# Is the $G_2$ Lie algebra useful for anything?

Seems like all the simpler Lie algebras have a use in one or another branch of theoretical physics. Even the exceptional $$E_8$$ comes up in string theory. But $$G_2$$? 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 $$G_2$$ representation.

• Related: Qmechanic's comment in physics.stackexchange.com/q/65979 May 28 '13 at 5:33
• 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 ;) Jul 4 '13 at 16:27
• There's an interesting historical survey on G2 which includes some of elementary discussion of the early applications to physics in "Agricola, I. (2008). Old and New on the Exceptional Group G2, Notices of the American Mathematical Society, vol 55, no8, pp 922-929". Apr 30 '20 at 12:59

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.
• 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. Sep 25 '11 at 20:56
• 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. Jul 20 '17 at 3:42

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...)