Special geometry of euclidean supersymmetry II. Hypermultiplets and the $c$-map
Résumé
We construct two new versions of the c-map which allow us to obtain the target manifolds of hypermultiplets in euclidean theories with rigid Script $N = 2$ 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-Kähler 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-)Kähler manifold $M$ is para-hyper-Kähler.