Noncommutative Deformations of Locally Symmetric K\"ahler manifolds

TL;DR

This paper develops algebraic recurrence relations to construct deformation quantizations with separation of variables for locally symmetric K"ahler manifolds, providing explicit star products for specific cases like ${ m C}P^N$ and Grassmann manifolds.

Contribution

It introduces a new algebraic recurrence approach to deformation quantization on locally symmetric K"ahler manifolds, enabling explicit calculations of star products.

Findings

Explicit star products for ${ m C}P^N$ and one-dimensional symmetric K"ahler manifolds.

Recurrence relations for Grassmann manifold $G_{2,2}$.

Concrete algebraic formulas for deformation quantization.

Abstract

We derive algebraic recurrence relations to obtain a deformation quantization with separation of variables for a locally symmetric K\"ahler manifold. This quantization method is one of the ways to perform a deformation quantization of K\"ahler manifolds, which is introduced by Karabegov. From the recurrence relations, concrete expressions of star products for one-dimensional local symmetric K\"ahler manifolds and ${\mathbb C}P^N$ are constructed. The recurrence relations for a Grassmann manifold $G_{2,2}$ are closely studied too.