Spectral geometry with exceptional symmetry and charged Higgs fields

We lay the groundwork for a general approach to nonassociative spectral geometry as an extension of Connes’ noncommutative geometry, by constructing finite-dimensional, discrete spectral geometries exhibiting exceptional symmetry and gauge-covariant Dirac operators. We identify the central challenges in generalizing to the nonassociative setting as: (i) the construction of generalized Dirac operators, and (ii) the formulation of nonassociative bimodules, which serve as analogues of spaces of differential forms. We propose a strategy to address these challenges and implement it explicitly for alternative algebras, chosen for their relative computational simplicity. As a concrete example, we present a spectral geometry corresponding to the internal space of a G2 × G2 gauge theory with charged scalar fields, where the scalar representations are constrained by novel conditions stemming from the associative properties of the coordinate algebra and its representation. This construction motivates a new definition of bimodules over nonassociative algebras in general, and introduces a novel form of bimodule tailored to semisimple octonion algebras specifically.