Heterotic Standard Model

I was happy to see the following article by Braun, He, Ovrut, and Pantev:

• hep-th/0501070

The authors have made an extensive search for a conventional Standard Model obtained from the heterotic strings on a Calabi-Yau manifold, and they claim that they have virtually found a unique solution. They argue that their model is the first pure Standard Model obtained from string theory - and I am sure that Christos Kokorelis with his intersecting brane models would disagree.



As a conservative person, I must say that this kind of models is what still looks as a most satisfactory answer of string theory to the real world, even though some issues - like the smallness of the cosmological constant and various unwanted operators from supersymmetry breaking - are not answered yet.



What are the features of their model?

• We have observed the neutrino oscillations and masses at the sub-eV scale implied by the seesaw mechanism, so it's natural to have right-handed neutrinos and something like GUT scale physics
• A generation of fermions, including the right-handed neutrino, then transforms as the 16 spinor under a SO(10) GUT-like group
• In string theory, one can obtain this SO(10) group from the E_8 in the heterotic string, simply by taking the heterotic strings on a Calabi-Yau with a SU(4) bundle that breaks the gauge group from E_8 to SO(10)
• SO(10) should then be broken to the Standard Model. The large Higgs representations etc. have always been bad, and string theory has a natural breaking pattern using the Wilson lines
• We really need two Wilson lines in a Z_3 x Z_3 discrete subgroup of SO(10), and their centralizer is the Standard Model group times an extra U(1) that counts B-L (baryon number minus lepton number)
• Such a breaking by the Wilson lines reduces the symmetry while it keeps the fermion spectrum
  
  

Now, in order to be able to define these Wilson lines, we need a Calabi-Yau manifold with a Z_3 x Z_3 fundamental group. Well, they have found an example in the previous paper:

• It is an elliptic fibration over the half-K3 surface dP_9
• Its values of h_{1,1} and h_{1,2} are both equal to 3
• Because both of these numbers are very small (3,3), in some sense it is a "simple" Calabi-Yau and a very attractive one for me
• Also, because 3=3, I guess that this Calabi-Yau is its own self-mirror, is that correct? At least the Hodge diamond seems to imply it
• It has the right fundamental group allowing you to break SO(10) to the Standard Model via the Wilson lines - a pretty special thing, they say
  
  

OK, so this is a nice smooth Calabi-Yau space and one can use it both in weakly coupled as well as strongly coupled (Horava-Witten) heterotic strings. One obtains a N=1 supersymmetric Standard Model with the observable sector:

• The gauge bosons of SU(3) x SU(2) x U(1) x U(1)_{B-L}
• Three generations of quarks and leptons with right-handed neutrinos
• The number of generations 3 requires that the 3rd Chern class of the SU(4) bundle must be +54 or -54
  
  
• Two pairs of Higgs doublets
• Six geometric moduli and a small number of vector bundle moduli
• No exotic matter fields (charged under the SM group)

The hidden sector, responsible e.g. for SUSY breaking via gaugino condensation, has

• A small number of vector bundle moduli
• The gauge group Spin(12) at weak coupling, or E_7 x U(6) at strong coupling
• For the weak coupling, there are also 2 matter field multiplets in 12 of Spin(12), and no matter fields at the strong coupling

This kind of an old-fashioned model does not solve some problems, at least not explicitly, but in some sense I still find it more attractive than the various "postmodern" models that have been studied in the recent years. The word "postmodernism" is especially meant to include the anthropically large numbers of solutions, but also various attempts to squeeze completely new physics that should be visible at the LHC.



I wonder whether there are dual descriptions to this vacuum. For example, because the Calabi-Yau three-fold is an elliptic fibration, one should be able to use the heterotic/F-theory duality to get an equivalent description of this model as F-theory on a K3-fibration over dP_9, is that correct?