The full derivation of the (minimal) Standard Model, minimally coupled with gravity from a noncommutative manifold can be found in

Abstract:We present an effective unified theory based on noncommutative geometry for the standard model with neutrino mixing, minimally coupled to gravity. The unification is based on the symplectic unitary group in Hilbert space and on the spectral action. It yields all the detailed structure of the standard model with several predictions at unification scale. Besides the familiar predictions for the gauge couplings as for GUT theories, it predicts the Higgs scattering parameter and the sum of the squares of Yukawa couplings. From these relations one can extract predictions at low energy, giving in particular a Higgs mass around 170 GeV and a top mass compatible with present experimental value. The geometric picture that emerges is that space-time is the product of an ordinary spin manifold (for which the theory would deliver Einstein gravity) by a finite noncommutative geometry F. The discrete space F is of KO-dimension 6 modulo 8 and of metric dimension 0, and accounts for all the intricacies of the standard model with its spontaneous symmetry breaking Higgs sector.

The high Higgs mass was lowered to a more realistic value in

A.H. Chamseddine and A. Connes. *Resilience of the spectral Standard Model*. JHEP 104 (2012)

Abstract:We show that the inconsistency between the spectral Standard Model and the experimental value of the Higgs mass is resolved by the presence of a real scalar field strongly coupled to the Higgs field. This scalar field was already present in the spectral model and we wrongly neglected it in our previous computations. It was shown recently by several authors, independently of the spectral approach, that such a strongly coupled scalar field stabilizes the Standard Model up to unification scale in spite of the low value of the Higgs mass. In this letter we show that the noncommutative neutral singlet modifies substantially the RG analysis, invalidates our previous prediction of Higgs mass in the range 160-180 Gev, and restores the consistency of the noncommutative geometric model with the low Higgs mass.

A mathematical framework required because of the abscence of the so-called **first-order condition **was published in

Abstract:We extend inner fluctuations to spectral triples that do not fulfill the first-order condition. This involves the addition of a quadratic term to the usual linear terms. We find a semi-group of inner fluctuations, which only depends on the involutive algebra A and which extends the unitary group of A. This has a key application in noncommutative spectral models beyond the Standard Model, of which we consider here a toy model.

with the application to noncommutative models beyond the Standard Model in

Abstract:The assumption that space-time is a noncommutative space formed as a product of a continuous four dimensional manifold times a finite space predicts, almost uniquely, the Standard Model with all its fermions, gauge fields, Higgs field and their representations. A strong restriction on the noncommutative space results from the first order condition which came from the requirement that the Dirac operator is a differential operator of order one. Without this restriction, invariance under inner automorphisms requires the inner fluctuations of the Dirac operator to contain a quadratic piece expressed in terms of the linear part. We apply the classification of product noncommutative spaces without the first order condition and show that this leads immediately to a Pati-Salam SU(2)_R x SU(2)_L x SU(4) type model which unifies leptons and quarks in four colors. Besides the gauge fields, there are 16 fermions in the (2,2,4) representation, fundamental Higgs fields in the (2,2,1), (2,1,4) and (1,1,1+15) representations. Interestingly we encounter a new phenomena where the Higgs fields in the high energy sector are composite and depend quadratically on the fundamental Higgs fields. The Pati-Salam symmetries are broken spontaneously at high energies to those of the Standard Model.