Vicente Cortés

Special geometry of euclidean supersymmetry 1. Vector multiplets

We construct the general action for abelian vector multiplets in rigid 4-dimensional euclidean (instead of minkowskian) = 2 supersymmetry, i.e., over space-times with a positive definite instead of a lorentzian metric. The target manifolds for the scalar fields turn out to be para-complex manifolds endowed with a particular kind of special geometry, which we call affine special para-K?hler geometry. We give a precise definition and develop the mathematical theory of such manifolds. The relation to the affine special K?hler manifolds appearing in minkowskian = 2 supersymmetry is discussed. Starting from the general five-dimensional vector multiplet action we consider dimensional reduction over time and space in parallel, providing a dictionary between the resulting euclidean and minkowskian theories. Then we reanalyze supersymmetry in four dimensions and find that any (para-)holomorphic prepotential defines a supersymmetric lagrangian, provided that we add a specific four-fermion term, which cannot be obtained by dimensional reduction. We show that the euclidean action and supersymmetry transformations, when written in terms of para-holomorphic coordinates, take exactly the same form as their minkowskian counterparts. The appearance of a para-complex and complex structure in the euclidean and minkowskian theory, respectively, is traced back to properties of the underlying R-symmetry groups. Finally, we indicate how our work will be extended to other types of multiplets and to supergravity in the future and explain the relevance of this project for the study of instantons, solitons and cosmological solutions in supergravity and M-theory.

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 Script

Cones over pseudo-Riemannian manifolds and their holonomy

N = 2

Polyvector Super-Poincare Algebras

supersymmetry. While the minkowskian para-

Special geometry of Euclidean supersymmetry III: the local r-map, instantons and black holes

c

AFFINE HYPERSPHERES ASSOCIATED TO SPECIAL PARA-KÄHLER MANIFOLDS

-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

Killing spinors are Killing vector fields in Riemannian Supergeometry

c

Half-flat structures and special holonomy

-maps, by proving that the cotangent bundle

Realisation of special Kahler manifolds as parabolic spheres

N = T*M

Handbook of pseudo-Riemannian geometry and supersymmetry

of any affine special (para-)Kahler manifold