Exact solutions in Einstein–Yang–Mills–Dirac systems

For the first time, exact solutions for the full self-consistent Einstein–Yang–Mills–Dirac systems with gauge groups SU(2) and SU(4) in Robertson–Walker space–time R×S3 are presented, which are symmetric under the action of the group SO(4) of spatial rotations. The approach is based on the dimensional reduction method for gauge, spinor, and gravitational fields and relates symmetric solutions in EYMD theory to certain solutions of an effective dynamic system. The solutions are interpreted as solutions of the cosmological type with an oscillating Yang–Mills field, which exchanges energy with a spinor field. The explicit form of the solution for the spinor field shows that its energy changes the sign in the process of the evolution of the Yang–Mills field from one vacuum to another.