Matrix Geometries Emergent from a Point

TL;DR

This paper introduces a categorical framework for finite noncommutative geometries, demonstrating how matrix geometries like the Moyal plane can emerge from a single point through metric fluctuations.

Contribution

It presents a novel categorical approach using spectral triples to unify metric fluctuations and unitary equivalences in noncommutative geometry.

Findings

Matrix geometry emerges from a single point via fluctuations.

Categorical approach unifies different geometric transformations.

Discussion of geometric properties of the Moyal plane.

Abstract

We describe a categorical approach to finite noncommutative geometries. Objects in the category are spectral triples, rather than unitary equivalence classes as in other approaches. This enables to treat fluctuations of the metric and unitary equivalences on the same footing, as representatives of particular morphisms in this category. We then show how a matrix geometry (Moyal plane) emerges as a fluctuation from one point, and discuss some geometric aspects of this space.