Dixon-Rosenfeld Lines and the Standard Model

Abstract

We present three new coset manifolds named Dixon-Rosenfeld lines that are similar to Rosenfeld projective lines except over the Dixon algebra $\mathbb{C}\otimes \mathbb{H}\otimes \mathbb{O}$. Three different Lie groups are found as isometry groups of these coset manifolds using Tits' formula. We demonstrate how Standard Model interactions with the Dixon algebra in recent work from Furey and Hughes can be uplifted to tensor products of division algebras and Jordan algebras for a single generation of fermions. The Freudenthal-Tits construction clarifies how the three Dixon-Rosenfeld projective lines are contained within $\mathbb{C}\otimes \mathbb{H}\otimes J_2(\mathbb{O})$, $\mathbb{O}\otimes J_2(\mathbb{C}\otimes \mathbb{H})$, and $\mathbb{C}\otimes \mathbb{O}\otimes J_2(\mathbb{H})$.

arXiv comment46 pages with references, 1 figure, 3 tables

References in corpus (14)

• Three generations, two unbroken gauge symmetries, and one eight-dimensional algebra
• Octonions, E6, and Particle Physics
• Generations: Three Prints, in Colour
• Deducing the symmetry of the standard model from the automorphism and structure groups of the exceptional Jordan algebra
• $SU(3{)}_C\times SU(2{)}_L\times U(1{)}_Y\left(\times U(1{)}_X\right)$ as a symmetry of division algebraic ladder operators
• Charge quantization from a number operator
• Octonions, exceptional Jordan algebra and the role of the group F_4 in particle physics
• Division algebraic symmetry breaking
• Towards a unified theory of ideals
• The Standard Model particle content with complete gauge symmetries from the minimal ideals of two Clifford algebras
• One generation of standard model Weyl representations as a single copy of $\mathbb{R}\otimes \mathbb{C}\otimes \mathbb{H}\otimes \mathbb{O}$
• SO(9) characterisation of the Standard Model gauge group
• An Exceptional G(2) Extension of the Standard Model from the Correspondence with Cayley-Dickson Algebras Automorphism Groups
• Octonionic Planes and Real Forms of G2, F4 and E6

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