Twists of quaternionic Kähler manifolds

Abstract

Haydys showed that to any hyperkähler manifold, equipped with a Killing field Z̃ that preserves one of its Kähler structures and rotates the other two, one can associate a quaternionic Kähler manifold of the same dimension, which has positive scalar curvature and also carries a Killing field Z. This HK/QK correspondence was extended to indefinite hyperkähler manifolds and quaternionic Kähler manifolds of negative scalar curvature by Alekseevsky, Cortés, and Mohaupt. It was later described by Macia and Swann in terms of elementary deformations and the twist construction, originally introduced by Swann.

In this dissertation, we use the twist realisation of the HK/QK correspondence to write down an elegant formula relating the Riemann curvature of the quaternionic Kähler manifold to that of the hyperkähler manifold. In particular, the Weyl curvature of the quaternionic Kähler manifold (which is of hyperkähler type) can be interpreted as a sum of two abstract curvature tensors, one coming from the curvature on the hyperkähler side of the correspondence, and one coming from a standard abstract curvature tensor constructed out of the twist form. We furthermore use the twist construction to show that the Lie algebra of Hamiltonian Killing fields of the quaternionic Kähler manifold commuting with Z is at least a central extension of the Lie algbera of Hamiltonian Killing fields on the hyperkähler side that preserve the HK/QK data. As an application of these general results, we prove that that the 1-loop deformation of Ferrara--Sabharwal metrics with quadratic prepotential, obtained using the HK/QK correspondence by Alekseevsky, Cortés, Dyckmanns, and Mohaupt, have cohomogeneity 1 in every dimension.

In addition to the above, we also complete the twist-based picture of the HK/QK correspondence by identifying certain canonical twist data on the quaternionic Kähler manifolds and showing that the QK/HK correspondence can be realised as the twist of an elementary deformation of the quaternionic Kähler manifold with respect to this twist data. More generally, we construct 1-loop deformations of quaternionic Kähler manifolds as twists of elementary deformations of the quaternionic Kähler manifold directly. In doing so, we prove an analogue of Macia and Swann's theorem where instead of a hyperkähler manifold, we have a quaternionic Kähler manifold.

In order to be able to efficiently carry out these constructions, we also develop an alternative local formulation of the twist construction which requires weaker hypotheses than that of Swann. The description of 1-loop deformations in terms of a local twist map is finally used to construct geometric flow equations on the space of quaternionic Kähler structures on an open ball.