Rod Structures and Patching Matrices: a review

Abstract

I review the twistor theory construction of stationary and axisymmetric, Lorentzian signature solutions of the Einstein vacuum equations and the related toric Ricci-flat metrics of Riemannian signature, \cite{W,MW,F,FW}. The construction arises from the Ward construction \cite{W2} of anti-self-dual Yang-Mills fields as holomorphic vector bundles on twistor space, with the observation of Witten \cite{LW} that the Einstein equations for these metrics include the anti-self-dual Yang-Mills equations. The principal datum for a solution is the holomorphic patching matrix $P$ for a holomorphic vector bundle on a reduced twistor space, and $P$ is typically simpler than the corresponding metric to write down. I give a catalogue of examples, building on earlier collections \cite{F,AG}, and consider the inverse problem: how far does the rod structure of such a metric, together with its asymptotics, determine $P$?