I am trying to recover the transformation properties of a Yang-Mills field and I'm not sure if I am wrong or if I am misunderstanding what is meant by a Yang-Mills field.
Suppose I had a principle $G$ bundle $P \xrightarrow\pi M$ and a connection 1-form $\omega$. Given a local section $\sigma_i : U_i \to P$, I can do two things with my connection
Pull $\omega$ down to $U_i$ by defining $\omega_i = \sigma_i^*(\omega)$
I can use my section to trivialize $\pi^{-1}(U_i)$ by $h_i :U \times G \to \pi^{-1}(U_i) $ taking $(m,g) \mapsto \sigma_i(m) \cdot g $. With this I can pull back $\omega$ to $U \times G$ by $\omega \mapsto h_i^* \omega$.
As I understand it, the Yang-Mills field is $\omega_i$, i.e. the object defined on the local patch $U_i$. The claim is that under change of section the Yang-Mills field transforms as
$$\omega_j = Ad_{\Omega^{-1}}\omega_i + (R_\Omega)^*\theta$$
Where $\sigma_j = \sigma_i\cdot\Omega$ is a new section on $U_j$, $(R_\Omega)^*$ is the pushforward arising from the right $G$ action, and $\theta$ is the Maurer-Cartan form on $G$.
However if I try and compute the transformation of the what I've been calling the Yang-Mills field I get something else.
The action of $\omega_j$ on $v\in T_mU_j$ is $\omega_j(v) = \sigma_j^*\omega(v) = \omega({\sigma_j}_* v) = \omega((\sigma_i \cdot \Omega)_* v) = (R_\Omega^*\omega)({\sigma_i}_* v) = Ad_{\Omega^{-1}}[\omega({\sigma_i}_* v)] = Ad_{\Omega^{-1}}[\omega_i (v)]$
Where the first equality follows from the definition of $\omega_j$, the next from the definition of the pushforward, the next follows from $\sigma_j = \sigma_i\cdot\Omega$. The next equality follows from associativity of pushforward, the next equality follows from the definition of a connection 1 form, and the final equality follows from the definition of $\omega_i$.
All this seems to give me that $\omega_j = Ad_{\Omega^{-1}}\omega_i$.
On the other hand I know that
$$h_i^*\omega = \omega_i + \theta$$
Again, $\theta$ is the Maurer-Cartan 1-form on G. In this case under change of section the $\omega_i$ should transform as I worked out before and $\theta$ should transform as any form defined on G, namely $\theta \mapsto (R_\Omega)^*\theta$. this gives me that
$$h_j^*\omega = Ad_{\Omega^{-1}}\omega_i + (R_\Omega)^*\theta$$
But this is precisely what is claimed to be the transformation properties of the Yang-Mills field. So am I making a mistake in my calculations, or am I misunderstanding which field is the Yang-Mills field?
