Quaternion-Kähler four-manifolds and Przanowski's function

Abstract

Quaternion-Kähler four-manifolds, or equivalently anti-self-dual Einstein manifolds, 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, which one can always find, gives rise to a conformally invariant differential operator acting on sections of a line bundle. Special cases of the associated generalised Laplace operator are the conformal Laplacian and the linearised Przanowski operator. We provide recursion relations that allow us to construct cohomology classes on twistor space from solutions of the generalised Laplace equation. Conversely, we can extract such solutions from twistor cohomology, leading to a contour integral formula for perturbations of Przanowski's function. Finally, we illuminate the relationship between Przanowski's function and the twistor description, in particular, we construct an algorithm to retrieve Przanowski's function from twistor data in the double-fibration picture. Using a number of examples, we demonstrate this procedure explicitly.