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)-K?hler geometry.
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机译：我们构造了N = 2个四维和五维超重力理论，它们在各种时空特征（t，s）中与矢量多重性耦合，其中t和s分别表示时间和空间维数。五维超重力理论t + s = 5是通过研究由Killing旋子方程产生的可积条件而构造的。五维超重力理论也可以通过将赫拉的十一维超重力减少三倍来获得。五维超重力在空间和类似时间的圆上的降维产生N = 2个具有（t？1，s）和（t，s？1）签名的四维超重力理论，它们表现出射影的特殊（para）-K几何形状。
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