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, $AdS_5\rightarrow dS_4$, or simply broken to $Spin(3,1)$ of Minkowski space. Another class of representations finds $Spin(2,2)$ and can give $AdS_3$ with various GUTs. An action for three generations of fermions in the Majorana-Weyl spinor ${ \bf 128}$ of $Spin(4,12)$ is found with $Spin(3)$ flavor symmetry inside $E_{ 8(-24)}$. The ${ \bf 128}$ of $Spin(4,12)$ can be regarded as the tangent space to a particular pseudo-Riemannian form of the octo-octonionic Rosenfeld projective plane $E_{ 8(-24)}/Spin(4,12)= (\mathbb{ O}_s\times\mathbb{ O})\mathbb{ P}^2$.
Categories: physics.gen-ph hep-ph hep-th