  We develop a global twistor correspondence for pseudo-Riemannian conformal
structures of signature (++--) with self-dual Weyl curvature. Near the
conformal class of the standard indefinite product metric on S^2 x S^2, there
is an infinite-dimensional moduli space of such conformal structures, and each
of these has the surprising global property that its null geodesics are all
periodic. Each such conformal structure arises from a family of holomorphic
disks in CP_3 with boundary on some totally real embedding of RP^3 into CP_3.
An interesting sub-class of these conformal structures are represented by
scalar-flat indefinite K\"ahler metrics, and our methods give particularly
sharp results in this more restrictive setting.
