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  Any use for $F_4$ in hep-th?

+ 18 like - 0 dislike

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.

This post has been migrated from (A51.SE)
asked Oct 7, 2011 in Theoretical Physics by Yuji (1,395 points) [ no revision ]
retagged Apr 19, 2014 by dimension10
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.

This post has been migrated from (A51.SE)
Wasn't Romans' F(4) the super algebra F(4)?

This post has been migrated from (A51.SE)
Hmm, you're probably right. The stringy $F_4$ was the exceptional Lie algebra, though. I can't seem to locate the paper, though.

This post has been migrated from (A51.SE)

1 Answer

+ 13 like - 0 dislike

$F_4$ 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,


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


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


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


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


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:


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


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:


This post has been migrated from (A51.SE)
answered Oct 7, 2011 by Luboš Motl (10,278 points) [ no revision ]

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