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

Taika Okuda, Akifumi Sako

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.

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 and are constructed by the method obtained in this study, and we verify that these star products satisfy the identities.
Paper Structure (15 sections, 14 theorems, 128 equations, 1 table)

This paper contains 15 sections, 14 theorems, 128 equations, 1 table.

Table of Contents

  1. Introduction
  2. Review of Noncommutative Kähler manifolds
  3. Noncommutative Kähler manifolds
  4. Deformation quantization with separation of variables for locally symmetric Kähler manifold
  5. Star product with separation of variables for a complex two-dimensional locally symmetric Kähler manifold
  6. Complex two-dimensional formula
  7. Another formula
  8. Examples
  9. Previous results for $\mathbb{C}^{2}$ and $\mathbb{C}P^{2}$
  10. Example of $\mathbb{C}^{2}$
  11. Example of $\mathbb{C}P^{2}$
  12. Discussions
  13. Some properties from Kähler geometry
  14. Calculations for both sides of \ref{['minus_thm_eq_ens']} in Subsection \ref{['NR_minus']}
  15. Hermiteness of $T_{n}$

Key Result

Theorem 2.1

Let $M$ be an $N$-dimensional Kähler manifold, $U$ be a holomorphic coordinate neighborhood on $M$, and $\omega$ be a Kähler form on $M$. Then, there is the left star-multiplication operator where $A_{n} := a_{n,\alpha}\left(f\right) D^{\alpha} \in \mathcal{S}$ are differential operators whose coefficients $a_{n,\alpha}\left(f\right) \in C^{\infty} \left(U\right)$ depend on $f$. $L_{f}$ is determ

Theorems & Definitions (23)

  • Definition 1.1
  • Definition 1.2
  • Theorem 2.1: KarabegovKara1
  • Definition 2.2
  • Proposition 2.3: Hara-SakoHS1HS2
  • Theorem 2.4: Hara-SakoHS1HS2
  • Proposition 2.5: Hara-SakoHS1HS2
  • Proposition 2.6: Hara-SakoHS1HS2
  • Theorem 3.1
  • Proof
  • ...and 13 more