Topological Investigation of the Fractionally Quantized Hall Conductivity

TL;DR

This paper explores the fractional quantum Hall effect using topological and geometric methods, relating conductivity to linking numbers and spin structures in a fiber bundle framework.

Contribution

It introduces a topological fiber bundle approach to analyze FQHE, connecting conductivity to linking numbers and spin structures, and relates it to Chern--Simons theory.

Findings

Conductivity expressed as a linking number of winding numbers.

Topological invariance of the fractional quantum Hall conductivity.

Explanation of odd denominators via spin structures.

Abstract

Using the fiber bundle concept developed in geometry and topology, the fractionally quantized Hall conductivity is discussed in the relevant many--particle configuration space. Electron-magnetic field and electron-electron interactions under FQHE conditions are treated as functional connections over the torus, the torus being the underlying two-dimensional manifold. Relations to the $(2+1)$--dimensional Chern--Simons theory are indicated. The conductivity being a topological invariant is given as $\frac{e^2}{h}$ times a linking number which is the quotient of the winding numbers of the self-consistent field and the magnetic field, respectively. Odd denominators are explained by the two spin structures which have been considered for the FQHE correlated electron system.