SO(9) characterisation of the Standard Model gauge group

Abstract: A recent series of works by M. Dubois-Violette, I. Todorov and S. Drenska characterised the SM gauge group GSM as the subgroup of SO(9) that, in the octonionic model of the later, 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 and this sense remained indirect. One of the goals of this paper is to provide as explicit description as possible and also clarify the underlying geometry. The other goal is to emphasise the role played by different complex structures in the spaces O and O2. We provide a new characterisation of GSM: The group GSM is the subgroup of Spin(9) that commutes with of a certain complex structure J in the space O2 of Spin(9) spinors. The complex structure J is parametrised by a choice of a unit imaginary octonion. This characterisation of GSM is essentially octonionic in the sense that J is restrictive because octonions are non-associative. The quaternionic analog of J is the complex structure in the space H2 of Spin(5) spinors that commutes with all Spin(5) transformations.