Categorical quantization on Kähler manifolds

YuTung Yau
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Abstract

Generalizing deformation quantizations with separation of variables of a K\"ahler manifold $M$, we adopt Fedosov's gluing argument to construct a category $\mathsf{DQ}$, enriched over sheaves of $\mathbb{C}[[\hbar]]$-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 $\mathsf{DQ}$, generalizing Chan-Leung-Li's notion [4] of quantizable functions. Upon evaluation of quantizable morphisms at $\hbar = \tfrac{\sqrt{-1}}{k}$, we obtain an enriched category $\mathsf{DQ}_{\operatorname{qu}, k}$. We show that, when $M$ is prequantizable, $\mathsf{DQ}_{\operatorname{qu}, k}$ is equivalent to the category $\mathsf{GQ}$ of holomorphic vector bundles over $M$ with morphisms being holomorphic differential operators, via a functor obtained from Bargmann-Fock actions.
凯勒流形上的分类量化
通过对K/"ahler流形$M$的变量分离的变形量子化的一般化,我们采用费多索夫的粘合论证来构造一个类别$\mathsf{DQ}$,它是对$M$上的赫(Hermitian)全态向量束类别的量子化。然后,我们定义了$\mathsf{DQ}$中对象间的可量子化态,并推广了陈亮丽的可量子化函数概念[4]。在$\hbar = \tfrac{\sqrt{-1}}{k}$处对可量子化态进行评估后,我们得到了一个丰富范畴$\mathsf{DQ}_{\operatorname{qu}, k}$。我们证明,当 $M$ 是可预量化的时候,$mathsf{DQ}_{\operatorname{qu}, k}$通过一个从巴格曼-福克作用得到的函子,等价于 $M$ 上全态向量束的类别 $mathsf{GQ}$,其态量是全态微分算子。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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