Twistor theory of hyper-K{ä}hler metrics with hidden symmetries

Details Twistor theory of hyper-K{ä}hler metrics with hidden symmetries Twistor theory of hyper-K{ä}hler metrics with hidden symmetries

2003-01-16

arxiv.org/abs/math/0301171v2

J.Math.Phys. 44 (2003) 3430-3454

Mathematics

Differential Geometry

Final version. To appear in the August 2003 special issue of JMP on `Integrability, Topological Solitons, and Beyond'

Scientific paper

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.

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