Index theory and topological phases of aperiodic lattices

TL;DR

This paper explores the application of noncommutative index theory to analyze topological phases in aperiodic lattices, providing mathematical tools for understanding invariants in quasicrystals and related materials.

Contribution

It introduces new methods for constructing unbounded Fredholm modules and analyzes index pairings for aperiodic lattice models, advancing the mathematical framework for topological materials.

Findings

Develops semifinite index pairings for aperiodic lattices

Constructs unbounded Fredholm modules for finite local complexity lattices

Provides a mathematical foundation for topological invariants in aperiodic systems

Abstract

We examine the noncommutative index theory associated to the dynamics of a Delone set and the corresponding transversal groupoid. Our main motivation comes from the application to topological phases of aperiodic lattices and materials, and applies to invariants from tilings as well. Our discussion concerns semifinite index pairings, factorisation properties of Kasparov modules and the construction of unbounded Fredholm modules for lattices with finite local complexity.