Fuzzy Geometries with an Internal Space

TL;DR

This paper explores a non-commutative spacetime model combining a spectral triple with an internal space, revealing gauge fields, geometry fluctuations, and novel induced bosonic terms.

Contribution

It introduces a new non-commutative geometric framework with an internal space, analyzing fluctuations and deriving induced bosonic terms.

Findings

Derived non-commutative gauge fields from fluctuations.

Calculated spacetime geometry fluctuations including charge-dependent derivatives.

Identified novel induced bosonic terms from fermion integration.

Abstract

The product of a non-commutative matrix spectral triple with a simple two-dimensional internal space is considered. This is interpreted as a non-commutative spacetime that contains one charged Dirac fermion and its antiparticle. The inner fluctuations of a vacuum Dirac operator are calculated, using the standard technique of Connes' one-forms. This results in the non-commutative analogue of a gauge field, as expected, and also fluctuations of the spacetime geometry. In addition, the fluctuations include a derivative operator that depends on the particle charge. The integral over the fermions in the model is calculated, leading to some novel induced bosonic terms.