SO(10) unification in noncommutative geometry

We construct an SO(10) grand unified theory in the formulation of noncommutative geometry. The geometry of space-time is that of a product of a continuous four-dimensional manifold times a discrete set of points. The properties of the fermionic sector fix almost uniquely the Higgs structure. The simplest model corresponds to the case where the discrete set consists of three points and the Higgs fields are ${16}_{\mathit{s}}$×16${\mathrm{¯}}_{\mathit{s}}$ and ${16}_{\mathit{s}}$×${16}_{\mathit{s}}$. The requirement that the scalar potential for all the Higgs fields not vanish imposes strong restrictions on the vacuum expectation values of the Higgs fields. We show that it is possible to remove these constraints by extending the number of discrete points to six and adding a singlet fermion and a ${16}_{\mathit{s}}$ Higgs field. Both models are studied in detail.