Quantisation of Twistor Theory by Cocycle Twist

Abstract

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 ${\mathbb {CP}^3}$ , compactified Minkowski space ${\mathbb {CM}^\#}$ 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 ${{\mathbb {CM}^\#}}$ pulls back to the basic instanton on ${S^4\subset{\mathbb {CM}^\#}}$ and that this observation quantises to obtain the Connes-Landi instanton on θ-deformed S ⁴ as the pull-back of the tautological bundle on our θ-deformed ${{\mathbb {CM}^\#}}$ . We likewise quantise the fibration ${{\mathbb {CP}^3} \rightarrow S^4}$ and use it to construct the bundle on θ-deformed ${{\mathbb {CP}^3}}$ that maps over under the transform to the θ-deformed instanton.