Quaternion-Kaehler four-manifolds and Przanowski's function

TL;DR

This paper demonstrates the integrability of Przanowski's equation for quaternion-Kaehler four-manifolds using twistorial methods, and develops tools to connect solutions with twistor space cohomology, enabling explicit reconstruction of Przanowski's function.

Contribution

It introduces a Lax Pair for Przanowski's equation, establishes a method to derive solutions from twistor cohomology, and provides an explicit algorithm to reconstruct Przanowski's function from twistor data.

Findings

Constructed a Lax Pair confirming integrability.

Developed a contour integral formula for perturbations.

Provided an explicit reconstruction algorithm from twistor data.

Abstract

Quaternion-Kaehler 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.