Twistor theory of hyper-Kahler metrics with hidden symmetries

We review the hierarchy for the hyper-Kahler equations and define a notion of symmetry for solutions of this hierarchy. A four-dimensional hyper-Kahler metric admits a hidden symmetry if it embeds into a hierarchy with a symmetry. It is shown that a hyper-Kahler 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 [Twistor Quotients of Hyper-Kahler Manifolds (World Scientific, River Edge, NJ, 2001)]. This leads to a construction for the solution to the hierarchy in terms of linear equations and variants of the generalized Legendre transform for the hyper-Kahler metric itself given by Ivanov and Rocek [Commun. Math. Phys. 182, 291 (1996)]. We show that the ALE spaces are examples of hyper-Kahler 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 our earlier work [Commun. Math. Phys. 213, 641 (2000)] for the four-dimensional hyper-Kahler equations to a generalization of the conformal anti-self-duality equations and briefly discuss hidden symmetries for these equations. (C) 2003 American Institute of Physics.