Classification of all constant solutions of ${\rm SU}(2)$ Yang-Mills equations with arbitrary current in pseudo-Euclidean space ${\mathbb R}^{p,q}$

Abstract

We present a classification and an explicit form of all constant solutions of the Yang-Mills equations with ${\rm SU}(2)$ gauge symmetry for an arbitrary constant non-Abelian current in pseudo-Euclidean space ${\mathbb R}^{p,q}$ of arbitrary finite dimension $n=p+q$. Using hyperbolic singular value decomposition and two-sheeted covering of orthogonal group by spin group, we solve the nontrivial system for constant solutions of the Yang-Mills equations of $3n$ cubic equations with $3n$ unknowns and $3n$ parameters in the general case. We present a new symmetry of this system of equations. All solutions in terms of the potential, strength, and invariant of the Yang-Mills field are presented. Nonconstant solutions of the Yang-Mills equations can be considered in the form of series of perturbation theory using all constant solutions as a zeroth approximation.