# Special Geometry and Space-time Signature

## TL;DR

This paper develops N=2 supergravity theories in various space-time signatures, exploring their construction via Killing spinor integrability and dimensional reduction, revealing new geometric structures in different signatures.

## Contribution

It introduces new N=2 supergravity models in diverse signatures and links them to special (para)-Kähler geometry through dimensional reduction methods.

## Key findings

- Constructed 5D supergravity theories with various signatures.
- Derived 4D theories with projective special (para)-Kähler geometry.
- Connected supergravity models to Hull's eleven-dimensional theories.

## Abstract

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\"ahler geometry.