The $n$-point Exceptional Universe

Abstract

We solve an open problem in spectral geometry: the construction of finite-dimensional, discrete geometries coordinatized by non-simple, exceptional Jordan algebras. The approach taken is readily generalisable to broad classes of nonassociative geometries, opening the door to the spectral geometric desciption of gauge theories with exceptional symmetries. We showcase a proof-of-principle 2-point geometry corresponding to the internal space of an $F_4 \times F_4$ gauge theory with scalar content restricted by novel conditions arising from the associative properties of the coordinate algebra. We then formally establish a setting for generalising to n-point exceptional Jordan geometries with distinct points coupled together via an action on 1-forms constructed as split Jordan bimodules.