Special Geometry of Euclidean Supersymmetry I: Vector Multiplets

TL;DR

This paper develops the mathematical framework for Euclidean N=2 supersymmetric vector multiplets, introducing affine special para-Kahler geometry and establishing a correspondence with Minkowskian theories, with implications for instantons and supergravity.

Contribution

It constructs the general Euclidean vector multiplet action, defines affine special para-Kahler geometry, and relates Euclidean and Minkowskian supersymmetric theories through dimensional reduction.

Findings

Euclidean scalar target manifolds are affine special para-Kahler

Supersymmetric Lagrangians are defined by para-holomorphic prepotentials with a four-fermion term

Euclidean and Minkowskian theories share similar structures in para-holomorphic coordinates

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.