The $K$-theoretic bulk-edge correspondence for topological insulators

TL;DR

This paper applies Kasparov theory to model the bulk-edge correspondence in topological insulators, using real $C^*$-algebras and unbounded Kasparov modules to encode the system dynamics.

Contribution

It introduces a $K$-theoretic framework for topological insulators that incorporates anti-linear symmetries via real $C^*$-algebras and constructs Kasparov modules linking bulk and edge systems.

Findings

Established a Kasparov product linking bulk and edge modules.

Extended $K$-theoretic methods to systems with anti-linear symmetries.

Provided a mathematical model for bulk-edge correspondence in topological insulators.

Abstract

We study the application of Kasparov theory to topological insulator systems and the bulk-edge correspondence. We consider observable algebras as modelled by crossed products, where bulk and edge systems may be linked by a short exact sequence. We construct unbounded Kasparov modules encoding the dynamics of the crossed product. We then link bulk and edge Kasparov modules using the Kasparov product. Because of the anti-linear symmetries that occur in topological insulator models, real $C^*$-algebras and $KKO$-theory must be used.