From Yang–Mills in de Sitter Space to Electromagnetic Knots

Abstract

We review analytic SU(2) Yang–Mills solutions with finite action on four-dimensional de Sitter space from a new perspective, by conformally mapping dS $$_{4}$$ to a finite Lorentzian cylinder $$(0,\pi ) \times {{S}^{3}}$$ . As a byproduct, all abelian (i.e. Maxwell) solutions are classified by SO(4) representations. Conformal equivalence of (two copies of half of) this cylinder to Minkowski space yields a complete set of rational Maxwell solutions on the latter, which are known as electromagnetic knots. Their properties are efficiently computed on de Sitter space. We close with a couple of explicit examples.