Spin (8,9,10), Octonions and the Standard Model

Abstract

"I will start by explaining how the (Weyl) spinor representations of the pseudo-orthogonal group Spin(2r+s,s) are the spaces of even and odd polyforms on Cr x Rs. Then, the triality identifies the Majorana-Weyl spinors of Spin(8) with octonions. Combining the two constructions one finds that the groups Spin(8+s,s) all have an octonionic description, with Weyl spinors of this group being a copy of O^(2^s). This also gives an octonionic description of the groups that can be embedded into Spin(8+s,s). Applying this construction to Spin(10,2) gives an octonionic description of Spin(10). The latter arises as the subgroup of Spin(10,2) that commutes with a certain complex structure on the space of its Weyl spinors O4. This gives a description of Weyl spinors of Spin(10) as O2_C, and an explicit description of the Lie algebra of Spin(10) as that of 2x2 matrices (of a special type) with complex (and octonionic) entries. It is well known from the SO(10) GUT that fermions of one generation of the SM can be described as components of a single Weyl spinor of Spin(10). Combining this with the previous construction one gets an explanation of why it is natural to identify elementary particles with components of two copies of complexified octonions. I explicitly describe the dictionary that provides this identification. I also describe how a choice of a unit imaginary octonion induces some natural complex structures on the space of Spin(10) Weyl spinors. For one of these complex structures, its commutant in Spin(10) is SU(2)_L x SU(2)_R x SU(3) x U(1). One thus gets a surprisingly large number of structures seen in the SM from very little input - a choice of a unit imaginary octonion. "