1-平移余等值面的积分与量子化
Integration and quantization of 1-shifted coisotropics
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- Université de Sherbrooke（舍布鲁克大学）
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中文总结 AI 辅助
本文将 Cattaneo–Felder 的余等值约化积分与量子化框架推广到 1-平移辛可微层中的 1-平移余等值面，建立积分定理并构造平坦 P_∞ 结构与弯曲 A_∞ 量子化，恢复约化泊松括号。
中文摘要 AI 辅助
为了量子化泊松流形中余等值子流形的约化，Cattaneo 和 Felder 构造了一个同伦泊松代数及其形变量子化。其零阶上同调为约化泊松代数，而完整的同伦结构即使在约化空间奇异或平凡时也保留了形式嵌入的信息。在积分方面，Cattaneo 证明了余等值子流形可积分为积分环境泊松流形的任意辛群胚的拉格朗日子群胚。现在公认这两种构造都属于平移辛几何：拉格朗日子群胚呈现了可微层上的 $1$-平移拉格朗日态射，而同伦泊松代数描述了相关的 $0$-平移泊松结构。然而，这样的态射仅构成 $1$-平移拉格朗日态射的一个特殊类别。我们将此图景推广到 $1$-平移拉格朗日态射，更一般地，推广到 $1$-平移辛可微层中的 $1$-平移余等值面。我们建立了其无穷小描述（以扭曲 Dirac 结构表示）的积分定理，并在不变函数上构造了自然的泊松括号。在正则性假设下，我们为相关的李群胚复配备了平坦的 $P_\infty$-结构和弯曲的 $A_\infty$-量子化，这些结构在同构意义下独立于辅助选择。零阶诱导括号恢复了约化泊松括号，且适当的同调消没条件给出了不变函数的形变量子化。这扩展了 Cattaneo–Felder 的图景到更一般的约化过程，其中相同的无穷小数据同时支配积分和量子化。
英文摘要
To quantize the reduction of a coisotropic submanifold of a Poisson manifold, Cattaneo and Felder constructed a homotopy Poisson algebra together with a deformation quantization. Its degree-zero cohomology is the reduced Poisson algebra, while the full homotopy structure retains information about the formal embedding even when the reduced space is singular or trivial. On the integration side, Cattaneo showed that a coisotropic submanifold integrates to a Lagrangian subgroupoid of any symplectic groupoid integrating the ambient Poisson manifold. It is now recognized that both constructions fit into shifted symplectic geometry: the Lagrangian subgroupoid presents a $1$-shifted Lagrangian morphism of differentiable stacks, and the homotopy Poisson algebra describes the associated $0$-shifted Poisson structure. However, such morphisms form only a special class of $1$-shifted Lagrangians. We extend this picture to $1$-shifted Lagrangians and, more generally, $1$-shifted coisotropics in $1$-shifted symplectic differentiable stacks. We establish an integration theorem for their infinitesimal description in terms of twisted Dirac structures and construct a natural Poisson bracket on invariant functions. Under regularity hypotheses, we equip the associated Lie algebroid complex with a flat $P_\infty$-structure and a curved $A_\infty$-quantization, independent of the auxiliary choices up to isomorphism. The induced bracket in degree zero recovers the reduced Poisson bracket, and suitable cohomological vanishing conditions yield a deformation quantization of invariant functions. This extends the Cattaneo--Felder picture to more general reduction procedures, with the same infinitesimal data governing integration and quantization.
