Kähler流形上的范畴量子化
Categorical quantization on Kähler manifolds
AI总结：
本文利用Fedosov粘合论证构造Kähler流形上Hermite全纯向量丛范畴的量子化范畴DQ，定义可量子化态射，并证明在预量子化条件下该范畴等价于全纯微分算子范畴GQ。
摘要：
作为对Kähler流形$M$上变量分离形变量子化的推广，我们采用Fedosov的粘合论证构造了一个范畴$\mathsf{DQ}$，该范畴富化于$M$上$\mathbb{C}[[\hbar]]$-模的层，作为$M$上Hermite全纯向量丛范畴的量子化，其中态射为hom-丛的光滑截面。随后我们定义了$\mathsf{DQ}$中对象之间的可量子化态射，推广了Chan-Leung-Li[4]关于可量子化函数的概念。在$\hbar = \tfrac{\sqrt{-1}}{k}$处对可量子化态射求值后，我们得到一个富化范畴$\mathsf{DQ}_{\operatorname{qu}, k}$。我们证明，当$M$可预量子化时，$\mathsf{DQ}_{\operatorname{qu}, k}$通过一个由Bargmann-Fock作用得到的函子，等价于$M$上全纯向量丛的范畴$\mathsf{GQ}$，其中态射为全纯微分算子。
英文摘要：
Generalizing deformation quantizations with separation of variables of a Kähler 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.
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