  We explore the quantum properties of self-dual gravity formulated as a large
$N$ two-dimensional WZW sigma model. Using a non-trivial classical background,
we show that a $(2,2)$ space-time is generated. The theory contains an infinite
series of higher point vertices. At tree level we show that, in spite of the
presence of higher than cubic vertices, the on-shell 4 and higher point
functions vanish, indicating that this model is related with the field theory
of closed N=2 strings. We examine the one-loop on-shell 3-point amplitude and
show that it is ultra-violet finite.
