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Is the G2 Lie algebra useful for anything?

+ 8 like - 0 dislike
63 views

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.


This post imported from StackExchange Physics at 2014-03-17 04:03 (UCT), posted by SE-user DarenW

asked Nov 16, 2010 in Mathematics by DarenW (40 points) [ revision history ]
retagged Mar 25, 2014 by dimension10
I added a couple of tags, hope you don't mind. Good question, by the way.

This post imported from StackExchange Physics at 2014-03-17 04:03 (UCT), posted by SE-user David Z
Thanks. I tried to add lie-algebra and group-representations but I am yet too meager of pointage to create new tags.

This post imported from StackExchange Physics at 2014-03-17 04:03 (UCT), posted by SE-user DarenW
No problem, I can take care of that for you.

This post imported from StackExchange Physics at 2014-03-17 04:03 (UCT), posted by SE-user David Z
Related: Qmechanic's comment in physics.stackexchange.com/questions/65979/…

This post imported from StackExchange Physics at 2014-03-17 04:03 (UCT), posted by SE-user Dimensio1n0
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 ;)

This post imported from StackExchange Physics at 2014-03-17 04:03 (UCT), posted by SE-user Alex Nelson

2 Answers

+ 8 like - 0 dislike

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...)

This post imported from StackExchange Physics at 2014-03-17 04:03 (UCT), posted by SE-user Predrag Cvitanović
answered Nov 17, 2010 by Predrag Cvitanović (80 points) [ no revision ]
Welcome! I'm a big fan of your books, and you blew my mind a talk of yours I saw at a March meeting a few years ago on turbulence!

This post imported from StackExchange Physics at 2014-03-17 04:03 (UCT), posted by SE-user j.c.
Neat stuff! And I have a new entry in my list of books to buy soon. Atomic physics beats string theory any day for impressing me with usefulness of advanced theoretical ideas.

This post imported from StackExchange Physics at 2014-03-17 04:03 (UCT), posted by SE-user DarenW
+ 7 like - 0 dislike

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.
This post imported from StackExchange Physics at 2014-03-17 04:03 (UCT), posted by SE-user Matt Reece
answered Nov 16, 2010 by Matt Reece (1,630 points) [ no revision ]
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.

This post imported from StackExchange Physics at 2014-03-17 04:03 (UCT), posted by SE-user Ron Maimon

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