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.

https://iopscience.iop.org/article/10.1088/1126-6708/2004/03/028/pdf

'Special geometry of Euclidean supersymmetry I: Vector multiplets'

Using constraints from supersymmetry and string perturbation theory, we determine the string loop corrections to the hypermultiplet moduli space of type II strings compactified on a generic Calabi-Yau threefold. The corresponding quaternion-Kahler manifolds are completely encoded in terms of a single function. The latter receives a one-loop correction and, using superspace techniques, we argue for the existence of a non-renormalization theorem excluding higher loop contributions.

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String loop corrected hypermultiplet moduli spaces

We study loop corrections to the universal dilaton supermultiplet for type IIA strings compactified on Calabi-Yau threefolds. We show that the corresponding quaternionic kinetic terms receive non-trivial one-loop contributions proportional to the Euler number of the Calabi-Yau manifold, while the higher-loop corrections can be absorbed by field ��

/pdf/string-loop-corrections-to-the-universal-hypermultiplet-26ojsept4t.pdf

String loop corrections to the universal hypermultiplet

Gauged locally supersymmetric D=3 nonlinear sigma models

We present theories of N=2 hypermultiplets in four spacetime dimensions that are invariant under rigid or local superconformal symmetries The target spaces of theories with rigid superconformal invariance are (4n)-dimensional {\it special} hyper-Kahler manifolds Such manifolds can be described as cones over tri-Sasakian metrics and are locally the product of a flat four-dimensional space and a quaternionic manifold The latter manifolds appear in the coupling of hypermultiplets to N=2 supergravity We employ local sections of an Sp$(n)\times{\rm Sp}(1)$ bundle in the formulation of the Lagrangian and transformation rules, thus allowing for arbitrary coordinatizations of the hyper-Kahler and quaternionic manifolds

Superconformal Hypermultiplets

Special geometry in hypermultiplets

Dilaton transformation under abelian and non-abelian t-duality in the path-integral approach

We give a detailed analysis of pairs of vector and hypermultiplet theories with N=2 supersymmetry in four spacetime dimensions that are related by the (classical) mirror map. The symplectic reparametrizations of the special Kahler space associated with the vector multiplets induce corresponding transformations on the hypermultiplets. We construct the Sp(1)$\times$Sp($n$) one-forms in terms of which the hypermultiplet couplings are encoded and exhibit their behaviour under symplectic reparametrizations. Both vector and hypermultiplet theories allow vectorial central charges in the supersymmetry algebra associated with integrals over the Kahler and hyper-Kahler forms, respectively. We show how these charges and the holomorphic BPS mass are related by the mirror map.

We present a convenient method for deriving the transformation of the dilaton under T-duality in the path-integral approach. Subtleties arising in performing the integral over the gauge fields are carefully analysed using Pauli-Villars regularization, thereby clarifying existing ambiguities in the literature. The formalism can not only be applied to the abelian case, but, and this for the first time, to the non-abelian case as well. Furthermore, by choosing a particular gauge, we directly obtain the target-space covariant expression for the dual geometry in the abelian case. Finally it is shown that the conditions for gauging non-abelian isometries are weaker than those generally found in the literature.

J. De Jaegher

Papers

Special geometry in hypermultiplets

Dilaton transformation under abelian and non-abelian t-duality in the path-integral approach

Special geometry in hypermultiplets

Dilaton transformation under abelian and non-abelian T-duality in the path integral approach