Solutions to pure Yang–Mills theory on four-dimensional de Sitter space

We consider pure SU(2) Yang–Mills theory on four-dimensional de Sitter space dS4 and construct a smooth and spatially homogeneous magnetic solution to the Yang–Mills equations. Slicing dS4 as R×S3, 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 τ ∈ R is given by B̃a = − 1 2Ia/(R2 cosh τ), where Ia 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 − 3 4 π. 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.