Hopf instantons and the Liouville equation in target space

TL;DR

This paper explores the relationship between Hopf instantons in Chern--Simons and Fermion theories and the Liouville equation, showing solutions are composed of Hopf maps and rational maps in target space.

Contribution

It generalizes previous results by linking Hopf instantons to the Liouville equation and characterizing solutions as compositions of Hopf and rational maps.

Findings

Instanton solutions obey the Liouville equation in target space.

Solutions are given by compositions of the standard Hopf map with rational maps.

The work extends understanding of topological solutions in gauge theories.

Abstract

We generalise recent results on Hopf instantons in a Chern--Simons and Fermion theory in a fixed background magnetic field. We find that these instanton solutions have to obey the Liouville equation in target space. As a consequence, these solutions are given by a class of Hopf maps that consist of the composition of the standard Hopf map with an arbitrary rational map.