Explicit non-Abelian monopoles and instantons inSU(N)pure Yang-Mills theory

Abstract

It is well known that there are no static non-Abelian monopole solutions in pure Yang-Mills theory on Minkowski space ${\mathbb{R}}^{3,1}$. I show that such solutions exist in $\mathrm{SU}(N)$ gauge theory on the spaces ${\mathbb{R}}^{2}\ifmmode\times\else\texttimes\fi{}{S}^{2}$ and $\mathbb{R}\ifmmode\times\else\texttimes\fi{}{S}^{1}\ifmmode\times\else\texttimes\fi{}{S}^{2}$ with Minkowski signature $(\ensuremath{-}+++)$. In the temporal gauge they are solutions of pure Yang-Mills theory on $\mathbb{T}\ifmmode\times\else\texttimes\fi{}{S}^{2}$, where $\mathbb{T}$ is $\mathbb{R}$ or ${S}^{1}$. Namely, imposing SO(3) invariance and some reality conditions, I consistently reduce the Yang-Mills model on the above spaces to a non-Abelian analog of the ${\ensuremath{\phi}}^{4}$ kink model whose static solutions give $\mathrm{SU}(N)$ monopole (-antimonopole) configurations on the space ${\mathbb{R}}^{1,1}\ifmmode\times\else\texttimes\fi{}{S}^{2}$ via the above-mentioned correspondence. These solutions can also be considered as instanton configurations of Yang-Mills theory in $2+1$ dimensions. The kink model on $\mathbb{R}\ifmmode\times\else\texttimes\fi{}{S}^{1}$ admits also periodic sphaleron-type solutions describing chains of $n$ kink-antikink pairs spaced around the circle ${S}^{1}$ with arbitrary $n>0$. They correspond to chains of $n$ static monopole-antimonopole pairs on the space $\mathbb{R}\ifmmode\times\else\texttimes\fi{}{S}^{1}\ifmmode\times\else\texttimes\fi{}{S}^{2}$ which can also be interpreted as instanton configurations in $2+1$ dimensional pure Yang-Mills theory at finite temperature (thermal time circle). I also describe similar solutions in Euclidean $\mathrm{SU}(N)$ gauge theory on ${S}^{1}\ifmmode\times\else\texttimes\fi{}{S}^{3}$ interpreted as chains of $n$ instanton--anti-instanton pairs.