Special geometry of Euclidean supersymmetry II: hypermultiplets and the   c-map
  We construct two new versions of the c-map which allow us to obtain the target manifolds of hypermultiplets in Euclidean theories with rigid N =2 supersymmetry. While the Minkowskian para-c-map is obtained by dimensional reduction of the Minkowskian vector multiplet lagrangian over time, the Euclidean para-c-map corresponds to the dimensional reduction of the Euclidean vector multiplet lagrangian. In both cases the resulting hypermultiplet target spaces are para-hyper-Kahler manifolds. We review and prove the relevant results of para-complex and para-hypercomplex geometry. In particular, we give a second, purely geometrical construction of both c-maps, by proving that the cotangent bundle N=T^*M of any affine special (para-)Kahler manifold M is para-hyper-Kahler. 
Vicente Cortes, Christoph Mayer, Thomas Mohaupt, Frank Saueressig