quantum field theory - Is gauge connection unique?

In QFT, given a gauge group and matter field, is the form of the gauge field unique? In other words, given a principal G-bundle and its associated vector bundle, is the construction of the principle G-connection unique?

This is related to the other question (here: Gauge Field Tensor from Wilson Loop) where the author implies that the gauge field is natural/unique given the matter field. May be it is, but I wanted to confirm (edit: From answer below, the gauge connection is not unique)

Because a gauge connection (or a gauge field) can exist independent of the vector matter field (as in pure gauge theories), non-uniqueness of the connection would imply a symmetry on the connection itself.

Answer

The gauge connection is not unique, and this has nothing to do with the presence of matter fields. Let $\Sigma$ be our space-time, $P$ a principal $G$-bundle, and $\mathcal{A}$ the space of connections on $P$. Then, gauge transformations $t : P \to G$, forming the group of gauge transformations $\mathcal{G}$ have an action on $\mathcal{A}$ given by

$$A \overset{t}{\mapsto} tAt^{-1} + t \mathrm{d}t$$

and the space of physically different connections is $\mathcal{A}/\mathcal{G}$.

Side note: Unfortunately, the naive way of taking this quotient does not quite succeed in producing a manifold we could integral the path integral over, since there are so-called reducible connections corresponding to "corners" in the resulting almost-manifold (I think it is technically an orbifold), and since the action of $\mathcal{G}$ on $\mathcal{A}$ is not free if the center of $G$ is non-trivial.