Special Geometry of Euclidean Supersymmetry I: Vector Multiplets

We construct the general action for Abelian vector multiplets in rigid4-dimensional Euclidean (instead of Minkowskian) N=2 supersymmetry, i.e., overspace-times with a positive definite instead of a Lorentzian metric. The targetmanifolds for the scalar fields turn out to be para-complex manifolds endowedwith a particular kind of special geometry, which we call affine specialpara-Kahler geometry. We give a precise definition and develop the mathematicaltheory of such manifolds. The relation to the affine special Kahler manifoldsappearing in Minkowskian N=2 supersymmetry is discussed. Starting from thegeneral 5-dimensional vector multiplet action we consider dimensional reductionover time and space in parallel, providing a dictionary between the resultingEuclidean and Minkowskian theories. Then we reanalyze supersymmetry in fourdimensions and find that any (para-)holomorphic prepotential defines asupersymmetric Lagrangian, provided that we add a specific four-fermion term,which cannot be obtained by dimensional reduction. We show that the Euclideanaction and supersymmetry transformations, when written in terms ofpara-holomorphic coordinates, take exactly the same form as their Minkowskiancounterparts. The appearance of a para-complex and complex structure in theEuclidean and Minkowskian theory, respectively, is traced back to properties ofthe underlying R-symmetry groups. Finally, we indicate how our work will beextended to other types of multiplets and to supergravity in the future andexplain the relevance of this project for the study of instantons, solitons andcosmological solutions in supergravity and M-theory.Comment: 74 page