Sums of two dimensional spectral triples

TL;DR

This paper constructs spectral triples from sums of two-dimensional modules over compact metric spaces, recovering the metric exactly or approximately, and analyzes their spectral properties, including explicit computations for the unit interval.

Contribution

It introduces a method to build spectral triples from two-dimensional modules that precisely or approximately recover the metric and topological features of compact metric spaces.

Findings

Spectral triples can exactly recover the metric on compact spaces.

The constructed spectral triples are finitely summable for any positive parameter.

Explicit computation for the unit interval shows the metric is recovered exactly.

Abstract

We study countable sums of two dimensional modules for the continuous complex functions on a compact metric space and show that it is possible to construct a spectral triple which gives the original metric back. This spectral triple will be finitely summable for any positive parameter. We also construct a sum of two dimensional modules which reflects some aspects of the topological dimensions of the compact metric space, but this will only give the metric back approximately. We make an explicit computation of the last module for the unit interval. The metric is recovered exactly, the Dixmier trace induces a multiple of the Lebesgue integral and the number N(K) of eigenvalues bounded by K behaves, such that N(K)/K is bounded, but without limit for K growing.