Sphalerons, instantons, and standing waves on $S^3 \otimes R$

TL;DR

This paper studies classical solutions like instantons, sphalerons, and standing waves in pure SU(2) Yang-Mills theory on a compact spherical space, providing new formulations and numerical solutions for these configurations.

Contribution

It reformulates known solutions in a convenient gauge, discusses stationary Minkowski solutions, and presents a numerical sphaleron solution in a finite spherical domain.

Findings

Reformulation of instantons and sphalerons in physical variables with gauge condition A_0=0

Discussion of standing wave solutions in Minkowski space

Numerical sphaleron solution in a finite spherical box

Abstract

We consider pure $SU(2)$ Yang-Mills theory when the space is compactified to a 3-dimensional sphere with finite radius. The Euclidean classical self-dual solutions of the equations of motion (the instantons) and the static finite energy solutions (the sphalerons) which have been found earlier are rewritten in handy physical variables with the gauge condition $A_0 = 0$. Stationary solutions to the equations of motion in the Minkowski space-time (the standing waves) are discussed. We briefly discuss also the theory defined in a flat finite spherical box with rigid boundary conditions and present the numerical solution describing the sphaleron.