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"The pure joy" \(E_8\) SUSY toroidal orbifold TOE

In the early 1970s, supersymmetry (SUSY) was discovered. By the end of the decade, it became a vital tool in model building. By 1981, the supersymmetric standard model was mostly understood and seen to be superior to the non-supersymmetric standard model in many respects.

Everybody is already in Mexico, Michal Tučný 1986

Grand unification (GUT) has been around from the early 1970s as well. And in 1985, SUSY GUTs were embedded in the first semi-realistic model that also included quantum gravity, namely the \(E_8\times E_8\) heterotic string theory. \(\ZZ_3\) orbifolds of tori have always been the "simplest" examples of Calabi-Yau threefolds that the unobserved 6 dimensions (among the 10 dimensions in total) could take.

In these new papers, a Mexican-Portuguese-Swedish team (Aranda, de Anda, Morais, Pasechnik) constructs the simplest, non-gravitational combination of the "too good to be an accident" traits of heterotic string theory on the toroidal orbifolds:

AdAMPmini: A Different Take on \(E_8\) Unification
AdAMPbig: Sculpting the Standard Model from low-scale Gauge-Higgs-Matter \(E_8\) Grand Unification in ten dimensions
The short paper may be considered a commercial for the long one.

OK, one starts with a supersymmetric \(E_8\) multiplet. That largest exceptional simple Lie group only has real representations and cannot be a realistic grand unified group. So it is broken and it is broken through an orbifold \(T^6 / (\ZZ_3\times \ZZ_3)\) with Wilson lines. That is a compactification that has been considered since the baby years of heterotic string theory, of course.

Although the setup is clearly stringy, either heterotic string theory or its strong coupling, the Hořava-Witten M-theory, they ignore all the characteristic stringy/M-stuff. So this is just a (heavily) string inspired model, with the same symmetries (including the "heterotic" \(E_8\) group as well as SUSY), the same total number of dimensions, the same compactification and orbifolding and Wilson lines as some of the simplest ones. But everything they do is pure quantum field theory – and they do it in a rather hardcore way. So there are tons of equations for dozens of beta-functions each of which has a dozen of terms, and many other things.

They claim that this produces a realistic model with the Standard Model, three generations of quarks and leptons (three arises naturally because the number of excess dimensions is three times two), some Higgs matter, and an automatic suppression of the proton decay, among other things. A fun feature is an entry in their embedding of groups:\[ SU(3)\!\times\! SU(2)\!\times\! U(1)\subset SU(5)\subset SO(10)\subset E_6\subset E_7\subset E_8 \] OK, we see that on top of the Standard Model group, they also use the minimal GUT group, the minimal GUT group which places one generation of fermions into an irreducible representation, and the simple exceptional Lie GUT group. But in between \(E_6\) and \(E_8\), you also see \(E_7\), the "intermediate size" \(E\)-type group. They actually organize the spectrum in terms of \(E_7\) representations, too. Note that all reps of \(E_7\) are real, just like in the case of \(E_8\); \(E_6\) is the only simple exceptional Lie group that has complex reps at all (the \(\ZZ_2\) symmetry of the \(E_6\) Dynkin diagram is exactly what is needed for the "duality" between the mutually complex conjugate reps; \(SU(3+)\) Dynkin diagrams have the same symmetry; the \(SO(8)\) Dynkin diagram has a greater \(S_3\) symmetry causing the "triality" of the reps).

I think that the people who have done some serious, high-precision F-theory model building will recognize the importance of \(E_7\) as an intermediate symmetry breaking pattern; see e.g. this 2009 blog post if you need to be reminded what constructions I am referring to.

Anyway, if this "most straightforward" way to get the realistic spectrum from something like a supersymmetric GUT in higher dimensions worked, it would mean that quantum gravity may be almost completely separated. Taking the field theory limit of the stringy/M tools would be equivalent to computing the non-gravitational part of the theory. This equivalence is natural but I would warn that it is not obviously true (at least I think it can be deviated from in both directions: purely stringy/M traits beyond a QFT may be needed for non-gravitational forces; but a part of the quantum gravity questions may depend on QFT fields and interactions, too).

P.S.: The long paper has a cute notational trait. The fields in all the equations are identified through a color according to the "type of representation": adjoint, lepton+Higgs, mirror lepton+Higgs, quark, mirror quark, and family-neutral exotics. I praise this feature and it should become standard. I am sure that such things increase the readability of the equations.

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