Explicit formula of deformation quantization with separation of variables for complex two-dimensional locally symmetric K\"{a}hler manifold

TL;DR

This paper derives an explicit formula for the star product in deformation quantization with separation of variables on complex two-dimensional locally symmetric Kähler manifolds, specifically for ^2 and  P^2.

Contribution

It provides a new explicit formula for the star product on these manifolds by solving recurrence relations, advancing the understanding of deformation quantization in complex geometry.

Findings

Explicit star product formula for complex 2D Kähler manifolds.

Verification of star product identities on ^2 and  P^2.

Solution of recurrence relations for separation of variables in deformation quantization.

Abstract

We give a complex two-dimensional noncommutative locally symmetric K\"{a}hler manifold via a deformation quantization with separation of variables. We present an explicit formula of its star product by solving the system of recurrence relations given by Hara-Sako. In the two-dimensional case, this system of recurrence relations gives two types of equations corresponding to the two coordinates. From the two types of recurrence relations, symmetrized and antisymmetrized recurrence relations are obtained. The symmetrized one gives the solution of the recurrence relation. From the antisymmetrized one, the identities satisfied by the solution are obtained. The star products for $\mathbb{C}^{2}$ and $\mathbb{C}P^{2}$ are constructed by the method obtained in this study, and we verify that these star products satisfy the identities.