The Quantum Twistor Bundle

Abstract

We investigate the quantum twistor bundle constructed as a $U(1)$-quotient of the quantum instanton bundle of Bonechi, Ciccoli and Tarlini. It is an example of a locally trivial noncommutative bundle fulfilling conditions of the framework recently proposed by Brzeziński and Szymański. In particular, we give a detailed description of the corresponding $C^{\ast }$-algebra of 'continuous functions' on its noncommutative total space. Furthermore, we analyse a different construction of a quantum instanton bundle due to Landi, Pagani and Reina, find a basis of its polynomial algebra and discover an intriguing and unexpected feature of its enveloping $C^{\ast }$-algebra.

The statement in Theorem 4.2. is wrong, the error in the proof lies in the analysis of the joint spectrum. An analysis of irreducible representations of the quantum symplectic sphere where made in the paper "Quantum quaternion spheres" by Bipul Saurabh, where we indeed see that x1 is nonzero