Fractional Quantum Hall Effect in a Curved Space: Gravitational Anomaly and Electromagnetic Response

TL;DR

The paper develops a Ward-identity–based framework to compute curvature-induced responses of Laughlin FQH states on curved surfaces, revealing a gradient expansion for the particle density that includes a gravitational anomaly term. It connects curvature response to flat-space electromagnetic observables through relationships between the odd viscosity $\eta(k)$, the structure factor $s(k)$, and the Hall conductance $\sigma_H(k)$, via an iterative analysis of the Ward identity and the two-point function of a Bose field $\varphi$. The authors derive explicit density corrections $\langle \rho \rangle = \rho_0 + \frac{1}{8\pi} R - \frac{b}{8\pi}(-l^2 \Delta_g) R$ with $b = \frac{1}{3} + \frac{\nu - 1}{4\nu}$ and show that the generating functional contains Polyakov's Liouville action $A^{(0)}$, reflecting the gravitational anomaly. Overall, the work proposes a universal, geometry-driven approach to bound and relate FQH response functions in curved space to observables in flat space, providing a principled link between topology, geometry, and electromagnetic response.

Abstract

We develop a general method to compute correlation functions of fractional quantum Hall (FQH) states on a curved space. In a curved space, local transformation properties of FQH states are examined through local geometric variations, which are essentially governed by the gravitational anomaly. Furthermore, we show that the electromagnetic response of FQH states is related to the gravitational response (a response to curvature). Thus, the gravitational anomaly is also seen in the structure factor and the Hall conductance in flat space. The method is based on iteration of a Ward identity obtained for FQH states.