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Revista de la Unión Matemática Argentina
versión impresa ISSN 00416932versión Online ISSN 16699637
Rev. Unión Mat. Argent. v.47 n.1 Bahía Blanca ene./jun. 2006
The special geometry of Euclidian supersymmetry: a survey
Vicente Cortés
Abstract: This is a survey about recent joint work with Christoph Mayer, Thomas Mohaupt and Frank Saueressig on the special geometry of Euclidian supersymmetry. It is based on the second of two lectures given at the II Workshop in Differential Geometry, La Falda, Córdoba, 2005.
The purpose of this note is to present some geometric structures and constructions which arise in the study of supersymmetric field theories on a Euclidian rather than Minkowskian spacetime, see [CMMS1, CMMS2].
The text is written for readers with a background in differential geometry. Nevertheless, a rough idea of supersymmetry and the restrictions it imposes on the geometry of the scalar manifold would be helpful for orientation. A short introduction to classical (rigid) supersymmetric field theories and to the notion of special geometry is provided in another contribution to this volume, based on the first of my two talks given at the II Workshop in Differential Geometry. Such theories are usually considered on the ddimensional Minkowski space. The case is particularly important since our observed spacetime is fourdimensional.
For various physical reasons, it is useful to consider also field theories on Euclidian space, even if our physical spacetime metric is not positive definite. In quantum field theory, for instance, one needs to understand instantons, since they contribute to the Feynman path integral. Instantons are particular solutions of the EulerLagrange equations of a Euclidian counterpart of the underlying Minkowskian classical field theory.
For the Euclidian 4space there exists an superPoincaré algebra and Euclidian vector multiplets can be defined [CMMS1]. (There exists no superPoincaré algebra on the Euclidian 4space.) The special geometry of the scalar manifold in supersymmetric field theories with Euclidian vector multiplets was determined in [CMMS1] and named (affine) special paraKähler geometry, since the role of the complex structure in special Kähler geometry is now played by a paracomplex structure , . Special paraKähler manifolds are discussed in section 1.
In section 2 we present several maps, which relate various special geometries associated to field theories in five, four and three dimensions. In particular, we discuss two purely geometric constructions of parahyperKähler manifolds, which correspond to the dimensional reduction of a Euclidian or Minkowskian theory in 4 dimensions to a Euclidian theory in 3 dimensions [CMMS2].
1. Special paraKähler manifolds
Definition 1. A paraKähler manifold is a pseudoRiemannian manifold endowed with a parallel skewsymmetric involution
A special paraKähler manifold is a paraKähler manifold endowed with a flat torsionfree connection satisfying
 where is the symplectic form associated to and
In particular, and is of split signature
Definition 2. A field of involutions on a manifold with integrable eigendistributions of the same dimension is called a paracomplex structure.
A manifold endowed with a paracomplex structure is called a paracomplex manifold.
A map between paracomplex manifolds is called paraholomorphic if
A paraholomorphic function is a paraholomorphic map with values in the ring of paracomplex numbers .
For any there exists an open neighbourhood and paraholomorphic functions
Such a system of paraholomorphic functions is called a system of paraholomorphic coordinates.
1.2. Extrinsic construction of special paraKähler manifolds. Consider the free module with its global linear paraholomorphic coordinates , its standard paraholomorphic symplectic form
Definition 3. Let be a paracomplex manifold of real dimension . A paraholomorphic immersion is called paraKählerian (respectively, Lagrangian) if is nondegenerate (respectively, if ).
It is easy to see that the metric induced by a paraKählerian immersion is paraKählerian. In the following, we will abbreviate paraKählerian Lagrangian immersion to PKLI.
Lemma 1. Let be a PKLI and the corresponding symplectic structure. Then where ,
By the lemma defines a system of local coordinates. Therefore, there exists a unique flat and torsionfree connection on for which and are affine functions.
Theorem 1. [CMMS1] Let be a PKLI with induced data Then is a special paraKähler manifold.
Conversely, any simply connected special paraKähler manifold admits a PKLI with induced data Moreover, the PLKI is unique up to an element of
Proof of "". Let be a PKLI with induced data We have to show that is special paraKähler. We know that is paraKähler and that is flat and torsionfree. By the lemma, the symplectic form has constants coefficients with respect to the affine coordinates Thus It remains to show that is symmetric. For a parallel oneform and vector fields on we calculate:
Therefore, it is sufficient to prove that is closed for and . Let us check this, for example, for . The function is the realpart of the paraholomorphic function So . Since and are closed, this shows that is closed.
