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

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An Erratum to this article was published on 26 June 2007

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.

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Authors and Affiliations

  1. Center for Theoretical Physics, University of Groningen, Nijenborgh 4, 9747, AG, Groningen, The Netherlands

    Eric Bergshoeff & Tim de Wit

  2. Instituut voor Theoretische Fysica, Katholieke Universiteit Leuven, Celestijnenlaan, 200D 3001, Leuven, Belgium

    Sorin Cucu, Jos Gheerardyn & Antoine Van Proeyen

  3. Dipartimento di Fisica Teorica, Università di Torino, and I.N.F.N., Sezione di Torino, via P. Giuria 1, 10125, Torino, Italy

    Jos Gheerardyn

  4. Institute for Theoretical Physics, Utrecht University, Leuvenlaan 4, 3508, TA, Utrecht, The Netherlands

    Stefan Vandoren

Authors
  1. Eric Bergshoeff
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  2. Sorin Cucu
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  3. Tim de Wit
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  4. Jos Gheerardyn
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  5. Stefan Vandoren
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  6. Antoine Van Proeyen
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Correspondence to Eric Bergshoeff.

Additional information

Communicated by G.W. Gibbons

An erratum to this article is available at http://dx.doi.org/10.1007/s00220-007-0266-7.

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Bergshoeff, E., Cucu, S., Wit, T. et al. The Map Between Conformal Hypercomplex/ Hyper-Kähler and Quaternionic(-Kähler) Geometry. Commun. Math. Phys. 262, 411–457 (2006). https://doi.org/10.1007/s00220-005-1475-6

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  • DOI: https://doi.org/10.1007/s00220-005-1475-6

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