Framing Anomaly in the Effective Theory of Fractional Quantum Hall Effect

TL;DR

This paper investigates the geometric effective action in fractional quantum Hall systems, emphasizing the importance of the framing anomaly in accurately determining gravitational responses, Hall conductivity, and viscosity.

Contribution

It demonstrates that including the framing anomaly in the effective theory is crucial for correct gravitational linear response functions in FQHE.

Findings

Framing anomaly affects the gravitational Chern-Simons term.

Fixes the thermal Hall conductivity value.

Impacts Hall viscosity calculations.

Abstract

We consider the geometric part of the effective action for Fractional Quantum Hall Effect (FQHE). It is shown that accounting for the framing anomaly of the quantum Chern-Simons theory is essential to the obtain correct gravitational linear response functions. In the lowest order in gradients the linear response generating functional includes Chern-Simons, Wen-Zee and gravitational Chern- Simons terms. The latter term has a contribution from the framing anomaly which fixes the value of thermal Hall conductivity and contributes to the Hall viscosity of the FQH states on a sphere. We also discuss the effects of the framing anomaly on linear responses for non-Abelian FQH states.