Singular instantons with SO(3) symmetry

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This article provides an explicit construction for a family of singular instantons on S⁴ S² with arbitrary real holonomy parameter α. This family includes the original α= 1/4, c₂ = 3/2 solution discovered by P. Forgacs, Z. Horvath, and L. Palla, and our approach is modeled on that of their 1981 paper. Our primary tool is the ansatz due to Corrigan, Fairlie, Wilczek, and 't Hooft that constructs a self-dual Yang-Mills connection using a positive real-valued harmonic super-potential. Here we reformulate this harmonic function ansatz in terms of quaternionic notation, and we show that it arises naturally from the Levi-Civita connection of a conformally Euclidean metric. To simplify the construction, we introduce an SO(3)-action on S⁴, and we show by dimensional reduction that the symmetric self-duality equation on S⁴ is equivalent to the vortex equations over hyperbolic space H². We thus obtain a similar harmonic function ansatz for hyperbolic vortices, which we also derive using conformal transformations of H². Using this ansatz, we construct the vortex equivalents of the symmetric 't Hooft instantons, and we prove using the equivariant ADHM construction that they provide a complete description of all hyperbolic vortices. We also analyze when two vortices constructed by this ansatz are gauge equivalent, obtaining the surprising result that two such vortices are completely determined by the gauge transformation between them.