I have a basic and perhaps admittedly misguided question about the loops in LQG. Broadly speaking I think I understand the logic that lead to them, namely that working in the ADM formalism (using the metric and time deriv of metric as canonical coordinates/momentum) turned out to be difficult because reasons and Ashketar found a better set of coordinates to try which brought the connection representation into the mix.
However using the connection representation didn't solve everything, namely still the Gauss constraint and maybe general gauge invariance (my understanding is limited here I confess), but one could construct objects which satisfied the gauge constraint by taking the trace of the holonomy i.e. the Wilson loop, and this had the added benefit of being a basis to expand the LQR wavefunctions in which helped 'explicitly solve the Gauss constraint'.
In the case of the Wilson loop, if i imagine the connection in question being the standard Christoffel symbol and talking about parallel transport I can see how taking the trace and so on will help everyone agree on the answer regardless of coordinate system (though we also require the underlying equations of motion to be diffeomorphism invariant which is a different notion, right?). However in this example I'm thinking of some curved geometry and parallel transport around a closed loop in real physical space.
My question is: is this the space that the loops in LQG live in? I understand that the loops in some sense 'label' the space of states, but at the end of the day if I want to evaluate a Wilson loop do I need to do an integral in real coordinate space? If so, is this is the same 'space' that LQG is trying to have arise naturally? Is this a source of problematic nonlinearity? Or is it enough to identify that this space of states has nice properties and who cares where it came from originally?