Twistor theory of hyper-K{\"a}hler metrics with hidden symmetries

We briefly review the hierarchy for the hyper-K"ahler equations and define a notion of symmetry for solutions of this hierarchy. A four-dimensional hyper-K"ahler metric admits a hidden symmetry if it embeds into a hierarchy with a symmetry. It is shown that a hyper-K"ahler 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 \cite{B00}. 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-K"ahler metric itself given by Ivanov & Rocek \cite{IR96}. We show that the ALE spaces are examples of hyper-K"ahler 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 \cite{DM00} for the four-dimensional hyper-K"ahler equations to a generalisation of the conformal anti-self-duality equations and briefly discuss hidden symmetries for these equations.