Boundary maps for $C^*$-crossed products with R with an application to the quantum Hall effect

TL;DR

This paper constructs a boundary map in K-theory for crossed product algebras with R, linking it to the Connes-Thom isomorphism, and applies it to prove the equality of bulk and edge conductivities in the quantum Hall effect.

Contribution

It provides an elementary construction of a boundary map on higher traces in the context of crossed products with R, connecting non-commutative geometry to quantum Hall physics.

Findings

Established the boundary map as dual to Connes' pairing.

Proved the equality of quantized bulk and edge conductivities.

Applied non-commutative Stokes theorem to physical models.

Abstract

The boundary map in K-theory arising from the Wiener-Hopf extension of a crossed product algebra with R is the Connes-Thom isomorphism. In this article the Wiener Hopf extension is combined with the Heisenberg group algebra to provide an elementary construction of a corresponding map on higher traces (and cyclic cohomology). It then follows directly from a non-commutative Stokes theorem that this map is dual w.r.t.Connes' pairing of cyclic cohomology with K-theory. As an application, we prove equality of quantized bulk and edge conductivities for the integer quantum Hall effect described by continuous magnetic Schroedinger operators.