Non-Perturbative SDiff Covariance of Fractional Quantum Hall Excitations

TL;DR

This paper develops a non-perturbative, unitary SDiff-equivariant quantum field theory for fractional quantum Hall excitations, revealing subtleties beyond traditional perturbative approaches.

Contribution

It introduces a non-perturbative Maxwell-Chern-Simons theory with SDiff symmetry, highlighting limitations of perturbative Lie algebra methods.

Findings

Constructed a non-perturbative SDiff-equivariant quantum field theory.

Identified non-differentiability issues in the non-perturbative construction.

Revealed subtleties in Hilbert space truncation when removing perturbative assumptions.

Abstract

Collective excitations of Fractional Quantum Hall (FQH) liquids at long wavelengths are thought to be of a generally covariant geometric nature, governed by area-preserving diffeomorphisms ($\mathrm{SDiff}$). But current analyses rely solely on the corresponding perturbative $w_\infty$ Lie algebra. We argue this is insufficient: We identify a non-perturbative construction of the effective Maxwell-Chern-Simons quantum field theory which carries unitary $\mathrm{SDiff}$ equivariance. But this turns out to be non-differentiable, suggesting underappreciated subtleties when the usual Hilbert space truncation is removed.