In Witten's paper [Three Dimensional Gravity Revisited](https://arxiv.org/pdf/0706.3359.pdf) and [Quantization of Chern-Simons Theory with Complex Gauge Group](https://projecteuclid.org/download/pdf_1/euclid.cmp/1104202513), he used a fact that for a principal $G$-bundle, the quantization of the Chern number is reduced to the case of its maximal compact subgroup.


For example, the reason why the Chern-level $k$ of the $\mathrm{SL}(2,\mathbb{C})$-Chern-Simons theory


$$S[A]=\frac{k}{4\pi}\int\_{M}\mathrm{Tr}\left(A\wedge dA+\frac{2}{3}A\wedge A\wedge A\right)$$


takes integer values ($k\in\mathbb{Z}$) is reduced to the case of the $\mathrm{SU}(2)$-Chern-Simons theory, because the group manifold $\mathrm{SL}(2,\mathbb{C})$ can be continuously shunk to $\mathrm{SU}(2)$.


Can anybody tell me how to prove this? Are there any references showing the proof that are easy to understand for physics students?



Why is the Chern Number Invariant under A Continuously Shrinking of the Structure Group?
I don't quite understand the quantization part of the question. 


However, the characteristic numbers are always given by pairing products of the (characteristic) cohomology classes with the fundamental class of the manifold $M$, say. Such classes are pullbacks of classes in the cohomology of $BG$, where $G$ is the structure group (where $M\to BG$ classifies the bundle).


Now, The cohomology of $BG$ is homotopy invariant in the sense that if $G\to H$ is a homomorphism which is also a homotopy equivalence of underlying spaces, then the map $H^\ast(BH) \to H^\ast(BG)$ is an isomorphism. 


If we take $H := SL\_n(\Bbb C)$ and $G = SU\_n(\Bbb C)$, we get the statement you are looking for.


