Exact gauge fields from anti-de Sitter space

TL;DR

The paper constructs exact SU(1,1) Yang-Mills solutions on Minkowski space by conformally mapping solutions from anti-de Sitter space, revealing a family of solutions with specific singularities and properties.

Contribution

It introduces a novel method of deriving exact SU(1,1) Yang-Mills solutions on Minkowski space via conformal mapping from anti-de Sitter space, expanding the class of known solutions.

Findings

Solutions are rational functions of Cartesian coordinates.

Configurations are singular on a dS_3 hyperboloid, leading to infinite action.

Constructed Abelian solutions with similar properties but zero action.

Abstract

In 1977 L\"uscher found a class of SO(4)-symmetric SU(2) Yang-Mills solutions in Minkowski space, which have been rederived 40 years later by employing the isometry $S^3\cong\mathrm{SU}(2)$ and conformally mapping SU(2)-equivariant solutions of the Yang-Mills equations on (two copies of) de Sitter space $\mathrm{dS}_4\cong\mathbb{R}{\times}S^3$. Here we present the noncompact analog of this construction via $\mathrm{AdS}_3\cong\mathrm{SU}(1,1)$. On (two copies of) anti-de Sitter space $\mathrm{AdS}_4\cong\mathbb{R}{\times}\mathrm{AdS}_3$ we write down SU(1,1)-equivariant Yang-Mills solutions and conformally map them to $\mathbb{R}^{1,3}$. This yields a two-parameter family of exact SU(1,1) Yang-Mills solutions on Minkowski space, whose field strengths are essentially rational functions of Cartesian coordinates. Gluing the two AdS copies happens on a $\mathrm{dS}_3$ hyperboloid in Minkowski space, and our Yang-Mills configurations are singular on a two-dimensional hyperboloid $\mathrm{dS}_3\cap\mathbb{R}^{1,2}$. This renders their action and the energy infinite, although the field strengths fall off fast asymptotically except along the lightcone. We also construct Abelian solutions, which share these properties but are less symmetric and of zero action.