Suppose we want to consider Chern-Simons theory on an (odd-dimensional) compact boundaryless smooth manifold $X$ for a Lie group (the "structure/gauge group") $G$.
Is it possible to do so without using a principal bundle on $M$ with connection? (Or perhaps this essentially uses the trivial principal bundle?)
As follows:
We take our "gauge field", labeled $A$, to be a section of $\Omega^1(X,\frak{g})$.
We define a "gauge transformation" by $g : X\rightarrow G$ on $A$ to be given by $A \mapsto \mathrm{Ad}_g A + (g^{-1})^*\theta$ where $\theta$ is the Maurer-Cartan form on $G$ (and $g^{-1}$ means pointwise $G$-inversion of $g$), following the formula at this Wikipedia page. (For matrix groups this is $A \mapsto gAg^{-1} + g^{-1}dg$.)
We take as our field Lagrangian $k \omega_A$ where $\omega_A$ is the Chern-Simons form (as defined at Wikipedia) for $A$, and the action $S$ is the integral over $X$ of the Lagrangian density. Here $k$ is a (real) constant.
We verify that although the Chern-Simons form isn't invariant under gauge transformations of $A$, it changes by a total derivative at least when the gauge transformation is connected to the identity (a "small" gauge transformation), so that the value of the Chern-Simons action integral $S$ over $X$ (which is boundaryless) is unchanged by such a gauge transformation.
So now we can consider the classical dynamics of $A$ on $X$ without reference to any principal bundle on $X$? Or, is this essentially just the case of using the trivial principal bundle $X\times G \xrightarrow{\mathrm{proj}_1} X$? Is there a way to extend the above to nontrivial principal bundles?
Now, if we consider the quantum action $\mathrm{exp}(\frac{i}{\hbar}S)$, we can verify that even if we now allow gauge transformations not connected to identity, the classical action $S$ shifts by an overall constant times some "winding number" (as stated on p. 354 in Witten's paper on QFT and the Jones polynomial), such that the quantum action is unchanged if the constant $k$ in our theory is appropriately quantized.
With the choice of $k$ made, if we want to consider the quantum path integral for our theory, would its space of field configurations $\mathcal{M}$ be the space of sections $\Omega^1(X,\frak{g})$, quotiented by gauge transformations (both large or small, i.e. whether or not connected to identity)?
Also, does anyone have a reference explaining (or can explain) how to show how $S$ changes under a "large" gauge transformation (not connected to identity) by an overall topological term? (And is this only for 3-manifolds, which is the only case considered in Witten's paper unless I've missed something.)
Finally, should we expect that the quantum path integral - i.e. the integral of the quantum action $\exp(iS[A]/\hbar)$ over the space $\mathcal{M}$ of field configurations (on $X$), for some appropriate measure on $\mathcal{M}$, which may depend on choices of metrics on $\frak{g}$, $G$, $X$, etc - should be a topological invariant of $X$?
More generally, should the path integral (over $\mathcal{M}$) of $\exp(iS[A]/\hbar)\,W(\gamma_1,\rho_1,A)\cdots W(\gamma_k,\rho_k,A)$ be an isotopy invariant of links $\gamma : (S^1)^{\sqcup k} \hookrightarrow X$? Here $\gamma : (S^1)^{\sqcup k} \hookrightarrow X$ is an embedded link in $X$ with assigned $G$-representations $\rho_i$ to each link component, and $W(\gamma_i,\rho_i,A)$ is the Wilson loop (i.e. holonomy for the associated principal bundle connection?) of $A$ around $\gamma_i$, traced over $\rho_i$. (To use Wilson lines, do we need to assume $G$ is a matrix group?)