Self-dual gravity as a two-dimensional theory and conservation laws

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Abstract
Starting from the Ashtekar Hamiltonian variables for general relativity, the self-dual Einstein equations (SDE) may be rewritten as evolution equations for three divergence-free vector fields given on a three-dimensional surface with a fixed volume element. From this general form of the SDE, it is shown how they may be interpreted as the field equations for a two-dimensional field theory. It is further shown that these 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 the 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.
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