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arXiv · 2506.21496

Spectral Geometry with Exceptional Symmetry and Charged Higgs Fields

Shane Farnsworth
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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.

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Shane Farnsworth. 2025-06-26. Spectral Geometry with Exceptional Symmetry and Charged Higgs Fields. https://arxiv.org/abs/2506.21496

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