Fractional Quantum Hall Effect for Extended Objects: from Skyrmionic Membranes to Dyonic Strings

TL;DR

This paper extends the concept of fractional quantum Hall effect from two dimensions to six, describing how extended objects like membranes and strings exhibit fractional charge and statistics, revealing higher-dimensional topological phenomena.

Contribution

It introduces a higher-dimensional FQHE framework involving membranes and strings, generalizing known 2D concepts to 6D with new gauge fields and topological effects.

Findings

Membranes carry fractional charge and statistics in 6D FQHE.

Skyrmionic membranes relate to higher-dimensional quantum Hall ferromagnetism.

Dyonic strings acquire electric charge via a generalized Witten effect.

Abstract

It is well known that in two spatial dimensions the fractional quantum Hall effect (FQHE) deals with point-like anyons that carry fractional electric charge and statistics. Moreover, in presence of a SO(3) order parameter, point-like skyrmions emerge and play a central role in the corresponding quantum Hall ferromagnetic phase. In this work, we show that in six spatial dimensions, the FQHE for extended objects shares very similar features with its two-dimensional counterpart. In the higher-dimensional case, the electromagnetic and hydrodynamical one-form gauge fields are replaced by three-form gauge fields and the usual point-like anyons are replaced by membranes, namely two-dimensional extended objects that can carry fractional charge and statistics. We focus on skyrmionic membranes, which are associated to a SO(5) order parameter and give rise to an higher-dimensional generalizaton of the quantum Hall ferromagnetism. We show that skyrmionic membranes naturally couple to the curved background through a generalized Wen-Zee term and can give us some insights about the chiral conformal field theory on the boundary. We then present a generalization of the Witten effect in six spatial dimensions by showing that one-dimensional extended monopoles (magnetic strings) in the bulk of the FQH states can acquire electric charge through an axion field by becoming dyonic strings.