Topological insulators and stable isomorphism versus isomorphism of vector bundles
Ralf Meyer
TL;DR
The note surveys the mathematical framework for topological insulators via $C^*$-algebras and $K$-theory, highlighting how real and quaternionic structures arise from time-reversal symmetry and how the Bloch bundle over $\mathbb{T}^{d}$ encodes topological data. It connects bulk invariants to protected boundary states through the Toeplitz extension and the index map, with the $F=\operatorname{sign}(H)$ construction yielding a $[p]\in K_0(\mathrm C^*(\mathbb{Z}^{d}))$ whose boundary index detects robust edge modes. The authors present explicit stability results (Bakuradze-Meyer) showing that stable isomorphism implies isomorphism for real bundles and for quaternionic bundles above rank thresholds, and discuss ongoing work in $G$-equivariant $K$-theory that introduces new phenomena when finite groups act. Altogether, the work clarifies when $K$-theory completely distinguishes topological phases and emphasizes the role of representation data in equivariant settings.
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.