Facts
- The mathematical framework of a gauge theory is that of principal $G$-bundles, where $G$ is a Lie group (representing some physical symmetry); let ${\pi}\colon{P}\to{M}$ be such a bundle.
- A connection 1-form on $P$ corresponds to a gauge field, and the curvature 2-form associated to it corresponds to the field strength tensor.
- A (local) section of $\pi$ serves as a (local) reference frame (alias, gauge) for an observer on $M$. In particular, if $s\colon{U}\to{P}$ is such a section, where $U$ is an open subset of $M$, then an observer on $U$ describes elements of $\Omega^{k}\left({\pi}^{-1}(U),\mathfrak{g}\right)$ by pulling them back by $s$.
- Specifically, let $A\in\Omega_{con}^{1}\left(P,\mathfrak{g}\right)$ be a connection 1-form and ${F^A}\in\Omega_{hor}^{2}\left(P,\mathfrak{g}\right)^{\operatorname{Ad}}$ the associated curvature 2-form. Then, our observer on $U$ interprets ${A_s}:={s^*}A$ and ${F_s^A}:={s^*}{F^A}$ as the gauge field and the field strength tensor, respectively, in that specific gauge, viz. $s$.
Conclusion: In a certain gauge, any connection 1-form and any curvature 2-form define, respectively, a unique element of $\Omega^{1}\left(M,\mathfrak{g}\right)$ and $\Omega^{2}\left(M,\mathfrak{g}\right)$.
Now, a physicist's approach on gauge theory is to define the space of gauge fields as $\Omega^{1}\left(M,\mathfrak{g}\right)$, when a particular gauge has (implicitly) been chosen. So my question is the following:
"does the whole $\Omega^{1}\left(M,\mathfrak{g}\right)$ qualify as the set of gauge fields, in the sence that there is one-to-one correspondence between $\Omega_{con}^{1}\left(P,\mathfrak{g}\right)$ and $\Omega^{1}\left(M,\mathfrak{g}\right)$, when a gauge is chosen? Or are there elements in $\Omega^{1}\left(M,\mathfrak{g}\right)$ that do not qualify as gauge fields, in the sence that they are not pullbacks of connection 1-forms by the chosen gauge?"
Probably, the question is of mathematical nature and can be answered by rigorous calculations. If one's mathematical background is such that they can provide a rigorous proof, then they are kindly requested to do so (or at least provide a sketch of that proof). However, if there is an intuitive way to approach the answer, I would be glad to know about that.