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October 16, 2025Open Access

Spectral Geometry with Exceptional Symmetry and Charged Higgs Fields

SFShane FarnsworthRegensburg University of Applied Sciences

Key Points

The study presents a framework for nonassociative spectral geometry, enhancing the understanding of gauge theories with charged scalar content.
An explicit construction showcases a geometry for a G2×G2 gauge theory, creating innovative definitions of bimodules over nonassociative algebras.
The work highlights novel conditions affecting the associative properties of coordinate algebra, impacting the design of Dirac operators.
This research calls for further exploration into bimodules over semi-simple octonion algebras, potentially advancing theoretical physics.

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 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.

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