A topological model of composite preons from the minimal ideals of two Clifford algebras

Niels G Gresnigt

doi:10.1016/j.physletb.2020.135687

Abstract

This paper demonstrates a direct correspondence between a recent algebraic characterization of leptons and quarks as basis elements of the minimal one-sided ideals of the complex Clifford algebras Cℓ(6) and Cℓ(4), shown earlier to transform as a single generation of leptons and quarks under the Standard Model's unbroken SU(3)c×U(1)em and SU(2)L gauge symmetries respectively, and a topological formulation of the Harari-Shupe preon model in which leptons and quarks are represented in terms of braids.It was previously shown that mapping a Witt basis of Cℓ(6) to particular braids in the circular Artin braid group B3c makes it possible to replicate the topological structure describing electrocolor symmetries in this preon model. This paper extends this curious correspondence, which involves only the minimal ideals of Cℓ(6) under SU(3)c×U(1)em, to include the SU(2)L chiral weak symmetry. This is achieved by mapping a Witt basis of an additional Cℓ(4) algebra to braids in B3, taken to be a subgroup of B3c. The braids corresponding to the charged vector bosons are determined, and it is demonstrated that chiral weak interactions can be described via the composition of braids.

Highlights

Grand unified theories (GUT) and preon models represent two approaches to motivating the Standard Model’s (SM) symmetry group, SU (3)c × SU (2)L × U (1)Y, and particle content from more fundamental principles
We have demonstrated that there exists a direct correspondence between the basis states of the minimal ideals of the complex Clifford algebras Cl(6) and Cl(4), and a simple topologically-based toy model in which leptons, quarks, and gauge bosons are represented as simple braids composed of three ribbons
Each ladder operator obtained from a Witt decomposition of Cl(6), and subsequently each basis state of the minimal left ideals Su and Sd, is identified with a simple braid in the circular braid group B3c, after which the resulting braiding is exchanged for twisting

Summary

Introduction

Grand unified theories (GUT) and preon models represent two approaches to motivating the Standard Model’s (SM) symmetry group, SU (3)c × SU (2)L × U (1)Y , and particle content from more fundamental principles The former approach merges the gauge groups of the SM into a single semi-simple Lie group. The chiral weak symmetry can be described via the minimal right ideals of Cl(4) [16] In these models, the finite particle content in the model is derived from the basis states of the finite-dimensional minimal left and right ideals of Clifford algebras.

Standard Model particle states from the minimal ideals of Clifford algebras

Mapping minimal ideal basis states to braided matter states

Electro-color twist structure

Weak braid structure

Charged vector bosons

Discussion