Origin of Ashtekar's Variables in a Pure-Connection Theory of Gravity

A pure-connection formulation of general relativity is presented in which the only dynamical variable present in the action is a connection one-form with values in the Lie algebra of the pseudounitary group SU(2,2). The action is quadratic in the curvature and is independent of any space-time metric. Although manifestly SU(2,2) gauge invariant, the symmetry group of the action can be broken down to that of the Lorentz group SL(2, $C$) yielding Jacobson and Smolin's covariant self-dual version (modulo a topological and cosmological term) of the Ashtekar formulation of general relativity.