Hyper-Kähler instantons, symmetries, and flat spaces

Abstract

We find all hyper-Kähler 4-manifolds admitting conformal Kähler structures with respect to either orientation, and we show that these structures can be expressed as a combination of twistor elementary states (and possibly a self-dual dyon) in locally flat spaces. The complex structures of different flat pieces are not compatible however, reflecting that the global geometry is not a linear superposition. For either orientation, the space must be Gibbons–Hawking (thus excluding the Atiyah–Hitchin metric), and, if the orientations are opposite, it must also be toric and have an irreducible Killing tensor. We also show that the only hyper-Kähler 4-metric with a non-constant Killing–Yano tensor is the half-flat Taub–NUT instanton.