Spectral Geometry with Exceptional Symmetry and Charged Higgs Fields

We lay the foundations for a general approach to nonassociative spectral geometry as an extension of Connes' noncommutative geometry by explaining how to construct finite-dimensional, discrete spectral geometries with exceptional symmetry, and gauge covariant Dirac operators. We showcase an explicit construction of a geometry corresponding to the internal space of a G₂× G₂ gauge theory with charged scalar content and scalar representations restricted by novel conditions arising from the associative properties of the coordinate algebra. Our construction motivates a new definition of bimodules over nonassociative algebras and a novel form of bimodule over semi-simple octonion algebras.