2015/05/25 by Popov, Alexander D.
#Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.1505.06634
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 d s2 =-d u2 + f2 d v2 +h2 d s2H2, where f and h are the functions of u and d s2H2 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. 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 ΩG as the target space. The sigma-model field equations can be 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.