Quantization of Kähler manifolds via differential operators
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913175347134464 |
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| author | Chan, Kwokwai Leung, Naichung Conan Li, Qin Yau, Yutung |
| author_facet | Chan, Kwokwai Leung, Naichung Conan Li, Qin Yau, Yutung |
| contents | In this paper, we study the quantization of classical observables (i.e., functions) on a Kähler manifold $X$ as differential operators acting on holomorphic sections of tensor powers $L^{\otimes k}$ of the pre-quantum line bundle $L$. We prove two global results as follows.
(1). For a general smooth function $f \in C^\infty(X)$, we construct higher order generalizations of Kostant-Souriau's pre-quantum differential operators using our Fedosov-type constructions of Bargmann-Fock sheaves in previous works. We prove that these differential operators are asymptotic to the Berezin-Toeplitz operators $T_{f,k}$ acting on the Hilbert space $H^0(X, L^{\otimes k})$ as $k \to \infty$.
(2). If a smooth function $f \in C^\infty(X)$ is furthermore the symbol of a level $k$ quantizable function , then we prove that the associated Berezin-Toeplitz operator $T_{f,k}$ is a holomorphic differential operator. Conversely, Berezin-Toeplitz operators that are holomorphic differential operators all arise in this way. This gives a complete characterization of when Berezin-Toeplitz operators are holomorphic differential operators.
To prove these results, we establish new orthogonality relations which generalize the classical Tuynman's Lemma, and employ various differential-geometric and analytic technqiues such as Hörmander's estimates. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_16889 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantization of Kähler manifolds via differential operators Chan, Kwokwai Leung, Naichung Conan Li, Qin Yau, Yutung Differential Geometry Mathematical Physics Complex Variables Quantum Algebra In this paper, we study the quantization of classical observables (i.e., functions) on a Kähler manifold $X$ as differential operators acting on holomorphic sections of tensor powers $L^{\otimes k}$ of the pre-quantum line bundle $L$. We prove two global results as follows. (1). For a general smooth function $f \in C^\infty(X)$, we construct higher order generalizations of Kostant-Souriau's pre-quantum differential operators using our Fedosov-type constructions of Bargmann-Fock sheaves in previous works. We prove that these differential operators are asymptotic to the Berezin-Toeplitz operators $T_{f,k}$ acting on the Hilbert space $H^0(X, L^{\otimes k})$ as $k \to \infty$. (2). If a smooth function $f \in C^\infty(X)$ is furthermore the symbol of a level $k$ quantizable function , then we prove that the associated Berezin-Toeplitz operator $T_{f,k}$ is a holomorphic differential operator. Conversely, Berezin-Toeplitz operators that are holomorphic differential operators all arise in this way. This gives a complete characterization of when Berezin-Toeplitz operators are holomorphic differential operators. To prove these results, we establish new orthogonality relations which generalize the classical Tuynman's Lemma, and employ various differential-geometric and analytic technqiues such as Hörmander's estimates. |
| title | Quantization of Kähler manifolds via differential operators |
| topic | Differential Geometry Mathematical Physics Complex Variables Quantum Algebra |
| url | https://arxiv.org/abs/2511.16889 |