SELF-DUAL GRAVITY AS A 2-DIMENSIONAL THEORY AND CONSERVATION-LAWS

Starting from the Ashtekar Hamiltonian variables for general relativit y, the self-dual Einstein equations (SDE) may be rewritten as evolutio n equations for three divergence-free vector fields given on a three-d imensional surface with a fixed volume element. From this general form of the SDE, it is shown how they may be interpreted as the field equa tions for a two-dimensional field theory. It is further shown that the se equations imply an infinite number of non-local conserved currents. A specific way of writing the vector fields allows an identification of the full SDE With those of the two-dimensional chiral model, with t he gauge group being the group of area-preserving diffeomorphisms of a two-dimensional surface. This gives a natural Hamiltonian formulation of the SDE in terms of that of the chiral model.