Braids, normed division algebras, and Standard Model symmetries

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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 B3 c, 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 B3 c 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 B3 c with trivial braiding in B2, something which is not possible for framed braids in general.

Original languageEnglish
Pages (from-to)212-221
Number of pages10
JournalPhysics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics
Publication statusPublished - 10 Aug 2018

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