Topological insulators and stable isomorphism versus isomorphism of vector bundles

TL;DR

This paper explores the mathematical structures of topological insulators, focusing on vector bundles and K-theory, and investigates conditions under which stable isomorphism implies isomorphism to better classify topological phases.

Contribution

It provides new insights into when stable isomorphism of vector bundles implies isomorphism, especially in the context of real and quaternionic bundles relevant to topological insulators.

Findings

Stable isomorphism does not always imply isomorphism for vector bundles.

Conditions are identified under which K-theory fully distinguishes topological phases.

Ongoing work extends results to G-equivariant K-theory for finite groups.

Abstract

This note gives an overview of the mathematical framework underlying topological insulators, highlighting the connection to K-theory and vector bundles. We see ``real'' and ``quaternionic'' vector bundles arise naturally in the presence of time-reversal symmetry. Our recent results about when stable isomorphism implies isomorphism are summarised, including some ongoing work for G-equivariant K-theory for finite groups. This clarifies when K-theory completely distinguishes topological phases.