Geometry of Spin(10) Symmetry Breaking

TL;DR

This paper characterizes the Standard Model gauge group within Spin(10) using the geometry of pure spinors, revealing new structural insights and connections to known gauge groups through complex and product structures.

Contribution

It introduces a novel geometric characterization of the Standard Model gauge group as a subgroup of Spin(10) based on pure spinor stabilizers and product structures.

Findings

GSM stabilizes a pure spinor Psi_1 and projectively stabilizes Psi_2.

The intersection of stabilizers yields the Standard Model gauge group.

The geometric approach links pure spinors to complex and product structures in R^{10}.

Abstract

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.