Hopf term induced by fermions

TL;DR

This paper derives an effective action for fermions coupled to a non-linear sigma model, showing how a quantum anomaly induces a Hopf term via a perturbative approach involving embedding and Chern-Simons terms.

Contribution

It introduces a novel method to derive the Hopf term from a fermionic quantum anomaly using embedding into a higher CP^M model and gradient expansion techniques.

Findings

The Hopf term arises from a global quantum anomaly in the fermion-NLSM system.

Embedding the CP^1 model into CP^M makes the anomaly perturbative.

The anomaly is expressed as a Chern-Simons term for an auxiliary gauge field.

Abstract

We derive an effective action for Dirac fermions coupled to O(3) non-linear sigma-model (NLSM) through the Yukawa-type interaction. The nonperturbative (global) quantum anomaly of this model results in a Hopf term for the effective NLSM. We obtain this term using the ``embedding'' of the CP$^1$ model into the CP$^M$ generalization of the model which makes the quantum anomaly perturbative. This perturbative anomaly is calculated by means of a gradient expansion of a fermionic determinant and is given by the Chern-Simons term for an auxiliary gauge field.