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)