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.

JF was supported by an Action de Recherche Concertée and by an Interuniversity Action Poles grant.
KK was supported by ERC Starting Grant 277570-DIGT
DP is a Royal Society Research Fellow