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Twistor theory of hyperK{ähler metrics with hidden symmetries  Maciej Dunajski
; Lionel J. Mason
;  Date: 
16 Dec 2002  Journal:  J.Math.Phys. 44 (2003) 34303454  Subject:  Differential Geometry; Exactly Solvable and Integrable Systems  math.DG grqc hepth nlin.SI  Abstract:  We briefly review the hierarchy for the hyperKähler equations and define a notion of symmetry for solutions of this hierarchy. A fourdimensional hyperKähler metric admits a hidden symmetry if it embeds into a hierarchy with a symmetry. It is shown that a hyperKähler metric admits a hidden symmetry if it admits a certain Killing spinor. We show that if the hidden symmetry is triholomorphic, 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 hyperKähler metric itself given by Ivanov & Rocek cite{IR96}. We show that the ALE spaces are examples of hyperKähler metrics admitting three triholomorphic 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 fourdimensional hyperKähler equations to a generalisation of the conformal antiselfduality equations and briefly discuss hidden symmetries for these equations.  Source:  arXiv, math.DG/0301171  Services:  Forum  Review  PDF  Favorites 


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