Let the Chern-Simons lagrangian for a group $G$ be,
$$L= k \epsilon^{\mu \nu \rho} Tr[A\_\mu \partial \_ \nu A\_\rho + \frac{2}{3} A\_\mu A\_\nu A\_\rho]$$


Then it is claimed that on "infinitesimal" variation of the gauge field ("connection") the lagrangian changes by,


$$\delta L = k \epsilon^{\mu \nu \rho} Tr[\delta A\_\mu F\_{\nu \rho}]$$


where the "curvature" $F\_{\mu \nu}$ is given as $\partial \_ \mu A\_\nu - \partial \_ \nu A\_\mu +[A\_\mu,A\_\nu]$


Under gauge transformations on $A$ by $g \in G$ it changes to say $A'$ whose $\mu$ component is given as $g^{-1}A\_\mu g+g^{-1}\partial \_ \mu g$ (This makes sense once a representation of $G$ has been fixed after which $A$ and $g$ are both represented as matrices on the same vector space)


Say the Lagrangian under the above gauge transformations change to $L'$ and then one has the relation,


$$L'-L=-k \epsilon^{\mu \nu \rho}\partial\_\mu Tr[\partial\_\nu g g^{-1}A\_\rho]-\frac{k}{3}\epsilon^{\mu \nu \rho} Tr[g^{-1}\partial\_\mu gg^{-1}\partial\_\nu gg^{-1}\partial\_\rho g]$$


The second term of the above expression is what is proportional to the "winding number density"
 of the Chern-Simons lagrangian and thats what eventually gets quantized. 


I would like to know the following things,


* Is there a neat coordinate free way of proving the above two variation change equations? Doing this in the above coordinate way is turning out to be quite intractable!
* Since the Lagrangian is just a complex number one can talk of the "real" and the "imaginary" part of it. But I get the feeling that at times a split of this kind is done at the level of the gauge field itself. Is this true and if yes the how is it defined? (Definitely there is lot of interest in doing analytic continuation of the "level" $k$)
* The Euler-Lagrange equations of this action give us only the "flat" connections and in that sense it is a topological theory since only boundary conditions seem to matter. Still all the flat connection configurations are not equivalent but are labelled by homomorphisms from the first fundamental group of the 3-manifold on which the theory is defined to $G$. How to see this? What is the background theory from which this comes? And why is this called "holonomy"? (I am familiar with "holonomy" as in the context of taking a vector and parallel transporting it around a loop etc)



Some basic questions about Chern-Simons theory
The secret to understanding the Chern-Simons functional over a 3-manifold is to realise that it's a 4-dimensional functional in disguise.


If $X$ is a closed, oriented 4-manifold, and $P \to X$ a principal $SU(n)$-bundle, one has the Chern-Weil formula for the second Chern number,
$$ c\_2(P)[X] = \frac{1}{8 \pi^2} \int\_X {Tr F\_A^2}, $$
where $F\_A$ is the curvature of a connection $A$ in $P$. If now $Y$ is a closed, oriented 3-manifold, and $P\_Y \to Y$ a principal $SU(n)$-bundle (necessarily trivial) then we can define the Chern-Simons functional
$$ cs: Connections(P\_Y) \to \mathbb{R}/\mathbb{Z} $$  

for a connection $A$ by choosing any compact oriented 4-manifold $X$ bounding $Y$ (one always exists), any $SU(n)$-bundle $P\_X \to X$ extending $P\_Y \to Y$ (one again exists), and any connection $B$ in $P\_X$ extending $A$ (ditto), and setting
$$ cs(A) = \frac{1}{8 \pi^2} \int\_X {Tr F\_A^2}. $$
This is well-defined (mod integers) because given another choice $(X',P\_{X'},B')$, we can glue together the two manifolds along $Y$ to get a closed 4-manifold (we reverse the orientation of one of them), and express the difference in values of $cs(A)$ as a Chern-Weil integral as above, which is an integer.


To obtain a formula for $cs(A)$, take $P\_X$ to be a trivial bundle, and take $B$ to be a connection that is trivial except over the collar $Y\times [-1,0]$. Over that collar, the bundle is $[-1,0]\times P\_Y$, and we can let $B$ the 4-dimensional connection $A + t(A-A\_0)$, where $A\_0$ is a reference connection trivialising $P\_Y$ (i.e., flat with trivial holonomy). Then we can explicitly perform the integral defining $cs(A)$ over the $[0,1]$-factor (try it!) and get a formula for real-valued lift of $cs(A)$ (here we use $A\_0$ to trivialise $P\_Y)$,
$$ CS(A) = \frac{1}{8 \pi^2} \int\_Y{ Tr (A \wedge dA + \frac{2}{3} A\wedge A\wedge A)}, $$
which is the formula you gave in indices (up to a factor).


From this formula, it's straightforward to compute the first variation of $CS$, and to interpret it in terms of $F\_A=dA+A\wedge A$. 


The gauge transformation formula says that if $g$ is an automorphism of $P\_Y$ of degree 
$d $ then $ CS(gA)-CS(A) = d$. This has a conceptual explanation: linearly interpolate from $A$ to $gA$ to obtain a connection over $Y\times [0,1]$ whose Chern-Weil integral is $CS(gA)-CS(A)$. Use $g$ as a gluing recipe to obtain a bundle over $Y\times S^1$ whose second Chern number is $d$ (this defines $d$, if you will), and apply the Chern-Weil formula. 


Reference: S.K. Donaldson, "Floer homology groups in Yang-Mills theory".


Very briefly on the second question: one can work with $SL(n,\mathbb{C})$-bundles instead. The Lie algebra is the complexfication of that of $SU(n)$, so (after the bundle is trivialised), connections have real and imaginary parts which are 1-forms valued in $su(n)$.


