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As Mariano points out in his comment, this follows from the definition of a connection on a principle $G$-bundle $\pi: P \to M$.

At every $p \in P$, the kernel of $\pi_* : T_pP \to T_{\pi(p)}M$ defines the vertical subspace of $T_pP$. Let's call it $V_p$. It is spanned by the fundamental vector fields of the $G$-action on $P$. Since this action is free, the fibres are principal homogeneous spaces and hence $V_p$ is isomorphic to the Lie algebra $\mathfrak{g}$. A connection (à la Ehresmann) is an equivariant choice of horizontal subspace $H_p$ complementary to $V_p$. Hence it can be defined as the kernel of a 1-form $\theta$ with values in the adjoint representation of $G$ (from equivariance of the horizontal subspace).Globally it is a 1-form on $P$ with values in the adjoint bundle, whose fibres are copies of the Lie algebra on which $G$ acts via the adjoint representation.

The gauge field in your question is then the pullback via a local section of that connection 1-form. Hence locally it is a 1-form on $M$ with values in the Lie algebra $\mathfrak{g}$.

So the reason the gauge field is $\mathfrak{g}$-valued is the equivariance of the of the connection (in the sense of Ehresmann).

If you then ask why one imposes equivariance, one answer is that it is the natural condition in this context, but perhaps someone else has a more convincing reason.

1

As Mariano points out in his comment, this follows from the definition of a connection on a principle $G$-bundle $\pi: P \to M$.

At every $p \in P$, the kernel of $\pi_* : T_pP \to T_{\pi(p)}M$ defines the vertical subspace of $T_pP$. Let's call it $V_p$. It is spanned by the fundamental vector fields of the $G$-action on $P$. Since this action is free, the fibres are principal homogeneous spaces and hence $V_p$ is isomorphic to the Lie algebra $\mathfrak{g}$. A connection (à la Ehresmann) is an equivariant choice of horizontal subspace $H_p$ complementary to $V_p$. Hence it can be defined as the kernel of a 1-form $\theta$ with values in the adjoint representation of $G$ (from equivariance of the horizontal subspace). Globally it is a 1-form on $P$ with values in the adjoint bundle, whose fibres are copies of the Lie algebra on which $G$ acts via the adjoint representation.

The gauge field in your question is then the pullback via a local section of that connection 1-form. Hence locally it is a 1-form on $M$ with values in the Lie algebra $\mathfrak{g}$.

So the reason the gauge field is $\mathfrak{g}$-valued is the equivariance of the of the connection (in the sense of Ehresmann).

If you then ask why one imposes equivariance, one answer is that it is the natural condition in this context, but perhaps someone else has a more convincing reason.