Loop groups in Yang-Mills theory

Abstract

We consider the Yang-Mills equations with a matrix gauge group $G$ on the de Sitter dS$_4$, anti-de Sitter AdS$_4$ and Minkowski $R^{3,1}$ spaces. On all these spaces one can introduce a doubly warped metric in the form $d s^2 =-d u^2 + f^2 d v^2 +h^2 d s^2_{H^2}$, where $f$ and $h$ are the functions of $u$ and $d s^2_{H^2}$ is the metric on the two-dimensional hyperbolic space $H^2$. We show that in the adiabatic limit, when the metric on $H^2$ is scaled down, the Yang-Mills equations become the sigma-model equations describing harmonic maps from a two-dimensional manifold (dS$_2$, AdS$_2$ or $R^{1,1}$, respectively) into the based loop group $\Omega G=C^\infty (S^1, G)/G$ of smooth maps from the boundary circle $S^1=\partial H^2$ of $H^2$ into the gauge group $G$. From this correspondence and the implicit function theorem it follows that the moduli space of Yang-Mills theory with a gauge group $G$ in four dimensions is bijective to the moduli space of two-dimensional sigma model with $\Omega G$ as the target space. The sigma-model field equations can be reduced to equations of geodesics on $\Omega G$, solutions of which yield magnetic-type configurations of Yang-Mills fields. The group $\Omega G$ naturally acts on their moduli space.