Twistor actions for gauge theory and gravity

This is a review of recent developments in the study of perturbative gauge theory and gravity using action functionals on twistor space. It is intended to provide a user-friendly introduction to twistor actions, geared towards researchers or graduate students interested in learning something about the utility, prospects, and shortcomings of this approach. For those already familiar with the twistor approach, it should provide a condensed overview of the literature as well as several novel results of potential interest. This work is based primarily upon the author's D.Phil. thesis. We first consider four-dimensional, maximally supersymmetric Yang-Mills theory as a gauge theory in twistor space. We focus on the perturbation theory associated to this action, which in an axial gauge leads to the MHV formalism. This allows us to efficiently compute scattering amplitudes at tree-level (and beyond) in twistor space. Other gauge theory observables such as local operators and null polygonal Wilson loops can also be formulated twistorially, leading to proofs for several correspondences between correlation functions and Wilson loops, as well as a recursive formula for computing mixed Wilson loop / local operator correlators. We then apply the twistor action approach to general relativity, using the on-shell equivalence between conformal and Einstein gravity. This can be extended to N=4 supersymmetry. The perturbation theory of the twistor action leads to formulae for the MHV amplitude with and without cosmological constant, yields a candidate for the Einstein twistor action, and induces a MHV formalism on twistor space. Appendices include discussion of super-connections and Coulomb branch regularization on twistor space.