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Seems like all the simpler Lie algebras have a use in one or another branch of theoretical physics. Even the exceptional E8 comes up in string theory. But G2? I've always wondered about that one.

I know only of one false alarm in the 1960s or 1970s before SU(3) quark theory was understood, some physicists tried to fit mesons into a G2 representation.

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  • $\begingroup$ I added a couple of tags, hope you don't mind. Good question, by the way. $\endgroup$ – David Z Nov 16 '10 at 4:11
  • $\begingroup$ Thanks. I tried to add lie-algebra and group-representations but I am yet too meager of pointage to create new tags. $\endgroup$ – DarenW Nov 16 '10 at 4:41
  • $\begingroup$ No problem, I can take care of that for you. $\endgroup$ – David Z Nov 16 '10 at 5:03
  • $\begingroup$ Related: Qmechanic's comment in physics.stackexchange.com/questions/65979/… $\endgroup$ – Abhimanyu Pallavi Sudhir May 28 '13 at 5:33
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    $\begingroup$ A review paper on G2 gauge theories arXiv:1210.7950, and topological aspects of G2 Yang-Mills theory arXiv:1210.5963...for some light reading ;) $\endgroup$ – Alex Nelson Jul 4 '13 at 16:27
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Yes.

G2 shows up often, starting with atomic physics (perhaps Racah is the first; see R. E. Behrends, J. Dreitlein, C. Fronsdal, and B. W. Lee, “Simple groups and strong interaction symmetries,” Rev. Mod. Phys. 34, 1 (1962).). You will find some refences in my 1976 Phys rev paper on cns.physics.gatech.edu/GroupTheory/refs . I have whole folder of physics G2 papers, but now I see I did not bother to enter G2 history into www.birdtracks.eu.

Nobody's perfect. Sorry

Predrag (for responses, email to dasgroup [snail] gatech.edu, I sometimes look at those. Pure accident I saw this question...)

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I don't know if these rise to the level of "useful," but:

  • Yang-Mills theory with gauge group $G_2$ is interesting because $G_2$ has trivial center. So people simulate it on a lattice, try to understand in what sense it might be confining, how string tensions scale, if it has a deconfinement phase transition, and so on. The idea is that looking at a group with no center provides an interesting window into which phenomena in gauge theories rely crucially on the existence of a center and which do not. One recent paper (selected more or less at random from a search; I don't know this literature well enough to make useful suggestions) is here.
  • M-theory compactified on seven-dimensional manifolds of $G_2$ holonomy gives rise to four-dimensional theories with ${\cal N} = 1$ supersymmetry. I don't know the earliest references (probably this knowledge goes back to early work on supergravity before M-theory), but one place to look might be this paper of Atiyah and Witten.
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  • $\begingroup$ The G2 manifolds are just the 7 dimensional analog of the Calabi Yau manifolds. I think that this was a folklore result, because the same analysis that selects out Calabi-Yaus (preserving a covariantly constant spinor) selects out G2s, so it was automatically known. $\endgroup$ – Ron Maimon Sep 25 '11 at 20:56
  • $\begingroup$ Specifically, they are the 7 dimensional analogue of Calabi-Yau 3-folds. For example, they both have two distinguished classes of calibrated submanifolds: associative and co-associate 3- and 4-folds for G2 manifolds, and SL 3-folds and J-holomorphic curves for Calabi-Yau 3-folds. People believe generally in a mirror symmetry and Gromov-Witten invariants for G2 manifolds too. $\endgroup$ – Jacob Gross Jul 20 '17 at 3:42

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