Loop groups in Yang–Mills theory

Abstract

We consider the Yang–Mills equations with a matrix gauge group G on the de Sitter dS4, anti-de Sitter AdS4 and Minkowski R3,1 spaces. On all these spaces one can introduce a doubly warped metric in the form ds2=−du2+f2dv2+h2dsH22, where f and h are the functions of u and dsH22 is the metric on the two-dimensional hyperbolic space H2. We show that in the adiabatic limit, when the metric on H2 is scaled down, the Yang–Mills equations become the sigma-model equations describing harmonic maps from a two-dimensional manifold (dS2, AdS2 or R1,1, respectively) into the based loop group ΩG=C∞(S1,G)/G of smooth maps from the boundary circle S1=∂H2 of H2 into the gauge group G. For compact groups G these harmonic map equations are reduced to equations of geodesics on ΩG, solutions of which yield magnetic-type configurations of Yang–Mills fields. The group ΩG naturally acts on their moduli space.