Hi all. The question I have should be a rather simple one, but I just can't think it through.


So the Chern-Simons action reads
\begin{equation}
S = \int\_M {\rm tr} (A\wedge dA + \frac{2}{3} A\wedge A \wedge A)
\end{equation}
where $M$ is 3-fold, and similarly for higher dimensional manifold.


Now, my question is:


\**since $A$, the connection 1-form is only defined patch by patch, what do we really mean by doing the integration? \**


It would be understandable if I write
\begin{equation}
S = \int\_M {\rm tr} \left[(A-A\_0)\wedge d(A-A\_0) + \frac{2}{3} (A-A\_0)\wedge (A-A\_0) \wedge (A-A\_0)) \right ]
\end{equation}
where $A\_0$ is some reference connection, since $A-A\_0$ is globally defined 1-form valued in ${\rm Lie}G$.


I see that under gauge transformation (or different chart),
\begin{equation}
CS(A^g) - CS(A) = d\alpha(A,g) + Q(g)
\end{equation}
where $Q(g)$ is closed. But I don't know how I can infer the validity of doing the integration from this gauge transformation.


Thank you!



How to understand Chern-Simons action
Often in the literature by "Chern-Simons theory" is meant by default $G$-[Chern-Simons theory](http://ncatlab.org/nlab/show/Chern-Simons%20theory) whose gauge group is a connected and simply connected semisimple compact group $G$, such as $G = SU$. In this case it so happens that all $G$-principal bundles on a 3-manifold $\Sigma\_3$ are trivializable, and hence one can identify the space of G-principal connections on $\Sigma\_3$ just with that of $\mathfrak{g}$-valued differential forms. So one gets away with the naive formula that you recall above.


In stark contrast to this is what may seem to be a simpler example, namely $U(1)$-Chern-Simons theory. Since $U(1)$ is not simply connected, clearly, there are of course non-trivial $U(1)$-principal bundles on $\Sigma\_3$, in general, and hence the above naive approach fails, as you notice.


In this case the correct Chern-Simons action is instead obtained this way: given a field configuration $\nabla$ which is a circle-principal connection, we can form its [differential cup-product](http://ncatlab.org/nlab/show/Beilinson-Deligne+cup-product) square in [ordinary differential cohomology](http://ncatlab.org/nlab/show/ordinary%20differential%20cohomology). This yields a $\mathbf{B}^2 U(1)$-[principal 3-connection](http://ncatlab.org/nlab/show/circle%20n-bundle%20with%20connection) $\nabla \cup \nabla$, often known as a bundle 2-gerbe with connection or else as a degree-4 cocycle in [Deligne cohomology](http://ncatlab.org/nlab/show/Deligne%20cohomology). This now has a connection 3-form and hence has a [volume holonomy](http://ncatlab.org/nlab/show/fiber%20integration%20in%20differential%20cohomology) over $\Sigma\_3$. And this now is the correct action functional for Chern-Simons theory. For more on this see at *[nLab:higher dimensional Chern-Simons theory](http://ncatlab.org/nlab/show/higher+dimensional+Chern-Simons+theory)*.


Secretly this higher principal connection structure also governs the first, seemingly simpler case. The action functional of Chern-Simons theory is always the volume holonomy of a 3-connection, the *[Chern-Simons circle 3-connection](http://ncatlab.org/nlab/show/Chern-Simons%20circle%203-bundle)*.


This is in fact the general abstract characterization of Chern-Simons theories and [all its higher (and lower) dimensional variants](http://ncatlab.org/nlab/show/infinity-Chern-Simons+theory+-+contents). A Chern-Simons-type action functional is always the volume holonomy of a refinement of a universal characteristic class to [ordinary differential cohomology](http://ncatlab.org/nlab/show/ordinary%20differential%20cohomology). Further remarks along these lines are for instance in 


Domenico Fiorenza, Hisham Sati, Urs Schreiber, *[A higher stacky perspective on Chern-Simons theory](http://ncatlab.org/schreiber/show/A%2Bhigher%2Bstacky%2Bperspective%2Bon%2BChern-Simons%2Btheory)*.


