Special Geometry of Euclidean Supersymmetry I: Vector Multiplets
Vicente Cortes, Christoph Mayer, Thomas Mohaupt, Frank Saueressig
TL;DR
The article establishes a complete framework for rigid 4D N=2 Euclidean vector multiplets by introducing affine special para-Kähler geometry as the natural scalar target space. It shows that para-holomorphic prepotentials define SUSY-invariant Lagrangians in Euclidean signature, provided a specific four-fermion term is included, and demonstrates that Euclidean actions and SUSY variations adopt the same structural form as their Minkowskian counterparts when expressed in para-holomorphic coordinates. A detailed dimensional reduction from 5D to 4D clarifies the correspondence between Minkowski and Euclidean theories, yielding an explicit dictionary and revealing a temporal r-map that links affine very special real manifolds to affine special para-Kähler manifolds. The work also emphasizes the central role of para-complex structures, para-Kähler potentials, and the non-compact R-symmetry ${ m SO}(1,1)_R$ in shaping the Euclidean scalar geometry and its SUSY content. These results set the stage for extending to other multiplets and supergravity, with immediate implications for instanton and soliton analyses in string/M-theory.
Abstract
We construct the general action for Abelian vector multiplets in rigid 4-dimensional Euclidean (instead of Minkowskian) N=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-Kahler geometry. We give a precise definition and develop the mathematical theory of such manifolds. The relation to the affine special Kahler manifolds appearing in Minkowskian N=2 supersymmetry is discussed. Starting from the general 5-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.