# Are Yang-Mills Fields sections of associated bundles to the orthonormal frame bundle?

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 ?

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.
• 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 – amilton moreira Oct 30 '19 at 12:41
• 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. – mike stone Oct 30 '19 at 19:27