6D Fractional Quantum Hall Effect

TL;DR

This work proposes a six-dimensional generalization of the fractional quantum Hall effect realized as membranes coupled to a three-form in a background four-form flux. The framework builds a bulk–edge correspondence between a 7D Chern-Simons-like theory for abelian three-forms and a 6D anti-chiral boundary theory of Euclidean strings, enabling a Laughlin-type wavefunction and a fractional conductivity tied to the flux and K-matrix data. It further develops the structure of the many-body wavefunction across regimes, analyzes limits such as zero-slope and large-rigid membranes, and demonstrates how lower-dimensional topological field theories (BF-like and CS-like) emerge upon compactification. The paper also embeds the 7D theory in M-theory, and speculative links to IIB/F-theory via a 10+2 bulk topological picture, suggesting a unified higher-dimensional topological framework for edge modes in string theory and condensed-matter analogs.

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.

Figures (5)

• Figure 1: A representation of the bulk-boundary geometry considered in this paper. We denote the 7D bulk by $M_{7}$, the 6D boundary by $\partial M_{7}$ and time running into the boundary by $t$.
• Figure 2: State insertion of a membrane wrapping a three-cycle $\Gamma$ with boundary $\partial\Gamma$.
• Figure 3: Depiction of a quantum Hall droplet of characteristic radius $\sim 1/\sqrt{G}$.
• Figure 4: Exchange of an anti-chiral two-form between two membranes wrapped on Riemann surfaces $\Sigma$ and $\Sigma^{\prime}$.
• Figure 5: Depiction of the 12D geometry associated with F-theory and IIB as an edge mode in the special case where the axio-dilaton is constant. Here, we have passed from the $(10,2)$ signature spacetime to one in which a dual winding coordinate $\widetilde{\theta}$ appears. This takes us to a dualized spacetime of signature $(11,1)$ and in which the standard elliptic fiber of F-theory appears geometrically.