Maciej Dunajski, Jan Gutowski, Wafic Sabra
We show how to lift solutions of Euclidean Einstein-Maxwell equations with non-zero cosmological constant to solutions of eleven-dimensional supergravity theory with non--zero fluxes. This yields a class of 11D metrics given in terms of solutions to $SU(\infty)$ Toda equation. We give one example of a regular solution and analyse its supersymmetry. We also construct and classify solutions to N=2 minimal gauged supergravity in four Euclidean dimensions (these are the same as Euclidean Einstein-Maxwell space-times with non-vanishing cosmological constant), where the Killing spinor equations admit more than one solution and so the supersymmetry is enhanced. If the Weyl tensor is anti-self-dual then the enhanced supersymmetric metrics are given by separable solutions to the $SU(\infty)$ Toda equation. Otherwise they are ambi-K\"ahler and are conformally equivalent to K\"ahler metrics of Calabi type or to product metrics on two Riemann surfaces.
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http://arxiv.org/abs/1301.1896
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