# Spectral estimators for finite non-commutative geometries

## TL;DR

This paper develops spectral methods to extract geometric information from finite non-commutative geometries, such as fuzzy spheres and tori, by analyzing the Dirac operator spectrum at finite energy scales.

## Contribution

It introduces a spectral variance measure for dimension, compares volume computation methods, and investigates geometry distances using spectral zeta functions in finite non-commutative spaces.

## Key findings

- Spectral variance effectively measures the dimension of fuzzy spaces.
- Volume can be computed from spectrum using Dixmier trace and Abel Stern methods.
- Spectral zeta functions distinguish different fuzzy geometries.

## Abstract

A finite non-commutative geometry consists of a fuzzy space together with a Dirac operator satisfying the axioms of a real spectral triple. This paper addreses the question of how to extract information about these geometries from the spectrum of the Dirac operator. Since the Dirac operator is a finite-dimensional matrix, the usual asymptotics of the eigenvalues makes no sense and is replaced by measurements of the spectrum at a finite energy scale. The spectral dimension of the square of the Dirac operator is improved to provide a new spectral measure of the dimension of a space called the spectral variance. Similarly, the volume of a space can be computed from the spectrum once the dimension is known. Two methods of doing this are investigated: the well-known Dixmier trace and a recent improvement due to Abel Stern. Finally, the distance between two geometries is investigated by comparing the spectral zeta functions using the method of Cornelissen and Kontogeorgis. All of these techniques are tested on the explicit examples of the fuzzy spheres and fuzzy tori, which can be regarded as approximations of the usual Riemannian sphere and flat tori. Then they are applied to characterise some random fuzzy spaces using data generated by a Monte Carlo simulation.