In mathematics, the Gibbons–Hawking ansatz is a method of constructing gravitational instantons introduced by . It gives examples of hyperkähler manifolds in dimension 4 that are invariant under a circle action.
Description Suppose that is an open subset of , and let denote the Hodge star operator on with respect to the usual (flat) Euclidean metric. is a harmonic function defined on such that the cohomology class is integral, i.e. lies in the image of . Then there is a -principal bundle equipped with a connection 1-form whose curvature form is . Then the Riemannian metric
is hyperkahler, and typically extends to the boundary of .
Examples Quaternions The usual (flat) metric on the quaternions is hyperkahler. It can be obtained as a result of the Gibbons-Hawking ansatz applied to the open subset and the harmonic function . ALE gravitational instantons The ALE gravitational instanton of type can be obtained by applying the Gibbons-Hawking ansatz to the open subset for distinct collinear points and the harmonic function . In the case , we recover the Eguchi-Hanson metric on . See also Gibbons–Hawking space Ooguri–Vafa metric
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