Abstract
In this paper a structural similarity between a recent braid- and division algebraic description of the unbroken internal symmetries of a single generation of Standard Model (SM) fermions is identified. This unexpected connection between two independently motivated models provides the first step towards unifying them into a unified theory based on braid groups and normed division algebras (NDA).Each of the four NDAs over the reals is shown to contain a representation of a circular braid group. For the complex numbers and the quaternions, the represented circular braid groups are B2 and B3c, precisely those used to represent leptons and quarks as framed braids in the Helon model of Bilson-Thompson. It is then shown that the twist structure of these framed braids representing fermions coincides exactly 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.This identification of basis states of minimal ideals with certain framed braids is possible because the braiding in B2 and B3c 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 B3c with trivial braiding in B2, something which is not possible for framed braids in general.
Highlights
The identification of basis states of minimal ideals with certain framed braids is possible because the braiding in B2 and B3c in the Helon model are interchangeable
This paper makes a first attempt at unifying the model of Bilson-Thompson based on framed braids with the model of Furey based on minimal ideals of normed division algebras (NDAs)
In 2016, Kauffman and Lomonaco showed that Clifford algebras contain representations of circular braid groups and highlighted the close connection between the quaternions and topology, and how braiding is fundamental to the structure of fermionic physics [16]
Summary
We give the briefest of overviews of the Helon model, sufficient for our purposes. The embedding of framed braids into ribbon networks make it possible to develop a unified theory of matter and spacetime in which both are emergent from the ribbon networks [5] Within ribbon networks, these braided structures correspond to local noiseless subsystems which have been shown to exist in background independent theories where the microscopic quantum states are defined in terms of the embedding of a framed, or ribbon, graph in a three manifold and in which the allowed evolution moves are the standard local exchange and expansion moves (Pachner moves). These braided structures correspond to local noiseless subsystems which have been shown to exist in background independent theories where the microscopic quantum states are defined in terms of the embedding of a framed, or ribbon, graph in a three manifold and in which the allowed evolution moves are the standard local exchange and expansion moves (Pachner moves) Such noiseless subsystems are given by braided sets of n edges joined at both ends by a set of connected nodes. Smolin and Wan [19] have shown that braid interactions in tetravalent such spin networks are understood in terms of dual Pachner moves
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