I have a question about the meaning of Yang-Mills Functional.

It is stated everywhere that the Yang-Mills Functional is a measure of energy. But the formal definition of the Yang-Mills Functional is:

To have a manifold $M$ together with a smooth vector bundle $E\longrightarrow M$

To a given connection $A$ over $E\longrightarrow M$, the Yang-Mills Functional assigns the integral of the norm of the curvature of $A$.

I understand the mathematical background, but **why does it represent energy?** Is there any intuitive way to explain this link?

Any idea or suggestion is welcome.

isa function, so the analogy is rather direct. More precisely, connections are in 1-1 correspondence with sections of a certain affine bundle. This is spelled out explicitly for the case of the Levi-Civita connection of (pseudo-)Riemannian geometry in the answers to this question. $\endgroup$