Geometry of Spin(10) Symmetry Breaking

We provide a new characterisation of the Standard Model gauge group GSM as a subgroup of Spin(10). The new description of GSM relies on the geometry of pure spinors. We show that GSM is the subgroup that stabilises a pure spinor Psi_1 and projectively stabilises another pure spinor Psi_2, with Psi_1, Psi_2 orthogonal and such that their arbitrary linear combination is still a pure spinor. Our characterisation of GSM relies on the facts that projective pure spinors describe complex structures on R^{10}, and the product of two commuting complex structures is a what is known as a product structure. For the pure spinors Psi_1, Psi_2 satisfying the stated conditions the complex structures determined by Psi_1, Psi_2 commute and the arising product structure is R^{10} = R^6 + R^4, giving rise to a copy of Pati-Salam gauge group inside Spin(10). Our main statement then follows from the fact that GSM is the intersection of the Georgi-Glashow SU(5) that stabilises Psi_1, and the Pati-Salam Spin(6) x Spin(4) arising from the product structure determined by Psi_1, Psi_2. We have tried to make the paper self-contained and provided a detailed description of the creation/annihilation operator construction of the Clifford algebras Cl(2n) and the geometry of pure spinors in dimensions up to and including ten.