"Fluctuat nec Mergitur" (a motto for strings* from Paris;-)

The non linear hi(gg)story of the noncommutative spectral geometric unification

Noncommutative geometry was shown to provide a promising framework for unification of all fundamental interactions including gravity [3], [5], [6], [12], [10]. Historically, the search to identify the structure of the noncommutative space followed the bottom-up approach where the known spectrum of the fermionic particles was used to determine the geometric data that defines the space. This bottom-up approach involved an interesting interplay with experiments. While at first the experimental evidence of neutrino oscillations contradicted the first attempt [6], it was realized several years later in 2006 ([12]) that the obstruction to get neutrino oscillations was naturally eliminated by dropping the equality between the metric dimension of space-time (which is equal to 4 as far as we know) and its KO-dimension which is only defined modulo 8. When the latter is set equal to 2 modulo 8 [2], [4] (using the freedom to adjust the geometry of the finite space encoding the fine structure of space-time) everything works fine, the neutrino oscillations are there as well as the see-saw mechanism which appears for free as an unexpected bonus. Incidentally, this also solved the fermionic doubling problem by allowing a simultaneous Weyl-Majorana condition on the fermions to halve the degrees of freedom.
The second interplay with experiments occurred a bit later when it became clear that the mass of the Brout-Englert-Higgs boson would not comply with the restriction (that m_H=170 Gev) imposed by the validity of the Standard Model up to the unification scale. This obstruction to lower m_H was overcome in [11] simply by taking into account a scalar field which was already present in the full model which we had computed previously in [10]. One lesson which we learned on that occasion is that we have to take all the fields of the noncommutative spectral model seriously, without making assumptions not backed up by valid analysis, especially because of the almost uniqueness of the Standard Model (SM) in the noncommutative setting...

One indication that there must be a new higher scale that effects the low energy sector is the small mass of the neutrinos which is explained through the see-saw mechanism with a Majorana mass of at least of the order of 10¹¹GeV... Another indication of the need to modify the SM at high energies is the failure (by few percent) of the three gauge couplings to be unified at some high scale which indicates that it may be necessary to add other matter couplings to change the slopes of the running of the RG equations. This leads us to address the issue of the breaking from the natural algebra A which results from the classification of irreducible finite geometries of KO-dimension 6 (modulo 8) [giving the total almost commutative spacetime a KO dimension 2]... to the algebra corresponding to the SM. This breaking was effected in [9], [8] using the [mathematical] requirement of the first order condition on the Dirac operator... which is ... the main reason behind the unique selection of the SM.

The existence of examples of noncommutative spaces where the first order condition is not satisfied such as quantum groups and quantum spheres provides a motive to remove this condition from the classification of noncommutative spaces compatible with unification [14], [15], [16], [17]. This study was undertaken in a companion paper [13] where it was shown that in the general case the inner fluctuations of D form a semigroup in the product algebra A⊗Aop, and acquire a quadratic part in addition to the linear part. Physically, this new phenomena will have an impact on the structure of the Higgs fields which are the components of the connection along discrete directions... We modify the classification of irreducible finite geometries in the absence of the first order condition and show that the resultant algebra is, almost uniquely, given by HR⊕HL⊕M4(C)... the resultant model is the Pati-Salam [21] ... type model [which] truncates correctly to the SM.

... we have here a clear advantage over grand unified theories which suffers of having arbitrary and complicated Higgs representations. In the noncommutative geometric setting, this problem is now solved by having minimal representations of the Higgs fields. Remarkably, we note that a very close model to the one deduced here is the one considered by Marshak and Mohapatra where the U(1) of the left-right model is identified with the B−L symmetry. They proposed the same Higgs fields that would result starting with a generic initial Dirac operator not satisfying the first order condition. Although the broken generators of the SU (4) gauge fields can mediate lepto-quark interactions leading to proton decay, it was shown that in all such types of models with partial unification, the proton is stable. In addition this type of model arises in the first phase of breaking of SO(10) to SU(2)R×SU(2)L×SU (4) and these have been extensively studied [1]. 

...

We conclude that the study of noncommutative spaces based on a product of a continuous four dimensional manifold times a finite space of KO-dimension 6, without the first order condition gives rise to almost unique possibility in the form of a Pati-Salam type model. This provides a setting for unification avoiding the desert and which goes beyond the SM. In addition one of the vacua of the Higgs fields gives rise at low energies to a Dirac operator satisfying the first order condition. In this way, the first order condition arises as a spontaneously broken phase of higher symmetry and is not imposed from outside.

Ali H. Chamseddine, Alain Connes and Walter D. van Suijlekom, Beyond the Spectral Standard Model: Emergence of Pati-Salam Unification, october 30th 2013

* = Solid Theoretical Research In Natural Geometric Structures