The following is based on a loose understanding of the nuts and bolts that go into Chern-Simons theory, so bear with any vagueness on my part.
Suppose I have a principal $G$-bundle $P\to M$ and I know that such bundles are classified by a finite set and in fact a cyclic group (from a calculation of $[M,BG]$). Assume also that the even-dimensional cohomology of $M$ is torsion or zero, so there's no chance of the usual Chern-Weil classes being non-zero in cohomology (this implies $M$ is odd-dimensional or non-orientable; let's assume the former with $\dim M = 2k+1$). Give $P$ a connection $A$ anyway, to see what we can do. Note that $A$ is not necessarily flat.
The question is whether we can detect the isomorphism class of $P$ using the geometry as encoded by the connection. One potential approach is as follows.
We can define the Chern-Simons forms of $A$, odd-degree differential forms on $P$, and integrate them over appropriate dimensional submanifolds of $M$. Exponentiating these integrals gives well-defined elements of $U(1)$ for each submanifold which I believe to be independent of the choice of $A$. My (first) question this:
Given the top-dimensional Chern-Simons form $CS_{2k+1}$, can $\exp(2\pi i\int_M CS_{2k+1}) \in U(1)$ be seen as living in a torsion subgroup corresponding to isomorphism classes of $G$-bundles?
Note that it is not likely that bundles $P$ are completely classified by one such invariant, but is it possible in certain cases?
More generally, can we detect isomorphism classes (at least partially) using torsion invariants derived from a connection?
There's a similar question that is about the broader problem of detecting torsion using some analytical objects: Can analysis detect torsion in cohomology?, but I'm interested in using connections.
