You should know that He, Braun, and Ovrut (and their co-author until recently, Pantev) are not the only ones who study the heterotic supersymmetric Standard Models with non-standard embedding of the bundles.

Another group works on these technically demanding, exciting, and generally promising issues. In December, and actually before the last BHOP paper, Vincent Bouchard and Ron Donagi constructed their SU(5)-based heterotic Standard Model starting from a Calabi-Yau whose fundamental group is "Z_2".

This apparently differs from the models of Braun et al. who use a Calabi-Yau whose fundamental group is "Z_3 x Z_3". We have dedicated many articles to the models by Braun et al. especially because this fundamental group looks rather natural to me.

However, there exists a certain amount of disagreement about the details. In reality, it only looks like a different emphasis to me. While the visible bundle of the models of Braun et al. was recently shown to be stable, no demonstrably stable bundle in the right cohomology class (to obey the equations of motion for the four-form dH) has been constructed for the hidden sector.

Braun et al. believe that this problem may be resolved. For example, they hope that some of their hidden bundles may still be stable because they satisfy the (necessary) Bogomolov inequality for certain values of the moduli. And if they are not stable, one can add anti-M5-branes and hope that their partial annihilation will still lead to a realistic model.

Bouchard, Donagi et al. have a different opinion. They think that the absence of a provably stable hidden sector bundle for the model of Braun et al. is a fatal problem. (Strong adjectives will keep you awake.) Instead, they work with an older Calabi-Yau three-fold whose fundamental group is "Z_2" instead of "Z_3 x Z_3". This requires a different treatment of grand unification.

Of course, both of these papers start with a visible stringy "E_8" gauge group that is broken to some subgroups. However, Braun et al. use the "SO(10)" subgroup whose centralizer in "E_8" is "SU(4)" as an intermediate step. The "SU(4)" is therefore the structure group of a bundle that breaks "E_8" to "SO(10)". This gives them the 16-dimensional fermion multiplets that include the right-handed neutrinos that lead to natural low neutrino masses via the seesaw mechanism. The "SO(10)" group is broken to the Standard Model via Wilson lines in a "Z_3 x Z_3" subgroup of "SO(10)" that is identified with the fundamental group of the Calabi-Yau: its centralizer is the Standard Model.

Bouchard and Donagi, on the other hand, only use the "SU(5)" subgroup of "E_8". Its centralizer within "E_8" is another "SU(5)". Therefore, they consider an "SU(5)" bundle. For an "SU(5)" gauge group, one can construct the Standard Model gauge group as a centralizer of a smaller group, namely a "Z_2", which is identified with the fundamental group of their manifold.

Tonight, these two authors will be joined by Mirjam Cvetič:

They also get rid of all exotics and their spectrum is the MSSM with 0, 1, or 2 Higgs doublets, depending on the values of the vector bundle moduli. I have not checked whether He, Ovrut, and Pantev agree that the BCD model gives a pure MSSM, too. Incidentally, some of these vector bundle moduli play the role of the right-handed neutrinos. As we mentioned, Bouchard, Cvetič, and Donagi compute the trilinear couplings. They also argue that proton remains stable and show how the mu-terms (and the Higgs masses) are obtained in their framework.

Although the structures of grand unification in these two classes of models differ, I can imagine that eventually such models may be shown to be connected and the "truth" may be located somewhere in between them. For example, if you add some anti-M5-branes to the Braun-He-Ovrut models, a partial annihilation could effectively transform their "SU(4)" bundle into a more general "SU(5)" bundle of BCD, together with some topology change of the Calabi-Yau manifold. But the Braun-He-Ovrut starting point could still be helpful to understand the existence of right-handed neutrinos etc. I believe that the transitions across the landscape should be better understood than they are today, and that the configuration space will turn out to be much more connected than the "landscape" ideology wants us to believe. The number of instabilities will increase and the number of stable points will decrease.

While I intuitively prefer the "SO(10)" grand unification, the existence of a demonstrably stable set of bundles is of course a big advantage of the BCD model. An "advantage" may be too weak a word. If BCD are correct, then BCD is simply a correct compactification while BHO is not.

Finally, there seems to be another subtle point of disagreement whose character does not seem to affect the technicalities. It is related to the questions about the anthropic reasoning. Burt Ovrut likes to assume that the number of relevant models is of order one, the currently existing models have a significant chance to be the correct ones, and the anthropic people don't know what they're talking about.

Ron Donagi, on the other hand, thinks that the currently existing models are just "toy models" and they will be supplemented with a large and possibly vast number (between 10 and 10^{100}) of similar models in the near future. See the page 1 of their December paper which even contains a smiling face.

Needless to say, my feelings about this philosophical issue are closer to Burt Ovrut's. If someone argues that even the "unique" model found by another group after years of research is non-existent, it seems a bit unnatural to assume that there actually exists zillions of consistent models of this type.

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