Special geometry and space–time signature

We construct N=2 four and five-dimensional supergravity theories coupled to vector multiplets in various space–time signatures (t,s), where t and s refer, respectively, to the number of time and spatial dimensions. The five-dimensional supergravity theories, t+s=5, are constructed by investigating the integrability conditions arising from Killing spinor equations. The five-dimensional supergravity theories can also be obtained by reducing Hull's eleven-dimensional supergravities on a Calabi–Yau threefold. The dimensional reductions of the five-dimensional supergravities on space and time-like circles produce N=2 four-dimensional supergravity theories with signatures (t−1,s) and (t,s−1) exhibiting projective special (para)-Kähler geometry.