Topological Investigation of the Fractionally Quantized Hall Conductivity

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.