The Geometry of Jordan Matrix Models

We investigate the spectral geometry of the exceptional Jordan algebra and its extensions. We examine the spectrum of the exceptional Jordan algebra over the sixteen-dimensional space of primitive idempotents, where it exhibits three real eigenvalues. We interpret the spectrum as coordinates for a coincident D-brane system where the real eigenvalues correspond to positions of three D0-branes on a line in the octonionic projective plane. The F4 gauge symmetry arises from the isometries of the octonionic projective plane. An argument is also given for the exceptional Jordan C*-algebra, where E6 symmetry arises. We conclude that M-theory, in Jordan matrix models, is inherently a sixteen-dimensional theory originating in octonionic matrix space. This matrix space demonstrates the existence of a Jordan algebraic Gel’fand-Naimark theorem, where a nonassociative geometry is produced from the spectrum of a nonassociative algebra.