Explicit formulas for noncommutative deformations of ${\mathbb C}{P^N}$CPN and ${\mathbb C}{H^N}$CHN

Abstract

We give explicit expressions of a deformation quantization with separation of variables for \documentclass[12pt]{minimal}\begin{document}${\mathbb C}P^N$\end{document}CPN and \documentclass[12pt]{minimal}\begin{document}${\mathbb C}H^N$\end{document}CHN. This quantization method is one of the ways to perform a deformation quantization of Kähler manifolds, which is introduced by Karabegov. Star products are obtained as explicit formulas in all order in the noncommutative parameter. We also give the Fock representations of the noncommutative \documentclass[12pt]{minimal}\begin{document}${\mathbb C}P^N$\end{document}CPN and \documentclass[12pt]{minimal}\begin{document}${\mathbb C}H^N$\end{document}CHN.