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It is easy to find various/many papers in HEP-lattice talk about "Non abelian string in QCD".

  1. What does it mean to say "non abelian string in QCD?" Does "non abelian string" happen for pure Yang-Mills (say $\mathrm{SU(N)}$ or $\mathrm{SO(N)}$) without any fermions? Or do we require additional fermions?

  2. Does "non abelian string" have any thing to do with the quantum statistics of strings are non abelian? Like non abelian Majorana for certain solid state systems?

  3. Should the string form a worldsheet in the spacetime, thus it should be descried by some 2-form field locally? Any precise math formulation?

P.s. Let us focus on the non-SUSY theory first. There is some question "How about SUSY?" that I removed just to get more focused.

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    $\begingroup$ To reopen this post (v1), consider to only ask one subquestion per post. $\endgroup$ – Qmechanic Jun 18 '17 at 4:48
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    $\begingroup$ all questions are related. Are you sure you are making the correct decision by a single force opinion? The only thing can be removed is probably SUSY version. I can focus on non-SUSY but everything else is totally related. $\endgroup$ – annie marie heart Jun 18 '17 at 5:01
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    $\begingroup$ Yes. For starters, explicit res. recom. qs are restricted on Phys.SE and can usually not be mixed with actual physics qs. $\endgroup$ – Qmechanic Jun 18 '17 at 5:11
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    $\begingroup$ @ Qmechanic, I just remove SUSY -- other parts if you read, they are basically related questions. If one intends to answer just the sub questions, one can focus on the 1 and 2. $\endgroup$ – annie marie heart Jun 18 '17 at 23:05
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    $\begingroup$ Related meta post: physics.meta.stackexchange.com/q/9943/2451 $\endgroup$ – Qmechanic Jun 22 '17 at 4:28
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Most of these questions don’t have answers as of now, but we can make some remarks:

  1. It is believed by arguments by t’Hooft, that large N gauge theories have dual descriptions as string theories. Only a few examples are known, the most famous being $\mathcal{N}=4$ SYM, whose string theory dual lives on 5d anti-de-Sitter space. One of the hopes is to find a similar description for QCD.
  2. The non-abelian refers to the non-abelian gauge group. Roughly the flux tubes connecting quarks are string-like. This doesn’t have much to do with the statistics of the strings (unless you introduce magnetic monopoles).
  3. Yes, the string should have a worldsheet description. The problem is that the spacetime on which the string moves is not the same as the spacetime on which the gauge theory lives. This is the hard part of the problem. The string should be described by a non-linear sigma model on some target space. There is currently no precise mathematical procedure for extracting this information from a gauge theory such as QCD.

Bonus: like one of the comments said, the supersymmetric case is actually simpler, since we can identify dual descriptions more easily and compute things to check these dualities.

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