Special geometry of Euclidean supersymmetry IV: the local c-map

A bstract We consider timelike and spacelike reductions of 4D, N = 2 Minkowskian and Euclidean vector multiplets coupled to supergravity and the maps induced on the scalar geometry. In particular, we investigate (i) the (standard) spatial c-map , (ii) the temporal c-map , which corresponds to the reduction of the Minkowskian theory over time, and (iii) the Euclidean c-map , which corresponds to the reduction of the Euclidean theory over space. In the last two cases we prove that the target manifold is para-quaternionic Kähler. In cases (i) and (ii) we construct two integrable complex structures on the target manifold, one of which belongs to the quaternionic and para-quaternionic structure, respectively. In case (iii) we construct two integrable para-complex structures, one of which belongs to the para-quaternionic structure. In addition we provide a new global construction of the spatial, temporal and Euclidean c -maps, and separately consider a description of the target manifold as a fibre bundle over a projective special Kähler or para-Kähler base.