'Special geometry of Euclidean supersymmetry I: Vector multiplets'

By Vicente Cortés, C. Mayer, T. Mohaupt, Frank Saueressig. 2003

253 citations recorded by the source corpus.

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.

Subfields: Astronomy and Astrophysics, Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/ee37239a020297ddf17a5c867e52ea2748b7dbe4

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Citations in the corpus

Half‐flat structures and special holonomy (2009, background): “6). Finally, we ﬁnd that one of the two open orbits of SL(6,R) on ∧3R6 carries aﬃne special paraKahler geometry, the geometry of N= 2 vector multiplets on Euclidian rather than Minkowskian space-time [CMMS]. The corresponding homogeneous projective special para-Ka¨hler manifold is the symmetric spac”
Riemannian Submersions from Quaternionic Manifolds (2008, background): “On the other hand, the quaternionic Kahler manifolds also have applications in physics as target spaces for nonlinear σ -models with supersymmetry [ 5 ].”
Quantum black holes, localization, and the topological string (2010, methodology): “note that in the Euclidean theory in four dimensions, the spinors should have a symplectic-Majorana condition imposed on them, while in Minkowski spacetime they can be majorana or symplectic-Majorana [75]. In addition, the Weyl projection is not compatible with the majorana condition in the Minkowsk”
Cones over pseudo-Riemannian manifolds and their holonomy (2007, background): “as in the previous sections but with diﬀerent structures coming up. These structures are related to a para-complex structure, and to a para-Sasakian structure. We recall the basic deﬁnitions given in [CMMS04] and [CLS06]. Deﬁnition 8.1. 1. Let V be a real ﬁnite dimensional vector space. A para-compl”
The geometry of modified Riemannian extensions (2009, background): “Moreover, Walker metrics are the underlying structure of many geometrical structures like para-Kaehler and hypersymplectic structures (see [3, 10, 11, 26, 29]).”
Maximally homogeneous para-CR manifolds (2006, unknown): “[1],[6],[7],[10],[17].”
Homogeneous para-Kähler Einstein manifolds (2009, methodology): “The notion of special para-K¨ahler structure on a manifold M had been defined in the important pa pe r [34].”