Twists of quaternionic Kähler manifolds

Haydys showed that to any hyperkahler manifold, equipped with a Killing field Z that preserves one of its Kahler structures and rotates the other two, one can associate a quaternionic Kahler 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 hyperkahler manifolds and quaternionic Kahler manifolds of negative scalar curvature by Alekseevsky, Cortes, 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 Kahler manifold to that of the hyperkahler manifold. In particular, the Weyl curvature of the quaternionic Kahler manifold (which is of hyperkahler type) can be interpreted as a sum of two abstract curvature tensors, one coming from the curvature on the hyperkahler 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 Kahler manifold commuting with Z is at least a central extension of the Lie algbera of Hamiltonian Killing fields on the hyperkahler 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, Cortes, 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 Kahler manifolds and showing that the QK/HK correspondence can be realised as the twist of an elementary deformation of the quaternionic Kahler manifold with respect to this twist data. More generally, we construct 1-loop deformations of quaternionic Kahler manifolds as twists of elementary deformations of the quaternionic Kahler manifold directly. In doing so, we prove an analogue of Macia and Swann's theorem where instead of a hyperkahler manifold, we have a quaternionic Kahler 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 Kahler structures on an open ball.