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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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Free nonabelian gauge theory is the limit of zero coupling. So it is true, in a sense, that free $SU(N_c)$ gauge theory is a theory of $N_c^2 -1$ free "photons." There's a crucial subtlety, though: it's a gauge theory, so we only consider gauge-invariant states to be physical. Thus, already in the free theory, there is a "Gauss law" constraint that renders calculations (e.g. of the partition function) nontrivial and the physics of nonabelian free theories different from that of abelian ones (at least in finite volume). See, for instance, this beautiful work of Aharony, Marsano, Minwalla, Papadodimas, and van Raamsdonk, which shows that many thermodynamic aspects of confinement appear already in free theories at finite volume.

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  • $\begingroup$ In that paper there is this concept of writing Gauss's law as a statement of representation theory which I could not understand. The statement somehow is that Gauss's law for $SU(N_c)$ gauge theory is that physical states are invariant under global $SU(N_c)$ transformations - this is something I could not understand and what it means in the language of representations. Though I could independently understand the Polya theory counting but their integral representation and their very crucial equations 3.12 to 3.20 were totally unclear to me! $\endgroup$
    – Student
    Dec 21, 2011 at 0:48
  • $\begingroup$ I have edited my question about the typo I had about what the effective gauge theory is at the $g=0$ point. Each independent generator of $SU(N_c)$ now becomes "free" and hence the gauge theory at $g=0$ is $U(1)^{N_C^2-1}$ But it somehow feels a bit weird that perturbation theory makes sense even when the point about which one is doing the perturbation and the actual theory are seeing two different gauge groups. Also can you kindly explain as to how the "Gauss's law" (in the above sense!) is preserved in the $g=0$ limit unlike the commutators? $\endgroup$
    – Student
    Dec 21, 2011 at 1:02
  • $\begingroup$ The "Gauss law" constraint is just that the states should be color singlets; physical states are always gauge invariant, in other words. I'm not sure I understand the rest of what you're asking here. $\endgroup$
    – Matt Reece
    Dec 21, 2011 at 5:02
  • $\begingroup$ The "Gauss's Law" that one learns in school is the statement that the electric flux through any surfaces is proportional to the total charge contained in it. Now how does that generalize to the statement that the all states in an Yang-Mill's theory are colour singlets? I am not sure how to rephrase this other part of the question - I was comparing perturbation theory in non-Abelian field theory with that in any other theory - $\endgroup$
    – Student
    Dec 21, 2011 at 22:26
  • $\begingroup$ - unlike in any other theory here it seems that one is doing perturbative expansion about a point ($g=0$) where infact the symmetry group of the theory is apparently different than that of the full theory! In hindsight that looks a bit freaky - its almost like saying that $SU(N_c)$ non-Abelian processes are only perturbative effects on a $U(1)^{N_c^2 -1}$ Abelian theory - though as you point out the $g=0$ theory still has many of the non-Abelian effects like confinement! (..though it has an Abelian gauge group!..) $\endgroup$
    – Student
    Dec 21, 2011 at 22:28

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