SO(10)
SO(10), also written as SO10, is a special orthogonal group of rank 5. More specifically, it is the group of rotations in 10-dimensional Euclidean space, represented by 10x10 orthogonal matrices with determinant 1. SO(10) is a Lie group, and its corresponding Lie algebra is so(10), or sometimes denoted 𝔰𝔬(10).
In particle physics and theoretical physics, SO(10) is significant as a Grand Unified Theory (GUT) gauge group. It is larger than the minimal SU(5) GUT and provides a natural framework for incorporating the Standard Model gauge group (SU(3)×SU(2)×U(1)) and right-handed neutrinos. The inclusion of right-handed neutrinos, along with the known quarks and leptons, allows for a more complete representation of fermions in a single irreducible representation of SO(10), typically the 16-dimensional spinor representation. This elegant unification is a primary motivation for considering SO(10) in GUT models.
SO(10) has several notable subgroups, including SU(5), the Pati-Salam group (SU(4)×SU(2)L×SU(2)R), and flipped SU(5). The symmetry breaking patterns of SO(10) determine the intermediate gauge groups that might exist at higher energy scales before the Standard Model emerges at lower energies. The study of these breaking patterns and the resulting particle spectra is a key aspect of SO(10) GUT model building.
The spinor representations of SO(10) are chiral (i.e., they are inequivalent to their complex conjugates), which is essential for explaining the observed chirality of the Standard Model fermions. This chirality arises naturally within the SO(10) framework, further supporting its relevance in particle physics. The study of fermion masses and mixing angles in SO(10) models is an active area of research.