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Donagi, Wijnholt: what 3 generations mean in F-theory

The best hep-th paper today is the second one,

Higgs bundles and UV completion in F-Theory (PDF)
Ron Donagi and Martijn Wijnholt look at the F-theoretical model building. As the first true professionals, these Gentlemen seriously study the question whether the models may have three generations and what it means. You can see that each segment of their writing that you read in detail seems to make sense, unlike some attempts that have tried to answer similar questions and that required many "clouds" saying that "here a miracle occurs".

Recall that in the local F-theory model building, the gauge groups are carried by real codimension two objects, i.e. by seven-branes of type IIB string theory. Such objects can carry exceptional symmetries, up to E_8 itself. In other words, the gauge theory is supported by a 7+1-dimensional spacetime manifold.




Three generations make things special

Donagi and Wijnholt realize that the compactifications - on Del Pezzo-like real-four-dimensional manifolds - are given by meromorphic Sl(5,C) Higgs bundles. The latter can be mapped to spectral covers in an auxiliary Calabi-Yau geometry. For generic moduli, they can practically prove that there can't exist any three-generation models. That's actually a good thing because these models wouldn't stabilize the moduli and the proton decay could be rapid.

Instead, they claim that there exist special points in the moduli space that solve a condition that mathematicians would refer to as the Lefschetz-Noether (LN) problem. At this LN locus, there actually exist new fluxes that can be turned on without breaking the supersymmetry. With these additional conditions satisfied, some moduli are automatically stabilized.

Interestingly enough, these "new fluxes" at the special points can't be inherited from the bulk which also means that these F-theory compactifications don't have ordinary heterotic duals. Analogous degrees of freedom actually do appear on the heterotic side, in the form of rigid bundles (I guess that they could be analogous to the "triples" by the seven authors), but they haven't been studied much so far.

This missing ingredient is one of the reasons why the heterotic model building as of today has mostly overlooked the bulk of the large type IIB landscape. ;-) While they can obtain three generations, the models they see by their method contain some exotic matter. Reduction of the spectrum is arguably equivalent to another layer of a Lefschetz-Noether problem and will probably stabilize additional moduli. Some mathematical tools to do so will have to be developed.

General message

At any rate, this is a great line of reasoning that would be completely overlooked by the "anything goes" anthropic pundits. By postulating that we will never learn anything new about the relationships between the properties of the real world as we know them from the existing low-energy effective theories, they would make it impossible to figure out how constraining the three generations are and how this number "three" is related to the stabilization of moduli, to the existence of dual descriptions, and to the choice of special solutions.

It seems very plausible to me that in the future, people will be able to show that a model with a hierarchy is essentially unique. The more papers of this kind I see, the more I am inclined to think that our Universe is very special. I am also aware of papers suggesting that our vacuum is rather generic but these papers seem more philosophical and "circular" and less analytical.

The gravity-electroweak gap may guarantee that the local reasoning (and perhaps the F-theoretical local reasoning) is legitimate, the local reasoning may imply that three generations and stable protons require us to solve the most constraining Lefschetz-Noether problem one can think of, and the solution to this problem may be unique, much like the global extension of the local model found in this way.

Whether this is the case or not can only be found by a careful and patient analysis of these exciting classes of models and by seriously considering all known symmetries, discrete parameters, and hierarchies.

And that's the memo.

Friday F-theory phenomenological papers

The F-theory phenomenological minirevolution continued on Friday, too. But one has to look at the hep-ph archive. Bouchard, Heckman, Seo, and Vafa study the dependence of fermion masses on the GUT coupling: it's different for quarks and for neutral leptons. The different hierarchical patterns seems to agree with the observations.

Randall and Simmons-Duffin, also of Harvard, also look at flavor physics in F-theory and interpret it perturbatively within an effective theory.

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