Quantization in mixed polarization via transverse Poincaré-Birkhoff-Witt theorem
Dan Wang, Yutung Yau
TL;DR
This work extends the Chan–Leung–Li polarized quantization to arbitrary non-singular polarizations by developing transverse jet bundles, a transverse PBW theorem, and a polarization-adapted Fedosov quantization. It constructs a deformation quantization together with a canonical filtered subalgebra that acts on P-polarized sections, and proves an isomorphism between the resulting level-k quantizable functions and the transverse differential operators on L^{⊗k}. The framework unifies deformation and geometric quantization beyond the Kähler setting and yields asymptotic Toeplitz-type operator expansions in real polarization, with explicit specializations to Kähler manifolds, cotangent bundles, and symplectic tori. The results provide a robust geometric-analytic toolkit for quantization with mixed or general polarizations, including concrete constructions, cohomological underpinnings, and practical asymptotics for Toeplitz-type operators in real polarization. This advances a cohesive bridge between Fedosov-type deformation quantization and geometric quantization in broader polarizations, with applications to symplectic tori and real-polarization Toeplitz theory.
Abstract
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].