Twistor Spaces, Mirror Symmetry and Self-dual Kahler Manifolds

We present the evidence for two conjectures related to the twistor string. The first conjecture states that two super-Calabi Yaus -- the supertwistor space and the superambitwistor space -- form a mirror pair. The second conjecture is that the B-model on the twistor space can be seen as describing a 4-dimensional gravitational theory, whose partition function should involve a sum over ''space-time foams'' related to D1 branes in the topological string