Quaternionic Kähler manifolds, constrained instantons and the magic square: I

The classification of homogeneous quaternionic manifolds has been done by Alekseevskii, Wolf et al. using transitive solvable group of isometries. These manifolds are not generically symmetric, but there is a subset of quaternionic manifolds that are symmetric and Einstein. A further subset of these manifolds are the magic square manifolds. We show that all the symmetric quaternionic manifolds including the magic square can be succinctly classified by constrained instantons. These instantons are mostly semilocal, and their constructions for the magic square can be done from the corresponding Seiberg-Witten curves for certain N=2 gauge theories that are in general not asymptotically free. Using these, we give possible constructions, such as the classical moduli space metrics, of constrained instantons with exceptional global symmetries. We also discuss the possibility of realising the Kähler manifolds in the magic square using other solitonic configurations in the theory, and point out an interesting new sequence of these manifolds in the magic square.