Type-II Calabi-Yau compactifications, T-duality and special geometry in general spacetime signature

We obtain the bosonic Lagrangians of vector and hypermultiplets coupled to four-dimensional $\mathcal{N}=2$ supergravity in signatures $(0,4)$, $(1,3)$ and (2,2) by compactification of type-II string theories in signatures (0,10), (1,9) and (2,8) on a Calabi-Yau threefold. Depending on the signature and the distinctions between type-IIA/IIA$^*$/IIB/IIB$^*$/IIB' the resulting scalar geometries are special K\"ahler or special para-K\"ahler for vector multiplets and quaternion-K\"ahler or para-quaternion K\"ahler for hypermultiplets. By spacelike and timelike reductions we obtain three-dimensional $\mathcal{N}=4$ supergravity theories coupled to two sets of hypermultiplets. We determine the c-maps relating vector to hypermultiplets, and show how the four-dimensional theories are related by spacelike, timelike and mixed, signature-changing T-dualities.