Spherically symmetric solutions of the euclidean Yang--Mills equations

L. V. Kapitanski ; O. A. Ladyzhenskaya

Zapiski Nauchnykh Seminarov POMI, Differential geometry, Lie groups and mechanics. Part VI, Tome 133 (1984), pp. 126-132

Résumé

We consider the euclidean Yang–Mills equations with the structural group $SU(2)$. The functionals of the Yang–Mills action and of topological charge are invariant under the transformations: $A_\mu(x)\,dx_\mu\to A_\mu(gx)\,d(gx)_\mu$, where $g$ runs over the set of quaternions with $|g|=1$, and $gx$ stands for the multiplication of quaternions $x=x_4+ix_1+jx_2+kx_3$. The $SU(2)$-symmetry allows us to use the Coleman's principle. Then, for gauge potentials $A_\mu$ we obtain the following spherically symmetric Anzatz: \begin{gather} A_\mu(x)=\frac{1}{|x|}f_\alpha(\ln|x|^2)\frac{1}{|x|}(\delta_{4\alpha}x_\mu-\delta_{4\mu}x_\alpha+\delta_{\alpha\mu}x_4+\varepsilon_{\alpha\mu\gamma4}x_\gamma), \end{gather} The Yang–Mills equations and the duality equations reduce to systems of ODE on the functions $f_\alpha^a(\mathcal T)$. We prove that for the Y–M equations every solution of the form (1) with finite action and positive (negative) charge is necessarilly a solution of the duality equations $F=*F$ (accordingly, $F=-*F$), and has a unit topological charge. Besides, we describe explicitly all the solutions of the form (1) for the duality equations; the 1-instanton solution of Belavin et al. is among them.