Recently I have been studying mathematical gauge theory and have come across some confusion regarding the connection between PFBs and physical gauge theories. I have a primarily mathematical background.
By my understanding, a gauge theory over a group $G$ essentially corresponds to a principal $G$-bundle over Minkowski space. However, since Minkowski space has the topology of $\mathbb{R}^4$, it is contractible and thus only admits trivial bundles, that is, the bundle is diffeomorphic to $\mathbb{R}^4\times G$ and thus admits a global section. But according to Wikipedia, Gribov ambiguities in nonabelian gauge theories arise because there are no global sections over the PFBs being used in such theories. Thus a contradiction arises between paragraphs.
I assume I am misunderstanding the $G$-bundle that nonabelian gauge theories rely on, however, I have been unable to find any resources suggesting otherwise, as Hamilton's Mathematical Gauge Theory seems to imply we are working on $G$-bundles over $\mathbb{R}^4$. I found this question, however, it does not resolve the contradiction with Gribov ambiguities.
My question then is how can we have a gauge theory with a Gribov ambiguity, since if we are working over $\mathbb{R}^4$ as our base space, any $G$-bundle will have a smooth global section?
