Fractional quantum Hall effect in higher dimensions

TL;DR

The paper develops a framework to define fractional quantum Hall states in higher dimensions by generalizing the parton construction bound to $U(1)$ gauge fields and enforcing anomaly cancellation with boundary spectator fields. The bulk action is derived from index-theorem densities (Dolbeault in higher dimensions and Dirac for spectators), yielding gauge and gravitational topological terms that determine transport properties. In 2+1D, the setup reproduces the Laughlin-like Hall conductivity $\sigma_H=\nu/(2\pi)$ and related gravitational couplings; in 4+1D, the leading Hall response scales as $1/m^2$ with corresponding Hall viscosity scaling as $1/m$, and dimensional reduction on specific manifolds is discussed. The approach provides a principled, anomaly-consistent route to higher-dimensional QHE, with explicit results on $S^2\times S^2$ and clear avenues for extension to arbitrary even dimensions and non-Abelian backgrounds, including potential realizations via synthetic dimensions.

Abstract

Generalizing from previous work on the integer quantum Hall effect, we construct the effective action for the analog of Laughlin states for the fractional quantum Hall effect in higher dimensions. The formalism is a generalization of the parton picture used in two spatial dimensions, the crucial ingredient being the cancellation of anomalies for the gauge fields binding the partons together. Some subtleties which exist even in two dimensions are pointed out. The effective action is obtained from a combination of the Dolbeault and Dirac index theorems. We also present expressions for some transport coefficients such as Hall conductivity and Hall viscosity for the fractional states.