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 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 Kähler or special para-Kähler for vector multiplets and quaternion-Kähler or para-quaternion Kähler for hypermultiplets. By spacelike and timelike reductions we obtain three-dimensional 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.