Pure Yang–Mills Solutions on dS4

Abstract

We consider pure SU(2) Yang–Mills theory on four-dimensional de Sitter space dS4 and construct smooth and spatially homogeneous classical Yang–Mills fields. Slicing dS4 as $$\mathbb{R} \times {{S}^{3}},$$ via an SU(2)-equivariant ansatz we reduce the Yang–Mills equations to ordinary matrix differential equations and further to Newtonian dynamics in a particular three-dimensional potential. Its classical trajectories yield spatially homogeneous Yang–Mills solutions in a very simple explicit form, depending only on de Sitter time with an exponential decay in the past and future. These configurations have not only finite energy, but their action is also finite and bounded from below. We present explicit coordinate representations of the simplest examples (for the fundamental SU(2) representation). Instantons (Yang–Mills solutions on the Wick-rotated $${{S}^{4}}$$ ) and solutions on AdS4 are also briefly discussed.