Anti-Self-Dual Conformal Structures in Neutral Signature

Abstract

We review the subject of four dimensional anti-self-dual conformal structures with signature (+ + - -). Both local and global questions are discussed. Most of the material is well known in the literature and we present it in a way which underlines the connection with integrable systems. Some of the results - e.g. the Lax pair characterisation of the scalar-flat Kahler condition and a twistor construction of a conformal structure with twisting null Killing vector - are new.

48 pages, 1 figure, contribution to the special volume `Recent developments in pseudo-Riemannian Geometry', ESI-Series on Mathematics and Physics. Revised manuscript with added references. A mistake pointed out by Gueo Grantcharov corrected