The Geroch group and non-Hausdorff twistor spaces

Abstract

By reducing the Ward correspondence, the authors show that there is a correspondence between stationary axisymmetric solutions of the vacuum Einstein equations and a class of holomorphic vector bundles, over a reduced twistor space, which is a compact one-dimensional, but non-Hausdorff, complex manifold. They show that the solutions generated by Ward's ansatze correspond to bundles which have a simpler behaviour on the 'real axis' in the reduced space. They identify the Geroch group (Kinnersley and Chitre's 'group K' (1978)) with a subgroup of the loop group of GL(2,C) and they describe its orbits. They also identify some of the subgroups which preserve asymptotic flatness.