Three generations, two unbroken gauge symmetries, and one eight-dimensional algebra

Abstract

A considerable amount of the standard model's three-generation structure can be realised from just the 8 C -dimensional algebra of the complex octonions. Indeed, it is a little-known fact that the complex octonions can generate on their own a 64 C -dimensional space. Here we identify an s u ( 3 ) ⊕ u ( 1 ) action which splits this 64 C -dimensional space into complexified generators of S U ( 3 ) , together with 48 states. These 48 states exhibit the behaviour of exactly three generations of quarks and leptons under the standard model's two unbroken gauge symmetries. This article builds on a previous one, [1], by incorporating electric charge.