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Let's have a look on a gauge theory on a trivial fiber bundle, as it is seen by mathematicians:

We have a trivial vector bundle $(E, \pi, M; V)$ with group structure. We denote the sections of $E$ by $\Gamma(E)$. Let there be a global trivialization $\Psi$. Then $\Gamma(E)$ can be identified with $C^\infty(M, V)$ and the set of all $G$-compatible connections can be identified with $\Omega^1(M, \mathfrak{g})$, where $\mathfrak{g}$ denotes the Lie-Algebra of the Gauge-Group G. Hence there is a 1-to-1 correspondence between states of the matterfield (i.e. sections) and connections on the bundle AND $C^\infty(M,V) \times \Omega^1(M, \mathfrak{g})$ for a given global trivialization. Hence the Configuration space should be given by $C = C^\infty(M,V) \times \Omega^1(M, \mathfrak{g})$.

If we change the local trivialization from $\Psi$ to $\Psi'$, this defines a transition function which acts as a gauge transformation on C. So we can relate the coordinate representations with this transition functions.

In physics now one says, that the configuration space is $C/\mathcal{G}$ where $\mathcal{G}$ is the set of all gauge transformations (compare arXiv:1512.02632 or the chapter on gauge theories in "topological solitons" by Manton/Sutcliffe). But in my opinion this makes no sense, since $C$ should already be the full configuration space.

To make my point more clear a little "paradoxon". Assume we have a $G$-vector bundle where the group structure can be reduced to a $H$-vector bundle. Then the configuration spaces of the $G$-vector bundle and the configuration space of the $H$-vector bundle would be physically different but mathematically there is a 1-to-1 correspondence between sections in the $G$-vector bundles and sections in the $H$-vector bundle and the same should be true for connections.

So what is the "true" configuration space and why?

Edit - I think I have the answer:

Considering the Yang-Mills Functional on a vector bundle one obtains, that, due to the symmetry of the Yang-Mills Functional, solutions are related by gauge transformations even if we've already chosen a local trivialization. This gauge group action corresponds to "active" bundle automorphisms, while the gauge group action on coordinate representations corresponds to the "passive" change of local trivializations.

Now, since all solutions are related via "active" Gauge-Transformations, the problem is reduced to the determination of the space $C/\mathcal{G}$. This was the mathematical point of view.

From the physical point of view we see, that we don't get a unique time evolution on $C$ but a unique time-evolution on $C/\mathcal{G}$, such that it makes sense to identify $C/\mathcal{G}$ with the configuration space of the physical system. Nevertheless for me the question remains, if the "size" of the equivalence classes in $C/G$ has any physical implication.

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