Hopf term induced by fermions

We derive an effective action for Dirac fermions coupled to O(3) non-linear σ-model (NLσM) through the Yukawa-type interaction. The nonperturbative (global) quantum anomaly of this model results in a Hopf term for the effective NLσM. We obtain this term using the “embedding” of the CP1 model into the CPM 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.