The bulk-edge correspondence for the quantum Hall effect in Kasparov theory

TL;DR

This paper establishes a mathematical framework using Kasparov theory to rigorously prove the bulk-edge correspondence in the quantum Hall effect, linking bulk and boundary topological invariants.

Contribution

It introduces a novel unbounded Kasparov module construction that explicitly represents bulk invariants as products of boundary invariants and extension classes.

Findings

Proves bulk-edge correspondence in K-theory for quantum Hall effect

Constructs explicit Kasparov modules linking bulk and boundary algebras

Potentially applicable to broader topological phases

Abstract

We prove the bulk-edge correspondence in $K$-theory for the quantum Hall effect by constructing an unbounded Kasparov module from a short exact sequence that links the bulk and boundary algebras. This approach allows us to represent bulk topological invariants explicitly as a Kasparov product of boundary invariants with the extension class linking the algebras. This paper focuses on the example of the discrete integer quantum Hall effect, though our general method potentially has much wider applications.