Chirality and fermion number in a knotted soliton background

TL;DR

This paper investigates how a Dirac fermion interacts with a knotted soliton in the Faddeev model, revealing that the fermion number is linked to the soliton's self-linking number through quantum anomalies.

Contribution

It introduces a novel connection between fermion number and the topological properties of knotted solitons via anomaly calculations.

Findings

Fermion number equals the soliton's self-linking number.

Quantum anomalies relate fermion chiral properties to soliton topology.

Potential for fermion-soliton interactions to produce effects similar to the Callan-Rubakov phenomenon.

Abstract

We consider the coupling of a single Dirac fermion to the three component unit vector field which appears as an order parameter in the Faddeev model. Classically, the coupling is determined by requiring that it preserves a certain local frame independence. But quantum mechanically the separate left and right chiral fermion number currents suffer from a frame anomaly. We employ this anomaly to compute the fermion number of a knotted soliton. The result coincides with the self-linking number of the soliton. In particular, the anomaly structure of the fermions relates directly to the inherent chiral properties of the soliton. Our result suggests that interactions between fermions and knotted solitons can lead to phenomena akin the Callan-Rubakov effect.