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Communications in Mathematical Physics

, Volume 262, Issue 2, pp 411–457 | Cite as

The Map Between Conformal Hypercomplex/ Hyper-Kähler and Quaternionic(-Kähler) Geometry

  • Eric Bergshoeff
  • Sorin Cucu
  • Tim de Wit
  • Jos Gheerardyn
  • Stefan Vandoren
  • Antoine Van Proeyen
Article

Abstract

We review the general properties of target spaces of hypermultiplets, which are quaternionic-like manifolds, and discuss the relations between these manifolds and their symmetry generators. We explicitly construct a one-to-one map between conformal hypercomplex manifolds (i.e. those that have a closed homothetic Killing vector) and quaternionic manifolds of one quaternionic dimension less. An important role is played by `ξ-transformations', relating complex structures on conformal hypercomplex manifolds and connections on quaternionic manifolds. In this map, the subclass of conformal hyper-Kähler manifolds is mapped to quaternionic-Kähler manifolds. We relate the curvatures of the corresponding manifolds and furthermore map the symmetries of these manifolds to each other.

Keywords

Neural Network Manifold Statistical Physic Complex System Nonlinear Dynamics 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Eric Bergshoeff
    • 1
  • Sorin Cucu
    • 2
  • Tim de Wit
    • 1
  • Jos Gheerardyn
    • 2
    • 3
  • Stefan Vandoren
    • 4
  • Antoine Van Proeyen
    • 2
  1. 1.Center for Theoretical PhysicsUniversity of GroningenGroningenThe Netherlands
  2. 2.Instituut voor Theoretische FysicaKatholieke Universiteit LeuvenCelestijnenlaanBelgium
  3. 3.Dipartimento di Fisica TeoricaUniversità di Torino, and I.N.F.N.TorinoItaly
  4. 4.Institute for Theoretical PhysicsUtrecht UniversityUtrechtThe Netherlands

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