### "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-dimensionwhich 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 lowerm_{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 thatwe 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 thatthere 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 10... 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.^{11}GeVThis 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 thatthe 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 solvedby 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 continuousfour dimensional manifold times a ﬁnite space of KO-dimension 6, without the ﬁrst ordercondition 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 aspontaneously 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

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