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Berezin–Toeplitz quantization in real polarizations with toric singularities
Communications in Number Theory and Physics ( IF 1.9 ) Pub Date : 2022-10-21 , DOI: 10.4310/cntp.2022.v16.n4.a6
Nai-Chung Conan Leung 1 , Yu-Tung Yau 1
Affiliation  

On a compact Kähler manifold $X$, Toeplitz operators determine a deformation quantization $(\mathrm{C}^\infty (X,\mathbb{C}) [\![\hbar]\!], \star)$ with separation of variables [10] with respect to transversal complex polarizations $T^{1,0} X, T^{0,1} X$ as $\hbar \to 0^{+}$ [15]. The analogous statement is proved for compact symplectic manifolds with transversal non-singular real polarizations [13]. In this paper, we establish the analogous result for transversal singular real polarizations on compact toric symplectic manifolds $X$. Due to toric singularities, half-form correction and localization of our Toeplitz operators are essential. Via norm estimations, we show that these Toeplitz operators determine a star product on $X$ as $\hbar \to 0^{+}$.

中文翻译:

具有复曲面奇点的实极化中的 Berezin-Toeplitz 量化

在紧致 Kähler 流形 $X$ 上,Toeplitz 算子确定变形量化 $(\mathrm{C}^\infty (X,\mathbb{C}) [\![\hbar]\!], \star)$关于横向复极化 $T^{1,0} X, T^{0,1} X$ 作为 $\hbar \to 0^{+}$ [ 15 ]的变量 [ 10 ] 的分离。对于具有横向非奇异实极化的紧辛流形,证明了类似的陈述[ 13 ]。在本文中,我们建立了横向奇异的类比结果紧复环辛流形 $X$ 上的实极化。由于复曲面奇点,我们的 Toeplitz 算子的半形式校正和本地化是必不可少的。通过范数估计,我们表明这些 Toeplitz 算子将 $X$ 上的星积确定为 $\hbar \to 0^{+}$。
更新日期:2022-10-21
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