Alfredo Aranda and Francisco de Anda posted an amusing 5-page-long hep-ph preprint

Complete \(E_8\) Unification in 10 Dimensionswhere they showed how one can get a damn realistic spectrum for a theory of everything while starting with the 10-dimensional \(E_8\) gauge superfield only, the superfield that lives e.g. on the domain walls of the 11-dimensional M-theory. The presence of an orbifold, \(T^6 / (\ZZ_6\times \ZZ_2)\) in this case, a typical stringy feature, is capable of circumventing the usual conclusion that you can't get a chiral spectrum with groups like \(E_8\).

Also, the model naturally produces three generations of quarks and leptons. To be precise, their model gives the MSSM spectrum (with two Higgs doublets), the usual completion by three right-handed neutrinos, two extra singlet scalars (flavons), and two vector-like triplets. A wonderful enriched Minimal Supersymmetric Standard Model, indeed.

Which groups are allowed as gauge groups for Grand Unification?

Here you have the Dynkin diagrams of simple compact Lie groups and you may ask: Which ones may be drawn so that you can see a left-right \(\ZZ_2\) symmetry? Yes, it is \(A_N\), \(D_N\), and \(E_6\), corresponding to groups \(SU(N+1)\) (except for \(SU(2)\) which only has real and pseudoreal reps), \(SO(2N)\) (and \(SO(8)\) has an enhanced symmetry to a triality), and \(E_6\). The left-right symmetry of a Dynkin diagram may be seen to give rise to an outer automorphism and a \(\ZZ_2\) one is nothing else than a complex conjugation of the representations. So this symmetry is both a sufficient and necessary condition for the corresponding Lie group to have complex reps (that are not equivalent to their complex conjugates).

Among the five exceptional groups, it is only \(E_6\) that can have complex representations, as an extended version of \(SO(10)\), also a nice grand unified group, or a group that may be imagined as an \(E_8\) broken in a complex way: \(E_6\times SU(3)\) is a maximal subgroup of \(E_8\). (\(F_4\times G_2\) is another maximal subgroup, one of a "real" type when it comes to reps.) So \(E_6\) is perhaps the largest gauge group that is considered as a gauge group for the grand unification. The quarks and leptons arise from the 27-dimensional fundamental representation etc.

But the larger group \(E_8\) still looks prettier, right? And beauty is really among the most important guides for any good theoretical physicist (which is why the sentence above written by Paul Dirac in 1951 is still kept on a blackboard in Moscow). \(E_8\) cannot be used as a grand unification group in the "mundane" quantum field theories directly but it is surely a starting point of more complete models, those in string theory. In particular, heterotic string theory and its strong coupling limit, the Hořava-Witten M=theory, have \(E_8\times E_8\) as their gauge group in ten dimensions.

Perturbative superstring theory predicts and requires 10 spacetime dimensions in total. 6 of them must be compactified. It is surely intriguing that 6 is a multiple of 3, the number of generations, so the generations could have something to do with pairs of compactified dimensions. Some string vacua use this observation to some extent but this Mexican one is the simplest and clearest one so far. On top of that, the centralizer of \(E_6\) within \(E_8\) is \(SU(3)\) which also hints at three generations as an extra feature beyond the QFT gauge group (such as \(E_6\)).

*This song by Three Sisters really sounds Mexican but believe me, the sentences are in Czech. OK, with some typical Mexican nouns and a slight Mexican accent.*

They compactify 6 of the dimensions on a flat manifold, the torus \(T^6\), but the torus is tilted so that it can have an isometry group, \(\ZZ_6\times \ZZ_2\) (yes, some hexagonal lattices must be involved), and we may quotient (orbifold) by it. With the orbifolds and some corresponding Wilson lines around non-contractible loops included (typical features in the string theory model building), we may obtain chiral generations so it doesn't matter that \(E_8\) itself has no complex reps. The orbifolding and Wilson lines breaks the gauge group to a group with complex reps.

And they may obtain a very minimalistic spectrum without having any other field in 10 dimensions. So aside from the graviton, only the 10D \(E_8\) gauge superfield living on the Hořava-Witten 10D boundary in M-theory, if you wish, is enough to produce the whole MSSM spectrum (plus flavons, possibly helpful to get realistic fermion masses and mixing, plus some triplets that may play a helpful role in dark matter or other things) out of the single superfield on a locally flat orbifold!

The model should be said to be "string-inspired" because they use features (numerous features) from string theory model building: 10 dimensions, 16 supercharges to start with (although 32 is present in M-theory without boundaries), orbifolds, Wilson lines. On the other hand, it may be said not to be a construction requiring "full string theory" because the excited strings are not used at all. After all, just the "field-theoretical" low-energy spectrum from the M-theory end-of-the-world domain walls is enough to obtain all the fields that we need to explain our low-energy experiments. From a certain perspective, this characteristic (which makes the model less dependent on any "strings") could be considered a virtue.

It's rather elegant, every genuine TOE builder may guess what is the relative merit or probability of this model relatively to other realistic string vacua. But if this model were on the right track, it would make sense to consider the extra triplets a prediction, probably the most observable prediction, of this model (and perhaps its variations or stringy completions). Of course, triplets have been searched by the LHC and predicted by various other string and string-inspired models (including those around Nanopoulos) but this model is almost childishly clean.

P.S.: This model is a variation of a similar model with Stephen King published two months ago. That one required extra fields to cancel anomalies, however (and due to the British King, it wasn't purely Mexican).

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