Quantisation of Twistor Theory by Cocycle Twist

We present the main ingredients of twistor theory leading up to and including the Penrose-Ward transform in a coordinate algebra form which we can then 'quantise' by means of a functorial cocycle twist. The quantum algebras for the conformal group, twistor space CP3, compactified Minkowski space CM# and the twistor correspondence space are obtained along with their canonical quantum differential calculi, both in a local form and in a global *-algebra formulation which even in the classical commutative case provides a useful alternative to the formulation in terms of projective varieties. We outline how the Penrose-Ward transform then quantises. As an example, we show that the pull-back of the tautological bundle on CM# pulls back to the basic instanton on S-4 subset of CM# and that this observation quantises to obtain the Connes-Landi instanton on theta-deformed S-4 as the pull-back of the tautological bundle on our theta-deformed CM#. We likewise quantise the fibration CP3 -> S-4 and use it to construct the bundle on theta-deformed CP3 that maps over under the transform to the theta-deformed instanton.