Does 2-form curvature $\Omega \in \Omega^2(P,\mathfrak{g})$ represent a physical quantity in gauge theory?

Question

In gauge theory, all measurable physical quantities remain invariant under a gauge transformation. I have always seen that the curvature 2-form $\Omega \in \Omega^2(P,\mathfrak{g})$ associated to a connection 1-form in a $G$-principal bundle represent the force field that you want to describe (for example $\Omega$ represents the electromagnetic field if $G=U(1)$, the strong force if $G=SU(3)$...) and so should be a physical quantity. But if you perform a gauge transformation, the curvature 2-form $\Omega$ change as $$ \Omega \rightarrow \Omega'=Ad_{g^{-1}} \circ \Omega$$

Question : Does 2-form curvature represent a physical quantity? If not, what is the object that represents the force fields?

Answer 1

1. For Abelian gauge groups, the adjoint representation is trivial so the curvature $F$ is itself gauge-invariant.

2. For non-Abelian groups, gauge-invariant local operators are given by Ad-invariant polynomial functions of the curvature, e.g. the integrand $\mathrm{tr}_\mathfrak{g}(F\wedge{\star}F)$, where $\mathrm{tr}_\mathfrak{g}$ is the trace in the adjoint also used in the construction of the Killing form. Non-local gauge invariant operators are the Wilson loops, i.e. the holonomies of the gauge field.