Particle models from special Jordan backgrounds and spectral triples

TL;DR

This paper introduces a new framework for spectral triples based on Jordan algebras, providing gauge-invariant bosonic configurations and addressing longstanding issues in noncommutative geometry.

Contribution

It develops a novel approach to spectral triples using Jordan coordinate algebras, differing from traditional associative noncommutative geometry, and solves the unimodularity problem for gauge fields.

Findings

Defined spectral triples from Jordan algebras

Constructed gauge-invariant bosonic configuration spaces

Achieved unimodular gauge fields in Jordan noncommutative geometry

Abstract

We put forward a definition for spectral triples and algebraic backgrounds based on Jordan coordinate algebras. We also propose natural and gauge-invariant bosonic configuration spaces of fluctuated Dirac operators and compute them for general, almost-associative, Jordan, coordinate algebras. We emphasize that the theory so obtained is not equivalent with usual associative noncommutative geometry, even when the coordinate algebra is the self-adjoint part of a $C^*$-algebra. In particular, in the Jordan case, the gauge fields are always unimodular, thus curing a long-standing problem in noncommutative geometry.