Noncommutative Differential Geometry on Infinitesimal Spaces

TL;DR

This paper develops a framework using noncommutative differential geometry to formalize discrete calculus on infinitesimal spaces, connecting poset approximations to classical manifolds and demonstrating convergence of finite difference methods.

Contribution

It introduces a novel approach to discretize differential calculus via noncommutative geometry, associating $C^*$-algebras and spectral triples to posets and manifolds.

Findings

Finite difference calculus recovered from spectral triples.

Convergence proven for $d$-lattice and torus cases.

Associations between posets, $C^*$-algebras, and manifolds established.

Abstract

In this paper, we use the language of noncommutative differential geometry to formalise discrete differential calculus. We begin with a brief review of inverse limit of posets as an approximation of topological spaces. We then show how to associate a $C^*$-algebra over a poset, giving it a piecewise-linear structure. Furthermore, we explain how dually the algebra of continuous function $C(M)$ over a manifold $M$ can be approximated by a direct limit of $C^*$-algebras over posets. Finally, in the spirit of noncommutative differential geometry, we define a finite dimensional spectral triple on each poset. We show how the usual finite difference calculus is recovered as the eigenvalues of the commutator with the Dirac operator. We prove a convergence result in the case of the $d$-lattice in $\mathbb{R}^d$ and for the torus $\mathbb{T}^d$.