On the hyperkaehler/quaternion Kaehler correspondence

TL;DR

This paper explores the relationship between hyperkaehler and quaternionic Kaehler manifolds, constructing associated line bundles and demonstrating their properties through examples like monopole and Higgs bundle moduli spaces.

Contribution

It introduces a holomorphic line bundle on twistor space and a twistor version of the hyperkaehler/quaternion Kaehler correspondence, with explicit computations and a meromorphic connection.

Findings

Constructed holomorphic line bundle on twistor space.

Computed examples including monopole and Higgs bundle moduli spaces.

Established a meromorphic connection as the quantum line bundle.

Abstract

A hyperkaehler manifold with a circle action fixing just one complex structure admits a natural a hyperholomorphic line bundle. This forms the basis for the construction of a corresponding quaternionic Kaehler manifold in the work of A.Haydys. We construct in this paper the corresponding holomorphic line bundle on twistor space and compute many examples, including monopole and Higgs bundle moduli spaces. We also show that the bundle on twistor space has a natural meromorphic connection which realizes it as the quantum line bundle for the hyperkaehler family of holomorphic symplectic structures. Finally we give a twistor version of the HK/QK correspondence.