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Mathematicians have done a complete classification of all possible Lie groups. Is there a set of conditions that allows us to identify which Lie groups from the classification can possibly act as a gauge group for a Yang-Mills theory?

My vague recollection from a book that I can't recall is that the direct/semi-direct product of simple compact Lie groups with an arbitrary number of $U(1)$ factors can serve as a gauge group in a QFT. Is this statement correct, and if so, what assumptions went into proving this?

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  • $\begingroup$ I am sorry, but is your question whether or not we can have exceptional gauge groups or symplectic ones for example in realistic quantum gauge theories? Did I understand correctly? $\endgroup$ – DiSp0sablE_H3r0 Oct 18 '20 at 12:12
  • $\begingroup$ Yes, that is somewhat correct. The questions can be extended to enquire about whether direct products of those Lie groups can be possible realistic gauge theories? $\endgroup$ – Tushar Gopalka Oct 18 '20 at 12:14
  • $\begingroup$ Would you accept an answer based on arxiv publications for beyond the standard model physics? $\endgroup$ – DiSp0sablE_H3r0 Oct 18 '20 at 12:15
  • $\begingroup$ @Tushar Gopalka: By a gauge theory you mean of a Yang-Mills theory? $\endgroup$ – Qmechanic Oct 18 '20 at 12:25
  • $\begingroup$ Yes, like Standard Model is SU(3)x SU(2) xU(1) ... And that satisfies the condition I have mentioned.. Product of simple compact Lie groups with a single factor of U(1). $\endgroup$ – Tushar Gopalka Oct 18 '20 at 12:28
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The following is just an example where you can use gauge groups other than $SU(N)$.

While the discovery of the Higgs was and still remains a huge step towards our better understanding of particle physics there still exists the question of whether or not the Higgs is an elementary particle or a bound state of a strongly coupled sector in higher energies. Note that the latter possibility is still not excluded by the LHC data.

The key component of such a model is a strongly sector that triggers $\chi SB$ in the fermion sector which resembles a lot QCD. This breaking has to generate at least four Nambu-Goldstone bosons.

The above is one of the requirements. There is another that the gauge group description has to be an asymptotically free theory for a certain choice of hypercolours and hyperflavours. In this context, the prefix hyper is to just denote the difference from the $SU(3)$ charges.

A final requirement was the existence of composite top partners.

Combining the above there was a group theoretic approach that discussed all possible scenarios for a composite Higgs model in this paper

As you can see from the above, the models that contain all the fermions in a single representation of the gauge group are described by the exceptional groups $F_4$ and $G_2$. There are also models with matter in two and more representations with a symplectic gauge group and a special orthogonal. The usual $SU(N)$ also appears in these theories.

The exceptional groups $E_{6,7,8}$ have applications in string theory and related topics thereof; the study of superconformal fixed points for example. Not sure if you are interested in these examples.

Hope this helps a bit!

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