哈密顿 $\coprod\limits_n BO(n)$-作用、分层 Morse 理论与 $J$-同态
A Hamiltonian $\coprod\limits_n BO(n)$-action, stratified Morse theory and the $J$-homomorphism
AI总结：
该论文通过谱层量子化哈密顿 BO(n)-作用，并用 J-同态丰富分层 Morse 理论，为精确 Lagrangian 子流形的 branes 结构分类提供关键步骤。
摘要：
我们使用谱层来量子化由 Bott 周期性自然产生的哈密顿 $\coprod\limits_n BO(n)$-作用在 $\varinjlim\limits_{N}T^*\mathbf{R}^N$ 上。我们利用 [GaRo] 中发展的对应范畴，通过 $J$-同态给出分层 Morse 理论的丰富化。这为 [Jin] 中关于 [JiTr] 中一个断言的证明提供了关键步骤：一个（浸入的）精确 Lagrangian 子流形 $L\subset T^*\mathbf{R}^N$ 上的 branes 结构局部系统的分类映射由稳定 Gauss 映射 $L\rightarrow U/O$ 与 $J$-同态的 delooping $U/O\rightarrow B\mathrm{Pic}(\mathbf{S})$ 的复合给出。我们特别强调所涉及范畴的函子性和（对称）幺半结构，并作为副产品，我们在对应的（对称）幺半 $\infty,2$-范畴中构造了若干具体的（交换）代数/模对象及它们之间的（右 lax）态射，推广了 [GaRo] 中基于 Segal 对象的构造，这可能本身就有意义。
英文摘要：
We use sheaves of spectra to quantize a Hamiltonian $\coprod\limits_n BO(n)$-action on $\varinjlim\limits_{N}T^*\mathbf{R}^N$ that naturally arises from Bott periodicity. We employ the category of correspondences developed in [GaRo] to give an enrichment of stratified Morse theory by the $J$-homomorphism. This provides a key step in the following work [Jin] on the proof of a claim in [JiTr]: the classifying map of the local system of brane structures on an (immersed) exact Lagrangian submanifold $L\subset T^*\mathbf{R}^N$ is given by the composition of the stable Gauss map $L\rightarrow U/O$ and the delooping of the $J$-homomorphism $U/O\rightarrow B\mathrm{Pic}(\mathbf{S})$. We put special emphasis on the functoriality and (symmetric) monoidal structures of the categories involved, and as a byproduct, we produce several concrete constructions of (commutative) algebra/module objects and (right-lax) morphisms between them in the (symmetric) monoidal $(\infty, 2)$-category of correspondences, generalizing the construction out of Segal objects in [GaRo], which might be of interest by its own.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
