Braided fermions from Hurwitz algebras

Abstract

Some curious structural similarities between a recent braid-and Hurwitz algebraic description of the unbroken internal symmetries for a single generations of Standard Model fermions were recently identified. The non-trivial braid groups that can be represented using the four normed division algebras are B2 and B c 3 , exactly those required to represent a single generation of fermions in terms of simple three strand ribbon braids. These braided fermion states can be identified with the basis states of the minimal left ideals of the Clifford algebra C (6), generated from the nested left actions of the complex octonions C ⊗ O on itself. That is, the ribbon spectrum can be related to octonion algebras. Some speculative ideas relating to ongoing research that attempts to construct a unified theory based on braid groups and Hurwitz algebras are discussed.

Topological preons from algebraic spinors

“…The connection between C (6) and C (4), with a topological representation of leptons and quarks in terms of 3-belts is therefore essentially unique. This is interesting because these Clifford algebras (and others) can be generated from the left and right actions of the octonions and quaternions on themselves respectively (or tensor products of division algebras acting on themselves) [5,[20][21][22].…”

Braids, 3-Manifolds, Elementary Particles: Number Theory and Symmetry in Particle Physics

“…Therefore, every S 2 of this manifold has no self-intersections. For the topology of the K3 surface with intersection form (7), this form has the desired form, but, as explained above, we will change the smoothness structure. The central idea is the usage of Casson handles CH for the 4-manifolds S 2 × S 2 \ pt, the onepoint complement of S 2 × S 2 .…”

“…In this paper, we will tackle this problem to get a geometrical/topological description of the standard model of elementary particle physics. Recently, there were efforts by Furey [2,3,4,5], Gresnigt [6,7,8] and Stoica [9] to use octonions and Clifford algebras to get a coherent model to describe the particle generations in the standard model. In the past, the stability of matter was related to topology like in the approach of Lord Kelvin [10] with knotted aether vortices.…”

Sedenions, the Clifford algebra $\mathbb{C}l(8)$, and three fermion generations

“…Finding a theoretical basis for the mathematical structure that underlies the SM remains a prominent challenge in physics. Instead of the common approach of embedding the SM gauge group into some larger group, recently there has been an interest in using the fundamental and generative structures of the four normed division algebras as a simple mathematical framework for particle physics [1,2,3,4,5,6,7,8].…”