Exact gauge fields from anti-de Sitter space

Abstract

In 1977 Lüscher found a class of SO(4)-symmetric SU(2) Yang–Mills solutions in Minkowski space, which have been rederived 40 years later by employing the isometry S3 ≅ SU(2) and conformally mapping SU(2)-equivariant solutions of the Yang–Mills equations on (two copies of) de Sitter space dS4≅R×S3. Here we present the noncompact analog of this construction via AdS3 ≅ SU(1, 1). On (two copies of) anti-de Sitter space AdS4≅R×AdS3 we write down SU(1,1)-equivariant Yang–Mills solutions and conformally map them to R1,3. This yields a two-parameter family of exact SU(1,1) Yang–Mills solutions on Minkowski space, whose field strengths are essentially rational functions of Cartesian coordinates. Gluing the two AdS copies happens on a dS3 hyperboloid in Minkowski space, and our Yang–Mills configurations are singular on a two-dimensional hyperboloid dS3∩R1,2. This renders their action and the energy infinite, although the field strengths fall off fast asymptotically except along the lightcone. We also construct Abelian solutions, which share these properties but are less symmetric and of zero action.