Quantum Hall Effect on the Hyperbolic Plane

TL;DR

This paper investigates the Quantum Hall Effect on the hyperbolic plane, establishing a topological framework that confirms the quantization of Hall conductivity through advanced algebraic and geometric methods.

Contribution

It introduces a twisted Kasparov map and links Hall conductivity to a topological index, extending QHE analysis to hyperbolic geometries.

Findings

Hall conductivity is a topological invariant.

Quantization of Hall conductivity is proven in hyperbolic geometry.

New algebraic tools are developed for QHE analysis.

Abstract

In this paper, we study both the continuous model and the discrete model of the Quantum Hall Effect (QHE) on the hyperbolic plane. The Hall conductivity is identified as a geometric invariant associated to an imprimitivity algebra of observables. We define a twisted analogue of the Kasparov map, which enables us to use the pairing between $K$-theory and cyclic cohomology theory, to identify this geometric invariant with a topological index, thereby proving the integrality of the Hall conductivity in this case.