AI 中文总结
本文构造了魏instein辛“范畴”的对偶双DG范畴，将其应用于几何量子化，还证明了超代数${\mathfrak {osp}}(1|2)$的作用及相关消失定理。
AI 中文摘要
在Weinstein的辛“范畴”中，对象是辛流形，态射是典范关系，这类关系可能无法复合，因此它只是名义上的范畴。我们构造了一个真正的微分分次（DG）范畴，它在名义上是Weinstein“范畴”的对偶。该范畴的对象是与整辛流形相关的预量子系统，态射是由预量子线丛扭曲的微分形式复形。我们还考虑了上同调范畴，它是一个普通的线性范畴。在每种情况下，Weinstein的态射都与对偶于这些形式的流相关；对于微分分次范畴，这些是迷向或拉格朗日流，由预量子线丛的截面扭曲；对于上同调范畴，这些流支撑在整拉格朗日子流形上，并配备预量子线丛的整体协变常截面。随后我们将方法应用于量子化，证明对于凯勒流形，该范畴中的量子化与全纯量子化相关。在此过程中，我们证明在预量子系统的情形下，由Brylinski、Mathieu、Guillemin及Tseng-Yau在辛情形中研究的${\mathfrak {sl}}(2,\R)$在微分形式上的Lefschetz作用，可提升为超代数${\mathfrak {osp}}(1|2)$在扭曲微分形式上的作用，其中${\mathfrak {sl}}(2,\R)$是其偶子代数。该超代数作用的首个应用是消失定理，它表明上同调范畴仅支撑在中间维度。
英文摘要
In Weinstein's symplectic "category", objects are symplectic manifolds and morphisms are canonical relations, which may not be composable; it is therefore only morally a category. We construct a true differential graded category which is morally dual to Weinstein's "category". The objects in this category are prequantum systems, associated to integral symplectic manifolds. The morphisms are a complex of differential forms twisted by a prequantum line bundle. We also consider the cohomology category, which turns out to be an (ordinary) linear category. In each case, Weinstein's morphisms are associated with currents dual to the forms. For the differential graded category, these are isotropic, or if preferred Lagrangian, currents, twisted by a section of the prequantum line bundle; in the case of cohomology, these currents are supported on {\em integral} Lagrangian submanifolds, equipped with a global covariant constant section of the prequantum line bundle. We then apply our methods to quantization, and show that for Kahler manifolds, a quantization in this category is related to holomorphic quantization. Along the way we show that in the case of prequantum systems, the Lefschetz action of ${\mathfrak {sl}}(2,\R)$ on differential forms, studied by Brylinski, Mathieu, Guillemin, and Tseng-Yau in the symplectic case, is promoted to an action of the superalgebra ${\mathfrak {osp}}(1|2)$ on twisted differential forms, with ${\mathfrak {sl}}(2,\R)$ as the even subalgebra. A first application of this superalgebra action is the vanishing theorem which shows the cohomology category is supported in middle dimension.
