Quantization in mixed polarization via transverse Poincaré-Birkhoff-Witt theorem
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911324009660416 |
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| author | Wang, Dan Yau, Yutung |
| author_facet | Wang, Dan Yau, Yutung |
| contents | On a prequantizable Kähler manifold $(M, ω, L)$, Chan-Leung-Li constructed a genuine (non-asymptotic) action of a subalgebra of the Berezin-Toeplitz star product on $H^0(M, L^{\otimes k})$ for each level $k$ [14]. We extend their framework to any non-singular polarization $P$ by developing a theory of transverse differential operators associated to $P$:
(1) For any pair of locally free $P$-modules $E, E'$, we construct a Poincaré-Birkhoff-Witt isomorphism for the bundle $\widetilde{D}(E, E')$ of transverse differential operators from $E$ to $E'$. When $E, E'$ are trivial rank-$1$ $P$-modules, this recovers the PBW theorem of Laurent-Gengoux-Stiénon-Xu [29] for the Lie pair $(TM_\mathbb{C}, P)$.
(2) Using these PBW isomorphisms, we show that the Grothendieck connections on the transeverse jet bundle of $L^{\otimes k}$ give rise to a deformation quantization $(C_M^\infty[[\hbar]], \star)$ together with a sheaf of subalgebras $C_{M, \hbar}^{<\infty}$ that acts on $P$-polarized sections of $L^{\otimes k}$. We obtain a geometric interpretation of $(C_{M, \hbar}^{<\infty}, \star)$ by evaluating at $\hbar = \tfrac{\sqrt{-1}}{k}$, yielding a sheaf $O_k^{(<\infty)}$, and proving that $O_k^{(<\infty)} \cong \widetilde{D}_{L^{\otimes k}}$ as sheaves of filtered algebras, where $\widetilde{D}_{L^{\otimes k}}$ is the sheaf of transverse differential operators on $L^{\otimes k}$. When $P$ is a Kähler polarization, this recovers the result of Chan-Leung-Li [14].
As an application, we study symplectic tori and derive asymptotic expansions for the Toeplitz-type operators in real polarization introduced in [35]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_15060 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantization in mixed polarization via transverse Poincaré-Birkhoff-Witt theorem Wang, Dan Yau, Yutung Symplectic Geometry Mathematical Physics Algebraic Geometry Differential Geometry On a prequantizable Kähler manifold $(M, ω, L)$, Chan-Leung-Li constructed a genuine (non-asymptotic) action of a subalgebra of the Berezin-Toeplitz star product on $H^0(M, L^{\otimes k})$ for each level $k$ [14]. We extend their framework to any non-singular polarization $P$ by developing a theory of transverse differential operators associated to $P$: (1) For any pair of locally free $P$-modules $E, E'$, we construct a Poincaré-Birkhoff-Witt isomorphism for the bundle $\widetilde{D}(E, E')$ of transverse differential operators from $E$ to $E'$. When $E, E'$ are trivial rank-$1$ $P$-modules, this recovers the PBW theorem of Laurent-Gengoux-Stiénon-Xu [29] for the Lie pair $(TM_\mathbb{C}, P)$. (2) Using these PBW isomorphisms, we show that the Grothendieck connections on the transeverse jet bundle of $L^{\otimes k}$ give rise to a deformation quantization $(C_M^\infty[[\hbar]], \star)$ together with a sheaf of subalgebras $C_{M, \hbar}^{<\infty}$ that acts on $P$-polarized sections of $L^{\otimes k}$. We obtain a geometric interpretation of $(C_{M, \hbar}^{<\infty}, \star)$ by evaluating at $\hbar = \tfrac{\sqrt{-1}}{k}$, yielding a sheaf $O_k^{(<\infty)}$, and proving that $O_k^{(<\infty)} \cong \widetilde{D}_{L^{\otimes k}}$ as sheaves of filtered algebras, where $\widetilde{D}_{L^{\otimes k}}$ is the sheaf of transverse differential operators on $L^{\otimes k}$. When $P$ is a Kähler polarization, this recovers the result of Chan-Leung-Li [14]. As an application, we study symplectic tori and derive asymptotic expansions for the Toeplitz-type operators in real polarization introduced in [35]. |
| title | Quantization in mixed polarization via transverse Poincaré-Birkhoff-Witt theorem |
| topic | Symplectic Geometry Mathematical Physics Algebraic Geometry Differential Geometry |
| url | https://arxiv.org/abs/2512.15060 |