Explicit formula of deformation quantization with separation of variables for complex two-dimensional locally symmetric Kähler manifold

Abstract

We give a complex two-dimensional noncommutative locally symmetric Kähler 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.

34 pages, no figure; In v2, typos and errors are corrected. Two subsections are added in Section 2. The organization of Section 4 is corrected from two to three subsections. Section 5 and many references are added

References in corpus (16)

• Deformation quantization of Poisson manifolds, I
• Toeplitz Quantization of Kähler Manifolds and $gl(N)$ $N\to \infty$
• Fuzzy CP2
• Fuzzy Complex Projective Spaces and their Star-products
• Generalized Coherent State Approach to Star Products and Applications to the Fuzzy Sphere
• Supersymmetric Nonlinear Sigma Models as Gauge Theories
• Berezin-Toeplitz quantization for compact Kaehler manifolds. A Review of Results
• Monopole Bundles over Fuzzy Complex Projective Spaces
• Phase Space Reduction for Star-Products: An Explicit Construction for CP^n
• Gauge Theoretical Construction of Non-compact Calabi-Yau Manifolds
• A New Noncommutative Product on the Fuzzy Two-Sphere Corresponding to the Unitary Representation of SU(2) and the Seiberg-Witten Map
• Quantum Equivalence of Auxiliary Field Methods in Supersymmetric Theories
• Berezin-Toeplitz quantization and Berezin transform
• Explicit Formulae for Noncommutative Deformations of CP^N and CH^N
• Gauge Theories in Noncommutative Homogeneous Kähler Manifolds
• Supersymmetric nonlinear sigma models as anomalous gauge theories