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 CP^3, compactified Minkowski space CMh 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 CMh pulls back to the basic instanton on S^4\subset CMh 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 CMh. We likewise quantise the fibration CP^3--> S^4 and use it to construct the bundle on θ-deformed CP^3 that maps over under the transform to the θ-deformed instanton.

68 pages 0 figures. Significant revision now has detailed formulae for classical and quantum CP^3

References in corpus (6)

• String Theory and Noncommutative Geometry
• Instantons on noncommutative R^4, and (2,0) superconformal six dimensional theory
• Noncommutative Manifolds the Instanton Algebra and Isospectral Deformations
• Noncommutative Instantons and Twistor Transform
• Twisting of quantum differentials and the Planck scale Hopf algebra
• Principal fibrations from noncommutative spheres

Cited by in corpus (10)

• Twistor theory at fifty: from contour integrals to twistor strings
• Nonassociative Riemannian Geometry by Twisting
• Noncommutative Solitons and Quasideterminants
• Noncommutative principal bundles through twist deformation
• The ADHM Construction of Instantons on Noncommutative Spaces
• Bäcklund Transformations for Noncommutative Anti-Self-Dual Yang-Mills Equations
• Bäcklund Transformations and the Atiyah-Ward ansatz for Noncommutative Anti-Self-Dual Yang-Mills Equations
• An algebraic framework for noncommutative bundles with homogeneous fibres
• Differential and Twistor Geometry of the Quantum Hopf Fibration
• The Noncommutative Topology of Anti-Self-Dual Gauge Fields