Three generations of colored fermions with $S_3$ family symmetry from Cayley-Dickson sedenions

TL;DR

This paper proposes a novel algebraic framework using Cayley-Dickson sedenions to model three fermion generations with $SU(3)_C$ symmetry, addressing the longstanding question of the origin of three generations.

Contribution

It introduces a new algebraic approach employing sedenions and the $S_3$ group to represent three fermion generations within the Standard Model gauge symmetries.

Findings

Representation of one fermion generation via minimal left ideals of $ ext{Cl}(6)$

Generation of two additional fermion generations using $S_3$ automorphisms of $ extbf{S}$

Provides a self-contained algebraic model for three generations of fermions

Abstract

An algebraic representation of three generations of fermions with $SU(3)_C$ color symmetry based on the Cayley-Dickson algebra of sedenions $\mathbb{S}$ is constructed. Recent constructions based on division algebras convincingly describe a single generation of leptons and quarks with Standard Model gauge symmetries. Nonetheless, an algebraic origin for the existence of exactly three generations has proven difficult to substantiate. We motivate $\mathbb{S}$ as a natural algebraic candidate to describe three generations with $SU(3)_C$ gauge symmetry. We initially represent one generation of leptons and quarks in terms of two minimal left ideals of $\mathbb{C}\ell(6)$, generated from a subset of all left actions of the complex sedenions on themselves. Subsequently we employ the finite group $S_3$, which are automorphisms of $\mathbb{S}$ but not of $\mathbb{O}$ to generate two additional generations. Given the relative obscurity of sedenions, efforts have been made to present the material in a self-contained manner.