Braids, normed division algebras, and Standard Model symmetries

Abstract

This paper represents a first attempt at unifying two promising attempts to understand the origin of the internal symmetries of leptons and quarks. It is shown that each of the four normed division algebras over the reals admits a representation of a circular braid group. For the complex numbers and the quaternions, the represented circular braid groups are B 2 and B c 3 , precisely those used to construct leptons and quarks as framed braids in the Helon model of Bilson-Thompson. It is then shown that these framed braids coincide with the states that span the minimal left ideals of the complex (chained) octonions, shown by Furey to describe one generation of leptons and quarks with unbroken SU (3) c and U (1) em symmetry.The identification of basis states of minimal ideals with certain framed braids is possible because the braiding in B 2 and B c 3 in the Helon model are interchangeable. It is shown that the framed braids in the Helon model can be written as pure braid words in B c 3 with trivial braiding in B 2 , something which is not possible for framed braids in general.

The Standard Model particle content with complete gauge symmetries from the minimal ideals of two Clifford algebras

“…Besides giving a particularly aesthetic embedding of one generations of fermions into minimal ideals of C (10), our model also provides a clear pathway for extending earlier work [8] which established a curious structural similarity between the basis states of the minimal left ideals of C (6) and certain braids used in a topological model of leptons and quarks [9]. Incorporating the results of the present paper to extend these earlier results to include the weak interaction will be the focus of another paper [10].…”

“…The motivation for this work came from the results of [8] which showed that there is a one-to-one correspondence between the basis states of the minimal left ideals of C (6) and the braided states in the topological model proposed in [9]. 5 To be able to extend this curious structural similarity to include the weak force, the underlying electrocolor structure generated from C (6) should remain intact.…”

“…This is not the case in the C (10) representation in [7], but is the case for the construction considered here. The extension of [8] to include the weak force will be the focus of an upcoming paper [10].…”

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].…”

A Supersymmetry and Quantum Cryptosystem with Path Integral Approach in Biology

“…In the loop structure of telomeres [7], there exist layers of G-quadruplex (G4) [15]. Their structures are similar to the structure with the loop braid group in the standard model [16,17]. The G letter (guanine) can be suitably chosen as the hidden time scale in a biological clock.…”