What is non-Abelian about non-Abelian Chern-Simons' theory? One is aware that in the axial gauge (say the light-cone gauge $A_{-}=0$) non-supersymmetric Chern-Simons' theory is a quadratic theory. Hence in this gauge there are no gauge-gauge interactions. Then what is non-Abelian about this theory? 


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*Are there effects which exist inspite of the above which make the theory different from the Abelian one?

*I am aware that there are calculations of exact partition functions for Chern-Simons' theory on compact 3-manifolds. Is that the subtle point that the non-Abelian-ness will somehow manifest itself when on compact space-times? (..or on manifolds with boundary?..) That there is some reason such convenient axial gauges may not exist for compact space-times? (..though i can't see why one can't always choose $A_{-}=0$..because 3-manifolds are parallelizable there aren't any topological restrictions but there could be an issue with Gribov ambiguities..I don't know and would like to know about) 

*I guess that for supersymmetric Chern-Simons' theories this question itself doesn;t arise since I guess there are no gauge choices where the theory will become quadratic in the gauge fields. 
(- related i would love to know of any proof/argument as to why there can't be such a gauge choice for YM theory which will make it quadratic - axial gauge can kill the quartic term but I guess thats the best one can do..)  
 A: I don't know so much about the Chern-Simons theory and its use in supersymmetric theories, so I can not really help regarding all your questions. But the difference between abelian and non-abelian Chern-Simons term is crystal clear from its definition.
Usually, a Chern-Simons term reads in general (below for a $2+1$-dimensional problem)
$$L_{\text{CS}}=\dfrac{k}{4\pi}\int\text{Tr}\left\{ A\wedge dA+\dfrac{2}{3}A\wedge A\wedge A\right\} =\dfrac{k}{8\pi}\int\varepsilon^{ijk}\text{Tr}\left\{ A_{i}\left(\partial_{j}A_{k}-\partial_{k}A_{j}\right)+\dfrac{2}{3}A_{i}\left[A_{j},A_{k}\right]\right\} $$
such that, for an abelian gauge-field, one as
$$L_{\text{CS}}=\dfrac{k}{8\pi}\int\varepsilon^{ijk}\text{Tr}\left\{ A_{i}\left(\partial_{j}A_{k}-\partial_{k}A_{j}\right)\right\} $$
since $\left[A_{j},A_{k}\right]=0$ from the previous definition. In practise, it means that the gauge-field is in the Lie algebra related to the $\text{U}\left(1\right)$ Lie group.
So in short, the abelian Chern-Simons term is just a restriction of the Chern-Simons term for abelian gauge.
