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

2025/12/17 by Wang, Dan, Yau, Yutung
Mathematics · #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology #Geometric Analysis and Curvature Flows

paper · doi:10.48550/arxiv.2512.15060

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 H⁰(M, L^{⊗ 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 \widetildeD(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_ℂ, P). (2) Using these PBW isomorphisms, we show that the Grothendieck connections on the transeverse jet bundle of L^{⊗ k} give rise to a deformation quantization (C_M^∞[[ℏ]], ⋆) together with a sheaf of subalgebras C_{M, ℏ}^<∞ that acts on P-polarized sections of L^{⊗ k}. We obtain a geometric interpretation of (C_{M, ℏ}^<∞, ⋆) by evaluating at ℏ = \tfrac√(-1)k, yielding a sheaf O_k^(<∞), and proving that O_k^(<∞) ≅ \widetildeD_{L^{⊗ k}} as sheaves of filtered algebras, where \widetildeD_{L^{⊗ k}} is the sheaf of transverse differential operators on L^{⊗ 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].

Citations

• Quantization of Kähler manifolds via differential operators
• A new look at unitarity in quantization commutes with reduction for toric manifolds
• Quantum spaces associated to mixed polarizations and their limiting behavior on toric varieties
• Symmetry in Deformation quantization and Geometric quantization
• Fibering polarizations and Mabuchi rays on symmetric spaces of compact type
• Quantization of Kähler Manifolds via Brane Quantization
• Limit of geometric quantizations on Kähler manifolds with T-symmetry
• Quantization in fibering polarizations, Mabuchi rays and geometric Peter--Weyl theorem
• Geometric quantizations of mixed polarizations on Kähler manifolds with T-symmetry
• Geodesic rays in the space of Kähler metrics with T-symmetry
• Berezin-Toeplitz Quantization in Real Polarizations with Toric Singularities
• Deformation quantization via Toeplitz operators on geometric quantization in real polarizations
• Partial coherent state transforms, G \× T-invariant K "ahler\n structures and geometric quantization of cotangent bundles of compact Lie\n groups
• Polarized deformation quantization
• Rozansky-Witten invariants via Atiyah classes

Related

• Berezin-Toeplitz Quantization in Real Polarizations with Toric Singularities
• Deformation quantization via Toeplitz operators on geometric quantization in real polarizations
• The double Gelfand-Cetlin system, invariance of polarization, and the Peter-Weyl theorem
• A formal model of Berezin-Toeplitz quantization
• Quantization of Kähler manifolds via differential operators