Abstract
In the present work some examples of toric hyperkahler metrics in eight dimensions are constructed. First it is described how the Calderbank-Pedersen metrics arise as a consequence of the Joyce description of selfdual structures in four dimensions, the Jones-Tod correspondence and a result due to Tod and Przanowski. It is also shown that any quaternionic Kahler metric with T2 isometry is locally isometric to a Calderbank-Pedersen one. The Swann construction of hyperkahler metrics in eight dimensions is applied to them to find hyperkahler examples with U(1)×U(1) isometry. The connection with the Pedersen-Poon toric hyperkahler metrics is explained and it is shown that there is a class of solutions of the generalized monopole equation related to eigenfunctions of a certain linear equation. These hyperkahler examples are lifted to solutions of the D=11 supergravity and type IIA and IIB backgrounds are found by use of dualities. As before, all the description is achieved in terms of a single eigenfunction F. Some explicit F are found, together with the Toda structure corresponding to the trajectories of the Killing vectors of the Calderbank-Pedersen bases.
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Santillan, O., Zorin, A. Toric Hyperkahler Manifolds with Quaternionic Kahler Bases and Supergravity Solutions. Commun. Math. Phys. 255, 33–59 (2005). https://doi.org/10.1007/s00220-004-1250-0
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DOI: https://doi.org/10.1007/s00220-004-1250-0