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Any use for \(F_4\) in hep-th?

Yuji asked: 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.

LM answered: \(F_4\) is the centralizer of \(G_2\) inside an \(E_8\). In other words, \(E_8\) contains an \(F_4\times G_2\) maximal subgroup; the decomposition of the fundamental (=adjoint) representation under the group is simple:

\( {\bf 248} = ({\bf 52},{\bf 1}) \oplus ({\bf 1},{\bf 14}) \oplus ({\bf 26},{\bf 7}) \)
It's a direct sum of the two subgroups' adjoint representations and the tensor product of their fundamental representations. A reason I wrote it is to remind you of the 26-dimensional fundamental and 52-dimensional adjoint representation of \(F_4\). Incidentally, one also sees that the dimension of the adjoint representation of \(G_2\), 14, is also twice the dimension of its fundamental representation, 7.

The existence of this maximal subgroup is 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.

\(E_6\) has six simple roots (nodes). If you identify two pairs of nodes by the \({\mathbb Z}_2\) symmetry, you obtain two new nodes which are \(\sqrt{2}\)-times shorter (from the pairs), aside from the two original "unpaired" nodes on the \(E_6\) axis. In other words, you obtain the \(F_4\) Dynkin diagram. The \(G_2\) Dynkin diagram may similarly be obtained from the \(SO(8)\) Dynkin diagram by the \(S_3\)-identification based on triality of \(SO(8)\), an unusually large outer automorphism group.

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, analogously to \(E_8\) which may be built by adding a 128-dimensional chiral real spinor to the adjoint of \(SO(16)\); the only other – third – example of the same thing is getting \(SO(9)\) from \(SO(8)\) by adding a spinor, an operation that differs from the normal "vectorial" extension of \(SO(8)\) to \(SO(9)\) by triality), 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:
TRF: Is M-theory hiding Cayley plane fibers?
There's one funny episode from the history of maths concerning \(F_4\) in the discussions of holonomy. Aside from 7-dimensional \(G_2\)-holonomy manifolds and 8-dimensional \(spin(7)\)-holonomy manifolds, Berger originally conjectured that there were also 16-dimensional \(spin(9)\)-holonomy manifolds. But they were later proved to be trivial – either locally flat or locally isomorphic to the Cayley plane, \(F_4/spin(9)\). So \(F_4\) was the only new, non-trivial external "killer" that made people stop talking about the \(spin(9)\) holonomy. ;-)

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 \({\mathcal N}=2\) 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.

Recall that while \(G_2\) is the automorphism group of the octonions \({\mathbb O}\), the group \(F_4\) may similarly be defined as the automorphism group of the algebra \(J=A_3({\mathbb O})\) of \(3\times 3\) "Hermitian" octonion matrices (Hermitian conjugation involves both transposition and the sign flip for all 7 imaginary units) with the bilinear operation
\(\hat A \diamond \hat B \equiv \hat A\cdot \hat B +\hat B \cdot \hat A \)
Note that the matrices in the algebra have 3 real numbers on the diagonal and 3 inequivalent octonions above the diagonal (copied below it, with the complex conjugation), making the total number of real components \(3+3\times 8=27\). However, the identity matrix obviously has to be mapped onto itself under automorphisms; the remaining 26 components form the (previously advertised) fundamental representation of \(F_4\).

If you have a feeling that \(F_4\) only arises in applications that are related to string theory or at least supergravity, you're right: that's where you encounter exceptional mathematical structures in physics. If there were no string theory and not even supergravity, physicists wouldn't be forced to learn exceptional groups.

Well, the \(E_6\) grand unified theories are the only exception but they would still probably remain ignorant about the other four. \(E_6\) is unique among the exceptional groups when it comes to grand unified theory model building because it's the only one that has complex representations, representations inequivalent to their complex conjugates (which is needed to get chiral fermions and P- and CP-violation). This fact is actually equivalent to the existence of a symmetry of the \(E_6\) Dynkin diagram – and this symmetry of the diagram, as a white picture above showed, may be "orbifolded by" to get nothing else than the Dynkin diagram of \(F_4\), the main hero of this blog entry.

As you can see, the relationships between the exceptional groups and similar structures are diverse, rich, and sometimes unexpected. To describe some basic features of \(F_4\), I needed an \(E_8\) as well as \(G_2\), the centralizer of \(F_4\) in \(E_8\), as well as \(E_6\) from which \(F_4\) could have been built by the orbifolding of the Dynkin diagram.

Only \(E_7\) hasn't been mentioned as an important partner of \(F_4\) so far. Let me just assure you it was a random omission: \(E_7\) is exactly what you get by the Kantor-Koecher-Tits construction applied to the very algebra \(J=A_3({\mathbb O})\) whose automorphism group was said to be \(F_4\) above.

Many of these relationships have a physical interpretation we have already found; the other interesting relationships have a physical/stringy explanation, interpretation, or visualization that we will find in the future. ;-)

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