Hopf maps as static solutions of the complex eikonal equation

TL;DR

This paper shows that certain torus-shaped Hopf maps satisfy the static complex eikonal equation, revealing their geometric structure and significance in nonlinear field theories.

Contribution

It introduces a class of torus-shaped Hopf maps with arbitrary linking numbers that solve the static complex eikonal equation and explains their geometric origin.

Findings

Hopf maps with arbitrary linking numbers satisfy the static complex eikonal equation

The geometric structure behind these solutions is elucidated

These solutions are relevant as integrability conditions in nonlinear field theories

Abstract

We demonstrate that a class of torus-shaped Hopf maps with arbitrary linking number obeys the static complex eikonal equation. Further, we explore the geometric structure behind these solutions, explaining thereby the reason for their existence. As this equation shows up as an integrability condition in certain non-linear field theories, the existence of such solutions is of some interest.