The map between conformal hypercomplex/hyper-Kaehler and quaternionic(-Kaehler) geometry

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-Kaehler manifolds is mapped to quaternionic-Kaehler manifolds. We relate the curvatures of the corresponding manifolds and furthermore map the symmetries of these manifolds to each other.

54 pages, 2 figures; v2: small corrections, version to be published in CMP; v3: changes of statement on (3.5)

References in corpus (13)

• N=2 supergravity in five dimensions revisited
• HKT and OKT Geometries on Soliton Black Hole Moduli Spaces
• Superconformal Hypermultiplets
• Superconformal N=2, D=5 matter with and without actions
• Hypermultiplets, Hyperkahler Cones and Quaternion-Kahler Geometry
• The Geometry of (Super) Conformal Quantum Mechanics
• Rigid N=2 superconformal hypermultiplets
• Superconformal Sigma Models in Higher Than Two Dimensions
• Supergravity with Fayet-Iliopoulos terms and R-symmetry
• Consistent and inconsistent truncations. Some results and the issue of the correct uplifting of solutions
• Special geometries, from real to quaternionic
• Hypermultiplets and hypercomplex geometry from 6 to 3 dimensions
• Aspects of on-shell supersymmetry

Cited by in corpus (18)

• N=2 supergravity in five dimensions revisited
• String loop corrected hypermultiplet moduli spaces
• On the fakeness of fake supergravity
• D-instantons and twistors
• Twistors and Black Holes
• Membrane Instantons and de Sitter Vacua
• Linear perturbations of quaternionic metrics
• Six-dimensional superconformal couplings of non-abelian tensor and hypermultiplets
• Higher Derivative Corrections and Central Charges from Wrapped M5-branes
• Conifold singularities, resumming instantons and non-perturbative mirror symmetry
• Quaternion-Kahler spaces, hyperkahler cones, and the c-map
• Stringy cosmic strings in matter coupled N=2, d=4 supergravity
• Hypermultiplets and hypercomplex geometry from 6 to 3 dimensions
• More on the Triplet Killing Potentials of Quaternionic Kaehler Manifolds
• Wess-Zumino sigma models with non-Kahlerian geometry
• The identification of conformal hypercomplex and quaternionic manifolds
• An analogue of the Gibbons-Hawking Ansatz for quaternionic Kähler spaces
• New aspects of N=2 theories