SO(9) characterization of the standard model gauge group

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 O=C⊕C3 of the space of octonions into a copy of the complex plane plus the rest. This description, however, proceeded via the exceptional Jordan algebras J3(O),J2(O) 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 O and O2. 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 O2 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 H2 of Spin(5) spinors that commutes with all Spin(5) transformations.