Beyond the standard model with six-dimensional spinors

Abstract

6D spinors with $Spin(3,3)$ symmetry are utilized to efficiently encode three generations of matter. $E_{8(-24)}$ is shown to contain physically relevant subgroups with representations for GUT groups, spacetime symmetries, three generations of the standard model fermions, and Higgs bosons. Pati-Salam, $SU(5)$, and $Spin(10)$ grand unified theories are found when a single generation is isolated. For spacetime symmetries, $Spin(4,2)$ may be used for conformal symmetry, $Ad{S}_5\to d{S}_4$, or simply broken to $Spin(3,1)$ of Minkowski space. Another class of representations finds $Spin(2,2)$ and can give $Ad{S}_3$ with various GUTs. An action for three generations of fermions in the Majorana-Weyl spinor $128$ of $Spin(4,12)$ is found with $Spin(3)$ flavor symmetry inside $E_{8(-24)}$. The $128$ of $Spin(4,12)$ can be regarded as the tangent space to a particular pseudo-Riemannian form of the octo-octonionic Rosenfeld projective plane $E_{8(-24)}/Spin(4,12)=({\mathbb{O}}_s\times \mathbb{O}){\mathbb{P}}^2$.

28 pages with references, 1 figure; v3: Abstract changed, various updates based on feedback from referees, addition of Lagrangian for 3 generations

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• Collineation groups of octonionic and split-octonionic planes
• A New Division Algebra Representation of $E_6$
• Division Algebras, Triality, and Exceptional Magic
• A Proposed $E_8\times E_8$ Kinematic Scaffolding for the Standard Model with Pre-Gravitation
• Flipped $SU(5)$ GUT with conformal gravity from a single supermultiplet