This is a very basic beginners question about Chern-Simons theory. The configurations that we sum over to get the partition function are given by a Lie-algebra valued 1-form $A$ on a topological 3-manifold, called the connection. More precisely, $A$ is a connection of a principal Lie-group bundle over the manifold.


What is the role of this principal bundle?


I didn't find this spelled out explicitly anywhere, but I assume one does not sum over different choices of principle bundles when calculating the partition function? Or are different choices of principle bundle already encoded in different choices of $A$?


So is Chern-Simons theory not only a topological field theory that one can put on any topological manifold, but one that we can additionally put on any topological manifold with a Lie-group principle bundle? Can one restrict to trivial principle bundles without loosing any of the physical interpretation? If one talks about the "3-manifold invariants" of the Chern-Simons theory, does that refer to the partition function on the trivial principle bundle over those manifolds? Or is the partition function independent of the choice of bundle?



Importance of the principal bundle in Chern-Simons theory
In quantum Chern-Simons theory with gauge group $G$ (compact Lie), a field on a 3-manifold $M$ is a principal $G$-bundle with a connection $A$. The partition function/path integral associated to $M$ is supposed to be the integral over the (generally infinite-dimensional) space ("stack") $\mathcal{F}$ of all principal $G$-bundles $P$ over $M$ equipped with a choice of connection $A$ (modulo gauge equivalence) of $\exp(iS(A))$, where $S$ is the classical Chern-Simons action.


Perhaps it is easier for pedagogical purposes to explore the simpler case of Chern-Simons theory when $G$ is finite; this is Dijkgraaf-Witten theory. In this case, one can make the notion of an integral over $\mathcal{F}$ perfectly rigorous. If $G$ is finite, then each bundle has a unique connection, so $\mathcal{F}$ is precisely $[M, BG]$, where $BG$ is the classifying space of $G$. The Lie algebra of $G$ is also trivial, so the action vanishes, and we're left with integrating (summing) $1$ over the space $[M,BG]$ with respect to some measure; the weight of a principal $G$-bundle $P$ in this case is $1/|\mathrm{Aut}(P)|$. In other words, the finite-group version of the Chern-Simons partition function is
$$Z(M) = \sum\_{P\in [M,BG]} \frac{1}{|\mathrm{Aut}(P)|}.$$


This is the literal analogue of Chern-Simons theory for a finite gauge group; however, a more interesting analogue is *twisted* Dijkgraaf-Witten theory, which might be what you were trying to get at. Recall that the action associated to the field $(P,A)$ over $M$ is $S(A) = \int\_M q(A)$; what matters is that the integrand $q(A)$ is a $3$-form. Since a Chern-Simons field on $M$ is a principal $G$-bundle with a choice of connection $A$, one might attempt to "canonically" associate to each principal $G$-bundle a $3$-form on $M$ in the finite group case; this $3$-form would be the replacement of $A$. Note that in this story, $A$ plays a somewhat different role.


Think of the $3$-form as a singular cochain on $M$, so integration is pairing the cochain with the fundamental class of the manifold $M$ (assume it's closed and oriented). Since a field on our manifold is still a principal $G$-bundle, determined by a map $f\_P:M\to BG$, we can associate to each bundle a $3$-dimensional cohomology class if we fix a choice of $\alpha\in \mathrm{H}^3(BG;\mathbf{C}^\times)$; then, the $3$-form "$q(A)$" associated to $P$ is $f\_P^\ast(\alpha)\in \mathrm{H}^3(M;\mathbf{C}^\times)$. (Really, one should fix a $3$-cocycle in $Z^3(BG;\mathbf{C}^\times)$.) The action associated to $P$ is then $\langle [M], f\_P^\ast(\alpha)\rangle$, and to obtain the quantum theory, one can now integrate over the space of all $G$-bundles (with the same measure as the untwisted case). Note that $A$ by itself doesn't appear in this story; only the analogue of the associated $3$-form does.


