SO(10) Unification in Non-Commutative Geometry

TL;DR

This paper develops an $SO(10)$ grand unified theory within non-commutative geometry, exploring how discrete space-time structures influence Higgs fields and fermion masses, and proposes extensions to relax model constraints.

Contribution

It constructs a novel $SO(10)$ GUT model in non-commutative geometry and analyzes how discrete space-time points affect Higgs and fermion mass structures.

Findings

Higgs structure is almost uniquely fixed by fermionic properties.

Constraints on Higgs VEVs can be relaxed by extending discrete points.

Models with added singlet fermions and Higgs fields are viable.

Abstract

We construct an $SO(10)$ grand unified theory in the formulation of non-com-\break mutative geometry. The geometry of space-time is that of a product of a continuos 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 ${\u {16}_s}\times \overline{\u {16}}_s$ and ${\u {16}_s}\times {\u {16}_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 and thus the fermion masses. We show that it is possible to remove these constraints by extending the number of discrete points and adding a singlet fermion and a ${\u {16}_s}$ Higgs field. Both models are studied in detail.