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 S4 ⊂ CM and that this observation quantises to obtain the Connes-Landi instanton on θ-deformed S4 as the pull-back of the tautological bundle on our θ-deformed CM. We likewise quantise the fibration CP3 → S4 and use it to construct the bundle on θ-deformed CP3 that maps over under the transform to the θ-deformed instanton.