Abstract
A supersymmetric solution of 5d supergravity may admit an ‘evanescent ergosurface’: a timelike hypersurface such that the canonical Killing vector field is timelike everywhere except on this hypersurface. The hyperkahler ‘base space’ of such a solution is ‘ambipolar’, changing signature from (+ + ++) to (− − −−) across a hypersurface. In this paper, we determine how the hyperkahler structure must degenerate at the hyper-surface in order for the 5d solution to remain smooth. This leads us to a definition of an ambipolar hyperkahler manifold which generalizes the recently-defined notion of a ‘folded’ hyperkahler manifold. We prove that such manifolds can be constructed from ‘initial’ data prescribed on the hypersurface. We present an ‘initial value’ construction of supersymmetric solutions of 5d supergravity, in which such solutions are determined by data prescribed on a timelike hypersurface, both for the generic case and for the case of an evanescent ergosurface.
Highlights
The result just summarized concerns configurations of 5d minimal supergravity that admit a supercovariantly constant spinor
We present an ‘initial value’ construction of supersymmetric solutions of 5d supergravity, in which such solutions are determined by data prescribed on a timelike hypersurface, both for the generic case and for the case of an evanescent ergosurface
If (g, F ) are smooth, admit a supercovariantly constant spinor, and there exists an evanescent ergosurface, the base space must satisfy our definition of an ambipolar hyperkahler manifold
Summary
We review ‘folded’ hyperkahler manifolds as defined in [1, 2]. While h is undefined at z = 0, the 2-forms X1, X2, X3 are smooth there. Θ is a contact form on S. and θ is a contact form on S From this canonical example, Hitchin [1] extracts a notion of a ‘folded’ hyperkahler manifold. A folded hyperkahler structure consists of a smooth 4-manifold. M, a smooth imbedded hypersurface S ⊂ M (the fold surface), three smooth, closed, 2-forms Xi on M, and a smooth diffeomorphism ι : M → M such that. The 2-forms Xi define a hyperkahler structure on M± with hyperkahler metric h± where h+ has signature (+ + + +) and h− has signature (− − − −); 3.
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