An analogue of the Gibbons-Hawking Ansatz for quaternionic Kähler spaces

Abstract

We show that the geometry of $4n$-dimensional quaternionic Kähler spaces with a locally free ${\mathbb{R}}^{n+1}$-action admits a Gibbons-Hawking-like description based on the Galicki-Lawson notion of quaternionic Kähler moment map. This generalizes to higher dimensions a four-dimensional construction, due to Calderbank and Pedersen, of self-dual Einstein manifolds with two linearly independent commuting Killing vector fields. As an application, we use this new Ansatz to give an explicit equivariant completion of the twistor space construction of the local c-map proposed by Roček, Vafa and Vandoren.

59 pages. v3: With a revised presentation. Also, the quaternionic coordinate on Swann bundle fibers has been replaced with its quaternionic conjugate in order to match the standard left/right-invariance conventions of principal bundles as well as to prompt some small esthetic improvements in a few formulas. The main results remain unchanged

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• Nonperturbative corrections to 4D string theory effective actions from SL(2,Z) duality and supersymmetry
• Quaternionic Kähler metrics associated with special Kähler manifolds
• Toric selfdual Einstein metrics on compact orbifolds
• A note on conformal symmetry in projective superspace