Carlos Castro Perelman (2013)

A Clifford *Cl *(5, *C*) unified gauge field theory formulation of conformal gravity and *U *(4) × *U *(4) ×*U *(4) Yang–Mills in 4D, is reviewed along with its implications for the Pati–Salam (PS) group *SU *(4) × *SU *(2)_{L} × *SU *(2)_{R}, and *trinification* grand unified theory models of three fermion generations based on the group *SU *(3)_{C} × *SU *(3)_{L} × *SU *(3)_{R}. We proceed with a brief review of a unification program of 4D gravity and *SU *(3) × *SU *(2) × *U *(1) Yang–Mills emerging from 8D pure quaternionic gravity. A realization of *E*_{8} in terms of the *Cl*(16) = *Cl*(8) ⊗ *Cl*(8) generators follows, as a preamble to F. Smith’s *E*_{8} and *Cl*(16) = *Cl*(8) ⊗ *Cl*(8) unification model in 8D. The study of chiral fermions and instanton backgrounds in **CP ^{2}** and

**CP**related to the problem of obtaining three fermion generations is thoroughly studied. We continue with the evaluation of the coupling constants and particle masses based on the geometry of bounded complex homogeneous domains and geometric probability theory. An analysis of neutrino masses, Cabbibo–Kobayashi–Maskawa quark-mixing matrix parameters, and neutrino-mixing matrix parameters follows. We finalize with some concluding remarks about other proposals for the unification of gravity and the Standard Model, like string, M, and F theories and noncommutative and nonassociative geometry.

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