# Any use for $F_4$ in hep-th?

In high energy physics, the use of the classical Lie groups are common place, and in the Grand Unification the use of $E_{6,7,8}$ is also common place.

In string theory $G_2$ is sometimes utilized, e.g. the $G_2$-holonomy manifolds are used to get 4d $\mathcal{N}=1$ susy from M-theory.

That leaves $F_4$ from the list of simple Lie groups. Is there any place $F_4$ is used in any essential way?

Of course there are papers where the dynamics of $d=4$ $\mathcal{N}=1$ susy gauge theory with $F_4$ are studied, as part of the study of all possible gauge groups, but I'm not asking those.

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If you search for `F(4)` in INSPIRE, you will see a number of papers. There is an old paper by Larry Romans with the construction of an $F_4$ gauged six-dimensional supergravity which was popular in its day. Also I seem to recall a paper with an $F_4$ string theory, probably prompted by the fact that the dimension of the fundamental representation of $F_4$ is 26. – José Figueroa-O'Farrill Oct 7 '11 at 17:15

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Wasn't Romans' F(4) the super algebra F(4)? – Yuji Oct 7 '11 at 17:33
Hmm, you're probably right. The stringy $F_4$ was the exceptional Lie algebra, though. I can't seem to locate the paper, though. – José Figueroa-O'Farrill Oct 7 '11 at 17:52

## 1 Answer

$F_4$ is the centralizer of $G_2$ inside an $E_8$. In other words, $E_8$ contains an $F_4\times G_2$ maximal subgroup. That's why by embedding the spin connection into the $E_8\times E_8$ heterotic gauge connection on $G_2$ holonomy manifolds, one obtains an $F_4$ gauge symmetry. See, for example,

http://arxiv.org/abs/hep-th/0108219

Gauge theories and string theory with $F_4$ gauge groups, e.g. in this paper

http://arxiv.org/abs/hep-th/9902186

depend on the fact that $F_4$ may be obtained from $E_6$ by a projection related to the nontrivial ${\mathbb Z}_2$ automorphism of $E_6$ which you may see as the left-right symmetry of the $E_6$ Dynkin diagram. This automorphism may be realized as a nontrivial monodromy which may break the initial $E_6$ gauge group to an $F_4$ as in

http://arxiv.org/abs/hep-th/9611119

Because of similar constructions, gauge groups including $F_4$ factors (sometimes many of them) are common in F-theory:

http://arxiv.org/abs/hep-th/9701129

More speculatively (and outside established string theory), a decade ago, Pierre Ramond had a dream

http://arxiv.org/abs/hep-th/0112261
http://arxiv.org/abs/hep-th/0301050

that the 16-dimensional Cayley plane, the $F_4/SO(9)$ coset (note that $F_4$ may be built from $SO(9)$ by adding a 16-spinor of generators), may be used to define all of M-theory. As far as I can say, it hasn't quite worked but it is interesting. Sati and others recently conjectured that M-theory may be formulated as having a secret $F_4/SO(9)$ fiber at each point:

http://motls.blogspot.com/2009/10/is-m-theory-hiding-cayley-plane-fibers.html

Less speculatively, the noncompact version $F_{4(4)}$ of the $F_4$ exceptional group is also the isometry of a quaternion manifold relevant for the maximal $N=2$ matter-Einstein supergravity, see

http://arxiv.org/abs/hep-th/9708025

In that paper, you may also find cosets of the $E_6/F_4$ type and some role is also being played by the fact that $F_4$ is the symmetry group of a $3\times 3$ matrix Jordan algebra of octonions.

A very slight extension of this answer is here:

http://motls.blogspot.com/2011/10/any-use-for-f4-in-hep-th.html

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