Corollary 1. Let be a paraholomorphic function defined on a open set satisfying the nondegeneracy condition
Then ,
Conversely, any special paraKähler manifold is locally of this form.
2. Maps between special geometries from dimensional reduction
Dimensional reduction is a procedure for the construction of a field theory in spacetime dimensions from one in dimensions.
In the context of special geometry of Euclidian supersymmetry it is natural to ask the following two questions:
 Is it possible to construct supersymmetric field theories with vector multiplets on 4dimensional Euclidian space from field theories on 5dimensional Minkowski space?
 Is it possible to construct Euclidian supersymmetric field theories in 3 dimensions out of supersymmetric field theories with vector multiplets in 4 dimensions?
The first question is given a detailed positive answers in [CMMS1], the second in [CMMS2]. We describe the corresponding geometrical constructions in the remaining two subsections.
2.1. Dimensional reduction from 5 to 4 dimensions. The allowed target geometry for the scalar fields in the relevant supersymmetric theories on 5dimensional Minkowski space is called (affine) very special, see [CMMS1].
It is defined by a real cubic polynomial with nondegenerate Hessian on some domain
We found that dimensional reduction of such a Minkowskian theory over time yields a Euclidian supersymmetric theory with vector multiplets such that the target is special paraKähler [CMMS1]. This means that we get a map:

which we call the pararmap.
Theorem 2. [CMMS1] There exists a map which associates a special paraKähler structure on the domain to any very special manifold , The special paraKähler structure is defined by the paraholomorphic function
which satisfies
This is the paraversion of the rmap:

introduced by B. de Wit and A. Van Proeyen in [DV] in the context of supergravity.
2.2. Dimensional reduction from 4 to 3 dimensions. We found two ways of constructing Euclidian supersymmetric field theories in 3 dimensions out of theories with vector multiplets in 4 dimensions [CMMS2].
One can start either with a Minkowskian theory and reduce over time or with a Euclidian theory. This gives us two maps:
which we call the paracmaps. They are paravariants of the cmap, worked out by Cecotti, Ferrara and Girardello in [CFG].
Definition 4. A parahyperKähler manifold is a pseudoRiemannian manifold with three pairwise anticommuting parallel skewsymmetric endomorphism fields , such that A pseudoRiemannian manifold is parahyperKähler if and only if its holonomy group
Here stands for the standard symplectic structure of . In particular, the dimension of any parahyperKähler manifold is divisible by 4.
The paracmaps. Now I describe, for instance, the parahyperKähler manifold associated to a special paraKähler manifold via the paracmap Let be the total space of the cotangent bundle and consider the decomposition , into horizontal and vertical subbundles with respect to the connection
This defines a canonical identification
With respect to the above identification, we define a pseudoRiemannian metric on by
Theorem 3. [CMMS2] For any special paraKähler manifold , is a parahyperKähler manifold.
Conclusion. The maps between special geometries induced by dimensional reduction are summarized in the following diagram:
The diagram is essentially commutative:
Theorem 4. [CMMS2] For any very special manifold the parahyperKähler manifolds and are canonically isometric.
[CFG] S. Cecotti, S. Ferrara, L. Girardello, Geometry of type II superstrings and the moduli of superconformal field theories, Internat. J. Modern Phys. A 4 (1989), no. 10, 24752529. [ Links ]
[CMMS1] V. Cortés, C. Mayer, T. Mohaupt and F. Saueressig, Special Geometry of Euclidean Supersymmetry I: Vector Multiplets, J. High Energy Phys. 028 (2004), no. 3, 73 pp. [arXiv:hepth/0312001]. [ Links ]
[CMMS2] V. Cortés, C. Mayer, T. Mohaupt and F. Saueressig, Special Geometry of Euclidean Supersymmetry II: hypermultiplets and the cmap, J. High Energy Phys. 025 (2005), no. 6, 37 pp. [arXiv:hepth/0503094]. [ Links ]
[DV] B. de Wit, A. Van Proeyen, Special geometry, cubic polynomials and homogeneous quaternionic spaces, Comm. Math. Phys. 149 (1992), no. 2, 307333 [arXiv:hepth/9112027]. [ Links ]
Vicente Cortés
Department Mathematik
Schwerpunkt Analysis und Differentialgeometrie
und
Zentrum für Mathematische Physik
Universität Hamburg
Bundesstrasse 55
D 20146 Hamburg
cortes@math.unihamburg.de
Recibido: 31 de agosto de 2005
Aceptado: 2 de octubre de 2006