Non-Perturbative SDiff Covariance of Fractional Quantum Hall Excitations

TL;DR

The paper tackles how fractional quantum Hall (FQH) excitations transform under area-preserving diffeomorphisms (SDiff), arguing that perturbative $w_\infty$-based descriptions are incomplete. It develops a non-perturbative Maxwell-Chern-Simons (MCS) quantum field theory with unitary $\mathrm{SDiff}$ equivariance, using Pickrell's construction to define a rigorous Hilbert space at finite coupling and showing that SDiff acts continuously but non-differentiably. The key finding is that excitations at finite coupling are generated by SDiff via $U_{\kappa}$ as $|\phi_{\kappa}\rangle = (1/\epsilon)(U_{\kappa}-\mathrm{id})|\Psi_0\rangle$, not by the conventional $\overline{\rho}_{\mathbf{k}}|\Psi_0\rangle$ action, implying a non-perturbative SDiff covariance beyond the $w_\infty$ picture. This shifts the theoretical lens toward SDiff-covariant, brane/gravity-inspired descriptions of FQH excitations and motivates further non-perturbative QFT analyses of the FQH spectrum and its symmetry structure.

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 FQH excitation phenomenology beyond the $w_\infty$ algebra.