On matrix geometry

TL;DR

This paper explores the foundations of matrix geometry, focusing on noncommutative spaces and their emergent Riemannian geometry, with implications for gravity in Yang-Mills matrix models and potential realizations of 4D geometries.

Contribution

It introduces a framework for embedded noncommutative spaces with emergent geometry, connecting matrix models to spectral geometry and 4D spacetime realization.

Findings

Noncommutative branes can have emergent Riemannian geometry.

The class of configurations is stable under small deformations.

A sketch for realizing 4D geometries in matrix models is provided.

Abstract

The foundations of matrix geometry are discussed, which provides the basis for recent progress on the effective geometry and gravity in Yang-Mills matrix models. Basic examples lead to a notion of embedded noncommutative spaces (branes) with emergent Riemannian geometry. This class of configurations turns out to be preserved under small deformations, and is therefore appropriate for matrix models. The relation with spectral geometry is discussed. A possible realization of sufficiently generic 4-dimensional geometries as noncommutative branes in D=10 matrix models is sketched.