Octions: An $E_8$ description of the Standard Model

Abstract

We interpret the elements of the exceptional Lie algebra ${\mathfrak{e}}_{8(-24)}$ as objects in the Standard Model, including lepton and quark spinors with the usual properties, the Standard Model Lie algebra $\mathfrak{su}(3)+\mathfrak{su}(2)+\mathfrak{u}(1)$, and the Lorentz Lie algebra $\mathfrak{so}(3,1)$. Our construction relies on identifying a complex structure on spinors and then working in the enveloping algebra. The resulting model naturally contains GUTs based on $SO(10)$ (Georgi--Glashow), $SU(5)$ (Georgi), and $SU(4)\times SU(2)\times SU(2)$ (Pati--Salam). We then briefly speculate on the role of the remaining elements of ${\mathfrak{e}}_8$, and propose a mechanism leading to exactly three generations of particles.

15 pages; to appear in J. Math. Phys

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• Three generations, two unbroken gauge symmetries, and one eight-dimensional algebra
• Division algebraic symmetry breaking
• Spin(11,3), particles and octonions
• One generation of standard model Weyl representations as a single copy of $\mathbb{R}\otimes \mathbb{C}\otimes \mathbb{H}\otimes \mathbb{O}$
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