Categorical quantization on Kähler manifolds

2024/08/30 by YuTung Yau, Yau, YuTung
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph) #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.2408.17201

openalex publication_date 2024/08/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Generalizing deformation quantizations with separation of variables of a Kähler manifold M, we adopt Fedosov's gluing argument to construct a category DQ, enriched over sheaves of ℂ[[ℏ]]-modules on M, as a quantization of the category of Hermitian holomorphic vector bundles over M with morphisms being smooth sections of hom-bundles. We then define quantizable morphisms among objects in DQ, generalizing Chan-Leung-Li's notion [4] of quantizable functions. Upon evaluation of quantizable morphisms at ℏ = \tfrac√(-1)k, we obtain an enriched category DQ_{qu, k}. We show that, when M is prequantizable, DQ_{qu, k} is equivalent to the category GQ of holomorphic vector bundles over M with morphisms being holomorphic differential operators, via a functor obtained from Bargmann-Fock actions.

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