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 dS_{4} and construct a smooth and spatially homogeneous magnetic solution to the Yang-Mills equations. Slicing dS_{4} as R×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 τ∈R is given by B[over ˜]_{a}=-1/2I_{a}/(R^{2}cosh^{2}τ), 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 -1/2j(j+1)(2j+1)π^{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.