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Let $\pi: P \to M$ be a principal bundle and $\omega$ a connection on it. Given a section $\sigma: M \to P$ we define Yang-Mills fields by $$A=\sigma^*\omega$$

Now since under Lorentz transformation $A$ should transform like vectors, are this Fields also, sections of associated bundles to $SO(1,3)$ orthonormal frame bundle ?

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The $A$ you define is a Lie-algebra valued 1-form on the base manifold $M$, and so is a local section of the cotangent bundle $T^*(M)$. Usually a physicist would make this clear by writing $$ A= \hat \lambda_a A^a_\mu dx^\mu $$ where $\hat \lambda_a$ is a generator of the Lie algebra. The index $\mu$ indicates the cotangent character of the gauge field.

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  • $\begingroup$ I know that $A$ is Lie-algebra valued 1-form. what i am asking if it is considered as an element of the associated bundle of the orthonormal bundle $\endgroup$ – amilton moreira Oct 30 at 12:41
  • $\begingroup$ I think all one-forms can be considered as sections of the dual of the orthonormal frame bundle. Just write $A= \hat \lambda_a A^a_m {\bf e}^{*m}$ where ${\bf e}^{*m}$ is the orthonormal co-frame ${\bf e}^{*m}= e^{*m}_\mu dx^\mu$ with $g_{\mu\nu}= \eta_{mn} e^{*m}_\mu e^{*n}_\nu$. So, in some sense, it is associated to the $O(3,1)$ principle bundle. $\endgroup$ – mike stone Oct 30 at 19:27

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