical viewpoint is that the adjoint rep gives the "correct" transformation, simply because of the way we have defined the field strength in non-abelian gauge theories. Specifically, the field strength is defined as (this definition comes from the differential geometry picture sketched above): $$ F_{\mu \nu} = \partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}-i[A_{\nu},A_{\mu}] $$ And if you perform a gauge transformation of this object (as is done e.g. here (2), you find that it transforms in the adjoint representation.

(1)Why quarks in the fundamental and gluons in the adjoint? (2)https://www.worldscientific.com/doi/pdf/10.1142/9789813234192_0001

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