The fermionic fields of one generation of the Standard Model (SM), including the Lorentz spinor degrees of freedom, can be identified with components of a single real 64-dimensional semi-spinor representation S+ of the group Spin(11, 3). We describe an octonionic model for Spin(11, 3) in which the semi-spinor representation gets identified with S+=O - O, where O,O are the usual and split octonions, respectively. It is then well known that choosing a unit imaginary octonion u∈Im(O) equips O with a complex structure J. Similarly, choosing a unit imaginary split octonion u∈Im(O) equips O with a complex structure J, except that there are now two inequivalent complex structures, one parameterized by a choice of a timelike and the other of a spacelike unit u. In either case, the identification S+=O- O implies that there are two natural commuting complex structures J,J on S+. Our main new observation is that the subgroup of Spin(11, 3) that commutes with both J,J on S+ is the direct product Spin(6) × Spin(4) × Spin(1, 3) of the Pati-Salam and Lorentz groups, when u is chosen to be timelike. The splitting of S+ into eigenspaces of J corresponds to splitting into particles and anti-particles. The splitting of S+ into eigenspaces of J corresponds to splitting of Lorentz Dirac spinors into two different chiralities. This provides an efficient bookkeeping in which particles are identified with components of such an elegant structure as O - O. We also study the simplest possible symmetry breaking scenario with the "Higgs"field taking values in the representation that corresponds to three-forms in R11,3. We show that this Higgs can be designed to transform as the bi-doublet of the left/right symmetric extension of the SM and thus breaks Spin(11, 3) down to the product of the SM, Lorentz, and U(1)B-L groups, with the last one remaining unbroken. This three-form Higgs field also produces the Dirac mass terms for all the particles.
Krasnov, K. (2022). Spin(11, 3), particles, and octonions. Journal of Mathematical Physics, 63(3), Article 031701. https://doi.org/10.1063/5.0070058