Spectral Geometry with Exceptional Symmetry and Charged Higgs Fields

TL;DR

This paper develops a framework for nonassociative spectral geometry extending Connes' approach, constructing finite geometries with exceptional symmetry and gauge covariant Dirac operators, relevant for advanced gauge theories.

Contribution

It introduces a method to build finite nonassociative spectral geometries with exceptional symmetry and charged scalar fields, expanding the scope of noncommutative geometry.

Findings

Constructed a geometry for a G_2×G_2 gauge theory with charged scalars.

Proposed a new definition of bimodules over nonassociative algebras.

Established conditions for scalar representations from algebra associativity.

Abstract

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 $G_2\times G_2$ 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.