6D Fractional Quantum Hall Effect

TL;DR

This paper generalizes the fractional quantum Hall effect to six dimensions, introducing a topological field theory involving membranes and three-form potentials, with implications for string theory and higher-dimensional physics.

Contribution

It develops a 6D fractional quantum Hall framework using a 7D topological field theory and connects it to string theory and F-theory, providing new insights into higher-dimensional topological phases.

Findings

Derived fractional conductivity in 6D system

Connected continued fractions to 6D superconformal classifications

Computed the Laughlin wavefunction analog in 6D

Abstract

We present a 6D generalization of the fractional quantum Hall effect involving membranes coupled to a three-form potential in the presence of a large background four-form flux. The low energy physics is governed by a bulk 7D topological field theory of abelian three-form potentials with a single derivative Chern-Simons-like action coupled to a 6D anti-chiral theory of Euclidean effective strings. We derive the fractional conductivity, and explain how continued fractions which figure prominently in the classification of 6D superconformal field theories correspond to a hierarchy of excited states. Using methods from conformal field theory we also compute the analog of the Laughlin wavefunction. Compactification of the 7D theory provides a uniform perspective on various lower-dimensional gapped systems coupled to boundary degrees of freedom. We also show that a supersymmetric version of the 7D theory embeds in M-theory, and can be decoupled from gravity. Encouraged by this, we present a conjecture in which IIB string theory is an edge mode of a 10+2-dimensional bulk topological theory, thus placing all twelve dimensions of F-theory on a physical footing.