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Godel Spacetimes, Abelian Instantons, the Graviphoton Background and other Flacuum Solutions

We study the relations between all the vacua of Lorentzian and Euclidean d=4,5,6 SUGRAs with 8 supercharges, finding a new limiting procedure that takes us from the over-rotating near-horizon BMPV black hole to the Godel spacetime. The timelike compactification of the maximally supersymmetric Godel solution of N=1,d=5 SUGRA gives a maximally supersymmetric solution of pure Euclidean N=2,d=4 with flat space but non-trivial anti-selfdual vector field flux (``flacuum'') that, on the one hand, can be interpreted as an U(1) instanton on the 4-torus and that, on the other hand, coincides with the graviphoton background shown by Berkovits and Seiberg to produce the C-deformation introduced recently by Ooguri and Vafa. We construct flacuum solutions in other theories such as Euclidean type IIA supergravity.