  This article investigates a new gauge theoretic approach to Einstein's
equations in dimension 4. Whilst aspects of the formalism are already explained
in various places in the mathematics and physics literature, our first goal is
to give a single coherent account of the theory in purely mathematical
language. We then explain why the new approach may have important mathematical
applications: the possibility of using the calculus of variations to find
Einstein 4-manifolds, as well as links to symplectic topology. We also carry
out some of the technical groundwork to attack these problems.
