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

TL;DR

This paper explores the deep geometric relationship between conformal hypercomplex/hyper-Kaehler manifolds and quaternionic(-Kaehler) manifolds, establishing a one-to-one correspondence and analyzing their curvature and symmetries.

Contribution

It explicitly constructs a one-to-one map between conformal hypercomplex and quaternionic manifolds, clarifying their geometric and symmetry relations.

Findings

Mapped conformal hypercomplex manifolds to quaternionic manifolds of one quaternionic dimension less

Related curvatures of the corresponding manifolds

Connected symmetries of these manifolds

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 '\xi-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.