Quantisation of Twistor Theory by Cocycle Twist

Abstract

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 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 ⊂ 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 CM # . 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.

The Adhm Construction of Instantons on Noncommutative Spaces

“…Although the products of functions and differential forms in the calculus are indeed twisted, one finds that the extra terms which appear in the twisted product all vanish in the expressions for the (anti-)commutators (cf. [6] for full details).…”

“…However, a good understanding of the noncommutativegeometric origins of this construction has so far been lacking and our goal is to shed some light upon this subject (although it is worth pointing out that a different approach to the twistor theory of R 4 , from the point of view of noncommutative algebraic geometry, was carried out in [14]). Using the noncommutative twistor theory developed in [6], we give an explicit construction of families of instantons on Moyal space-time, from which we recover the well-known noncommutative ADHM equations of Nekrasov and Schwarz.…”

Noncommutative Principal Bundles Through Twist Deformation

“…Next, we prove that G is a coalgebra morphism, i.e. (4) ) ⊗ h (6) h (5) ⊗ γ G(h (7) )γ S(h (1) ) ⊗ h (8) =γ S(h (2) ) ⊗ h (4) h (3) ⊗ γ G(h (5) )γ S(h (1) ) ⊗ h (6) =γ S(h (2) ) ⊗ h (4) h (3) ⊗ γ h (7) u γ (h (5) )γ S(h (6) ) ⊗ h (8) γ S(h (1) ) ⊗ h (9) = ∆ γ (h) , for all h ∈ H γ . The last equality follows from comparison with (3.8).…”

q-Fuzzy Spheres and Quantum Differentials on B q [SU 2] and U q (su 2)

“…It is easy to check using the usual properties σ i σ j = δ ij + ıǫ ijk σ k of the Pauli matrices that the monopole projector e is now deformed to e = 1+λ 2 − σ · x. It is also easy in the point of view of [7] to run the calculation that e is a projector backwards, i.e. we write e = 1 + λ − a b b * a so that (6) trace(e) = 1 + λ as a deformation of CP 1 , where we fix λ to be some real number.…”

“…Uniform projector construction of quantum spheres 2.1 We first recall the 'quantum logic' point of view on CP 1 used in [7]. Thus, specifying a line in C 2 is the same thing as specifying a matrix e of size 2 × 2 and obeying (2) e 2 = e, e † = e…”