The noncommutative geometry of wire networks from triply periodic surfaces

TL;DR

This paper explores the noncommutative geometric structure of wire networks derived from triply periodic minimal surfaces, specifically P, D, G surfaces, using the Harper Hamiltonian in magnetic fields.

Contribution

It introduces a classification of $C^*$-geometries associated with these wire networks through noncommutative geometry methods.

Findings

Classification of $C^*$-geometries for the wire networks

Application of Harper Hamiltonian in magnetic fields

Analysis of symmetric and self-dual graphs from triply periodic surfaces

Abstract

We study wire networks that are the complements of triply periodic minimal surfaces. Here we consider the P, D, G surfaces which are exactly the cases in which the corresponding graphs are symmetric and self-dual. Our approach is using the Harper Hamiltonian in a constant magnetic field. We treat this system with the methods of noncommutative geometry and obtain a classification for all the $C^*$ geometries that appear.