Twistor theory of hyper-Kähler metrics with hidden symmetries

We briefly review the hierarchy for the hyper-Kähler equations and define a notion of symmetry for solutions of this hierarchy. A four-dimensional hyper-Kähler metric admits a hidden symmetry if it embeds into a hierarchy with a symmetry. It is shown that a hyper-Kähler metric admits a hidden symmetry if it admits a certain Killing spinor. We show that if the hidden symmetry is tri-holomorphic, then this is equivalent to requiring symmetry along a higher time and the hidden symmetry determines a ‘twistor group’ action as introduced by Bielawski [3]. This leads to a construction for the solution to the hierarchy in terms of linear equations and variants of the generalised Legendre transform for the hyper-Kähler metric itself given by Ivanov & Rocek [17]. We show that the ALE spaces are examples of hyper-Kähler metrics admitting three tri-holomorphic Killing spinors. These metrics are in this sense analogous to the ’finite gap’ solutions in soliton theory. Finally we extend the concept of a hierarchy from that of [8] for the four-dimensional hyper-Kähler equations to a generalisation of the conformal anti-self-duality equations and briefly discuss hidden symmetries for these equations.