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 ⊂ Spin(10) is the group that stabilises a pure spinor Ψ1 and projectively stabilises another pure spinor Ψ2, with Ψ1,2 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 R10, and the product of two commuting complex structures is a what is known as a product structure. For the pure spinors Ψ1,2 satisfying the stated conditions the complex structures determined by Ψ1,2 commute and the arising product structure is R10=R6⊕R4, 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 Ψ1, and the Pati–Salam Spin(6) × Spin(4) arising from the product structure determined by Ψ1,2. 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.