Anti-self-dual fields and manifolds

Abstract

In this thesis we study anti–self–duality equations in four and eight dimensions on manifolds of special Riemannian holonomy, among these hyper–Kähler, Quaternion–Kähler and Spin(7)–manifolds.

We first consider the octonionic anti–self–duality equations on manifolds with holonomy Spin(7). We construct explicit solutions to their symmetry reductions, the non–abelian Seiberg–Witten equations, with gauge group SU(2). These solutions are singular for flat and Eguchi–Hanson backgrounds, however we find a solution on a co–homogeneity one hyper–Kähler metric with a domain wall, and the solution is regular away from the wall. We then turn to Quaternion–Kähler four–manifolds, which are locally determined by one scalar function subject to Przanowski’s equation. Using twistorial methods we construct a Lax Pair for Przanowski’s equation, confirming its integrability. The Lee form of a compatible local complex structure gives rise to a conformally invariant differential operator, special cases of the associated generalised Laplace operator are the conformal Laplacian and the linearised Przanowski operator. Using recursion relations we construct a contour integral formula for perturbations of Przanowski’s function. Finally, we construct an algorithm to retrieve Przanowski’s function from twistor data. At last, we investigate the relationship between anti–self–dual Einstein metrics with non–null symmetry in neutral signature and pseudo–, para– and null–Kähler metrics. We classify real–analytic anti–self–dual null–Kähler metrics with a Killing vector that are conformally Einstein. This allows us to formulate a neutral signature version of Tod’s result, showing that around non-singular points all real–analytic anti–self–dual Einstein metrics with symmetry are conformally pseudo– or para–Kähler.