An Algebraic Roadmap of Particle Theories, Part III: Intersections

TL;DR

This paper presents an algebraic approach to derive the Standard Model gauge symmetries from Spin(10) using a single constraint, extending to quaternionic and complex structures, and exploring intersections that mirror known symmetry breakings.

Contribution

It introduces a novel algebraic framework for deriving Standard Model symmetries via intersections of Lie algebra actions, bypassing detailed symmetry breaking pathways.

Findings

Single algebraic constraint links Spin(10) to Standard Model

Extension to quaternions and complex numbers reveals new symmetry breakings

Five-way intersection yields Standard Model gauge group

Abstract

In this article, we bypass the detailed symmetry breaking pathways established in [1]. Instead, a direct route from the Spin(10) model to the Standard Model is enabled via a single algebraic constraint. This single constraint, however, may be reconfigured as a requirement that three $\mathfrak{so}(10)$ actions coincide on a fixed space of multi-vector fermions. This $\mathfrak{so}(10) \mapsto \mathfrak{su}(3)_{C} \oplus \mathfrak{su}(2)_{L} \oplus{u}(1)_{Y}$ breaking (from a three-way intersection) mirrors, in certain ways, the $\mathfrak{so}(8) \mapsto \mathfrak{g}_2$ breaking (from a three-way intersection) in the context of octonionic triality. By extending this result to include quaternions and complex numbers, we find that a five-way intersection breaks $\mathfrak{so}(10) \mapsto \mathfrak{su}(3)_{C} \oplus \mathfrak{u}(1)_{Q}$. These are the Standard Model's unbroken gauge symmetries, post-Higgs-mechanism.