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Division Algebras and Supersymmetry I

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arxiv 0909.0551 v3 pith:HRPNEV2J submitted 2009-09-02 hep-th math.DGmath.RA

John C. Baez , John Huerta This is my paper · ORCID
classification hep-thmath.DGmath.RA
keywords algebrasdivisionsupersymmetrynumbersaccountcasescertaincomplex
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Supersymmetry is deeply related to division algebras. Nonabelian Yang-Mills fields minimally coupled to massless spinors are supersymmetric if and only if the dimension of spacetime is 3, 4, 6 or 10. The same is true for the Green-Schwarz superstring. In both cases, supersymmetry relies on the vanishing of a certain trilinear expression involving a spinor field. The reason for this, in turn, is the existence of normed division algebras in dimensions 1, 2, 4 and 8: the real numbers, complex numbers, quaternions and octonions. Here we provide a self-contained account of how this works.

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  1. The Dirac equation in (split-)octonions: origins, variants, and modern context

    math-ph 2026-06 conditional novelty 5.0 of 10

    Four approaches to the Dirac equation in (split-)octonions are classified; a 2024 split-octonionic equation is shown identical to the 2006 2-factor form by structure-preserving rotation and relabeling.

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