Chern-Simons Duality and the Quantum Hall Effect

TL;DR

This paper explores the duality transformations in Chern-Simons theories related to the quantum Hall effect, linking bulk and edge descriptions and extending the hierarchy of Hall conductivities.

Contribution

It introduces a duality transformation for Chern-Simons theories that parallels the scalar field duality at the edge, connecting bulk actions with edge excitations.

Findings

Generated hierarchy of bulk Chern-Simons actions from edge dualities.

Established a duality in CS theory analogous to R→1/R in scalar fields.

Linked bulk and edge descriptions via anomaly cancellation.

Abstract

In previous work on the quantum Hall effect on an annulus, we used $O(d,d;{\bf Z})$ duality transformations on the action describing edge excitations to generate the Haldane hierarchy of Hall conductivities. Here we generate the corresponding hierarchy of ``bulk actions'' which are associated with Chern-Simons (CS) theories, the connection between the bulk and edge arising from the requirement of anomaly cancellation. We also find a duality transformation for the CS theory exactly analogous to the $R\rightarrow \frac{1}{R}$ duality of the scalar field theory at the edge.