Quantization in mixed polarization via transverse Poincaré-Birkhoff-Witt theorem

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].

💡 Research Summary

The paper develops a unified framework for quantization on a pre‑quantizable symplectic manifold (M, ω) equipped with an arbitrary non‑singular complex polarization P, extending the recent work of Chan–Leung–Li that was limited to Kähler polarizations. The authors introduce the notions of transverse jet bundles and transverse differential operators associated to P. For any locally free P‑modules E and E′, they construct a Poincaré–Birkhoff–Witt (PBW) isomorphism
 pbw_{E,E′}: Sym Q ⊗ E* ⊗ E′ → \widetilde D(E,E′)
where Q = TM_ℂ / P is the quotient bundle. This PBW map provides a filtered C^∞(M)‑module isomorphism and generalizes the Lie‑pair PBW theorem of Laurent‑Gengoux‑Stiénon‑Xu. The construction depends only on auxiliary choices: a splitting of the exact sequence 0→P→TM_ℂ→Q→0, a torsion‑free connection on Q, and a flat P‑connection on E extending the given one. The PBW isomorphism yields a canonical Kapranov connection ∇_{K,E} on the transverse jet bundle, obtained by pulling back the Grothendieck connection via the PBW map.

Using these tools, the authors adapt Fedosov’s deformation quantization to the transverse setting. They consider the Weyl bundle W = d Sym T* M_ℂ