New knotted solutions of Maxwell's equations

TL;DR

This paper introduces a new method to generate topologically non-trivial solutions of Maxwell's equations using conformal transformations, and explores their properties including conserved charges and links to gauge theory.

Contribution

It develops a covariant formulation of Bateman's construction, introduces complex-parameter conformal transformations, and maps knotted solutions to flat connections in SU(2) gauge theory.

Findings

Generated a wide class of knotted solutions from basic electromagnetic fields.

Computed conserved charges and helicities for these solutions.

Established a link between electromagnetic knots and flat SU(2) connections.

Abstract

In this note we have further developed the study of topologically non-trivial solutions of vacuum electrodynamics. We have discovered a novel method of generating such solutions by applying conformal transformations with complex parameters on known solutions expressed in terms of Bateman's variables. This has enabled us to get a wide class of solutions from the basic configuration like constant electromagnetic fields and plane-waves. We have introduced a covariant formulation of the Bateman's construction and discussed the conserved charges associated with the conformal group as well as a set of four types of conserved helicities. We have also given a formulation in terms of quaternions. This led to a simple map between the electromagnetic knotted and linked solutions into flat connections of $SU(2)$ gauge theory. We have computed the corresponding CS charge in a class of solutions and it takes integer values.