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-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 parahyper-Kähler.

Abstract. We classify extended Poincaré Lie super algebras and Lie algebras of any signature (p, q), that is Lie super algebras and Z 2 -graded Lie algebras g = g 0 + g 1 , where g 0 = so(V ) + V is the (generalized) Poincaré Lie algebra of the pseudo Euclidean vector space V = R p,q of signature (p, q) and g 1 = S is the spinor so(V )-module extended to a g 0 -module with kernel V . The remaining super commutators {g 1 , g 1 } (respectively, commutators [g 1 , g 1 ]) are defined by an so(V )-equivariant linear mappingDenote by P + (n, s) (respectively, P − (n, s)) the vector space of all such Lie super algebras (respectively, Lie algebras), where n = p + q = dim V and s = p − q is the signature. The description of P ± (n, s) reduces to the construction of all so(V )-invariant bilinear forms on S and to the calculation of three Z 2 -valued invariants for some of them.This calculation is based on a simple explicit model of an irreducible Clifford module S for the Clifford algebra Cl p,q of arbitrary signature (p, q). As a result of the classification, we obtain the numbers L ± (n, s) = dim P ± (n, s) of independent Lie super algebras and algebras, which take values 0,1,2,3,4 or 6. Due to Bott periodicity, L ± (n, s) may be considered as periodic functions with period 8 in each argument. They are invariant under the group Γ generated by the four reflections with respect to the axes n = −2, n = 2, s − 1 = −2 and s − 1 = 2. Moreover, the reflection (n, s) → (−n, s) with respect to the axis n = 0 interchanges L + and L − :

We define and study projective special para-Kähler manifolds and show that they appear as target manifolds when reducing five-dimensional vector multiplets coupled to supergravity with respect to time. The dimensional reductions with respect to time and space are carried out in a uniform way using an ǫ-complex notation. We explain the relation of our formalism to other formalisms of special geometry used in the literature. In the second part of the paper we investigate instanton solutions and their dimensional lifting to black holes. We show that the instanton action, which can be defined after dualising axions into tensor fields, agrees with the ADM mass of the corresponding black hole. The relation between actions via Wick rotation, Hodge dualisation and analytic continuation of axions is discussed.

We introduce the notion of a special complex manifold: a complex manifold (M,J) with a flat torsionfree connection \nabla such that (\nabla J) is symmetric. A special symplectic manifold is then defined as a special complex manifold together with a \nabla-parallel symplectic form \omega . This generalises Freed's definition of (affine) special K\"ahler manifolds. We also define projective versions of all these geometries. Our main result is an extrinsic realisation of all simply connected (affine or projective) special complex, symplectic and K\"ahler manifolds. We prove that the above three types of special geometry are completely solvable, in the sense that they are locally defined by free holomorphic data. In fact, any special complex manifold is locally realised as the image of a holomorphic 1-form \alpha : C^n \to T^* C^n. Such a realisation induces a canonical \nabla-parallel symplectic structure on M and any special symplectic manifold is locally obtained this way. Special K\"ahler manifolds are realised as complex Lagrangian submanifolds and correspond to closed forms \alpha. Finally, we discuss the natural geometric structures on the cotangent bundle of a special symplectic manifold, which generalise the hyper-K\"ahler structure on the cotangent bundle of a special K\"ahler manifold.Comment: 24 pages, latex, section 3 revised (v2), modified Abstract and Introduction, version to appear in J. Geom. Phy

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-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 parahyper-Kähler.