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

Authors
  • Bergshoeff, Eric1
  • Cucu, Sorin2
  • Wit, Tim de1
  • Gheerardyn, Jos2, 3
  • Vandoren, Stefan4
  • Proeyen, Antoine Van2
  • 1 University of Groningen, Center for Theoretical Physics, Nijenborgh 4, Groningen, AG, 9747, The Netherlands , Groningen
  • 2 Katholieke Universiteit Leuven, Instituut voor Theoretische Fysica, Celestijnenlaan, Leuven, 200D 3001, Belgium , Celestijnenlaan
  • 3 Università di Torino, and I.N.F.N., Dipartimento di Fisica Teorica, Sezione di Torino, via P. Giuria 1, Torino, 10125, Italy , Torino
  • 4 Utrecht University, Institute for Theoretical Physics, Leuvenlaan 4, Utrecht, TA, 3508, The Netherlands , Utrecht
Type
Published Article
Journal
Communications in Mathematical Physics
Publisher
Springer-Verlag
Publication Date
Dec 20, 2005
Volume
262
Issue
2
Pages
411–457
Identifiers
DOI: 10.1007/s00220-005-1475-6
Source
Springer Nature
Keywords
License
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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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