Finite-action solutions of Yang-Mills equations on de Sitter dS$_4$ and anti-de Sitter AdS$_4$ spaces

Abstract

We consider pure SU(2) Yang-Mills theory on four-dimensional de Sitter dS$_4$ and anti-de Sitter AdS$_4$ spaces and construct various solutions to the Yang-Mills equations. On de Sitter space we reduce the Yang-Mills equations via an SU(2)-equivariant ansatz to Newtonian mechanics of a particle moving in ${\mathbb R}^3$ under the influence of a quartic potential. Then we describe magnetic and electric-magnetic solutions, both Abelian and non-Abelian, all having finite energy and finite action. A similar reduction on anti-de Sitter space also yields Yang-Mills solutions with finite energy and action. We propose a lower bound for the action on both backgrounds. Employing another metric on AdS$_4$, the SU(2) Yang-Mills equations are reduced to an analytic continuation of the above particle mechanics from ${\mathbb R}^3$ to ${\mathbb R}^{2,1}$. We discuss analytical solutions to these equations, which produce infinite-action configurations. After a Euclidean continuation of dS$_4$ and AdS$_4$ we also present self-dual (instanton-type) Yang--Mills solutions on these backgrounds.