Levi-Civita connections for a class of noncommutative minimal surfaces

TL;DR

This paper develops a framework for metric and torsion-free connections on noncommutative minimal surfaces, including fuzzy spheres, using hermitian modules and index calculus, advancing the understanding of noncommutative differential geometry.

Contribution

It introduces a method to construct metric and torsion-free connections on noncommutative minimal surfaces, expanding the toolkit for noncommutative geometry research.

Findings

Existence of metric connections on hermitian modules

Development of an index calculus for such modules

Existence of metric and torsion-free connections on noncommutative minimal surfaces

Abstract

We study connections on hermitian modules, and show that metric connections exist on regular hermitian modules; i.e finitely generated projective modules together with a non-singular hermitian form. In addition, we develop an index calculus for such modules, and provide a characterization in terms of the existence of a pseudo-inverse of the matrix representing the hermitian form with respect to a set of generators. As a first illustration of the above concepts, we find metric connections on the fuzzy sphere. Finally, the framework is applied to a class of noncommutative minimal surfaces, for which there is a natural concept of torsion, and we prove that there exist metric and torsion-free connections for every minimal surface in this class.