Solutions to Yang-Mills Equations on Four-Dimensional de Sitter Space

We consider pure SU(2) Yang-Mills theory on four-dimensional de Sitter space ${\mathrm{dS}}_4$ and construct a smooth and spatially homogeneous magnetic solution to the Yang-Mills equations. Slicing ${\mathrm{dS}}_4$ 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 double-well potential. Its local maximum yields a Yang-Mills solution whose color-magnetic field at time $\tau \in \mathbb{R}$ is given by ${\tilde{B}}_a=-\frac{1}{2}{I}_a/(R^2{\cosh }^2\tau )$, where $I_a$ for $a=1$, 2, 3 are the SU(2) generators and $R$ is the de Sitter radius. At any moment, this spatially homogeneous configuration has finite energy, but its action is also finite and of the value $-\frac{1}{2}j(j+1)(2j+1){\pi }^3$ in a spin-$j$ representation. Similarly, the double-well bounce produces a family of homogeneous finite-action electric-magnetic solutions with the same energy. There is a continuum of other solutions whose energy and action extend down to zero.