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 Kähler condition and a twistor construction of a conformal structure with twisting null Killing vector – are new.