Quantization in polarization via transverse Poincaré-Birkhoff-Witt theorem

Abstract

We construct, for every prequantizable symplectic manifold $(M,\omega ,L)$ equipped with a nonsingular polarization, a deformation quantization $(C_M^{\infty }[[\hslash ]],\star )$ together with a filtered sheaf of subalgebras $C_{M,\hslash }^{(<\infty )}$ that admits an exact action on polarized sections of $L^{\otimes k}$ for each $k\in {\mathbb{Z}}^{+}$. This extends the Kostant--Souriau action of polarization-preserving functions. The main geometric ingredient is a transverse PBW theory associated with a Nirenberg integrable distribution $P$. For any locally free $P$-modules $E$ and $E^{\prime }$, we construct a PBW isomorphism for the bundle of transverse differential operators from $E$ to $E^{\prime }$. This extends the PBW theorem for Lie pairs of Laurent-Gengoux--Stiénon--Xu to the setting of $P$-modules. After specializing $\hslash =\frac{\sqrt{-1}}{k}$, we obtain a sheaf $O_k^{(<\infty )}$ of level-$k$ quantizable functions and prove that its action identifies $O_k^{(<\infty )}$ with the sheaf of transverse differential operators on $L^{\otimes k}$. In the Kähler case, this recovers the construction of Chan--Leung--Li. Finally, for standard symplectic tori with real polarizations, we compare our construction with Toeplitz-type operators and obtain their semiclassical asymptotic expansions.

arXiv comment42 pages