SO(9) characterization of the standard model gauge group

Journal of Mathematical Physics, Volume 62, Issue 2, February 2021. A recent series of works characterized the Standard Model (SM) gauge group GSM as the subgroup of SO(9) that, in the octonionic model of the latter, preserves the split [math] of the space of octonions into a copy of the complex plane plus the rest. This description, however, proceeded via the exceptional Jordan algebras [math] and, in this sense, remained indirect. One of the goals of this paper is to provide as explicit a description as possible and also to clarify the underlying geometry. The other goal is to emphasize the role played by different complex structures in the spaces [math] and [math]. We provide a new characterization of GSM: The group GSM is the subgroup of Spin(9) that commutes with a certain complex structure JR in the space [math] of Spin(9) spinors. The complex structure JR is parameterized by a choice of a unit imaginary octonion. This characterization of GSM is essentially octonionic in the sense that JR is restrictive because octonions are non-associative. The quaternionic analog of JR is the complex structure in the space [math] of Spin(5) spinors that commutes with all Spin(5) transformations.