I hope this question will not be closed down as something completely trivial!

I did not think about this question till in recent past I came across papers which seemed to write down pretty much simple looking solutions to "free" Yang-Mill's theory on $AdS_{d+1}$. The solutions looks pretty much like the electromagnetic fields!

- Aren't classical exact solutions to Yang-Mill's theory very hard to find?
Don't they have something to do with what are called Hitchin equations?
(..I would be grateful if someone can point me to some expository literature about that..)

I would have thought that non-Abelian Yang-Mill's theory has no genuine free limit since it always has the three and the four point gauge vertices at any non-zero value of the coupling however arbitrarily small. This seemed consistent with what is called "background gauge field quantization" where one looks at fluctuations about a classical space-time independent gauge fields which can't be gauged to zero at will since they come into the gauge invariant quantities which have non-trivial factors of structure constants in them which are fixed by the choice of the gauge group and hence nothing can remove them by any weak coupling limit.

But there is another way of fixing the scale in which things might make sense - if one is working the conventions where the Yang-Mill's Lagrangian looks like $-\frac{1}{g^2}F^2$ then the structure constants are proportional to $g$ and hence a weak coupling limit will send all the gauge commutators to zero!

- So in the second way of thinking the "free" limit of a non-Abelian $SU(N_c)$ Yang-Mill's theory is looking like an Abelian gauge theory with the gauge group $U(1)^{N_c^2 -1}$. So is this what is meant when people talk of "free" non-Abelian Yang-Mills theory? (..which is now actually Abelian!..)

I would grateful if someone can help reconcile these apparently conflicting points of view.

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