A Hamiltonian $\coprod _{n}BO(n)$-action, stratified Morse theory and the $J$-homomorphism

Abstract

We use sheaves of spectra to quantize a Hamiltonian $\coprod _{n}BO(n)$-action on $\underset{N}{\underrightarrow{\lim }}{T}^{\ast }{\mathbf{R}}^N$ 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 $L\subset T^{\ast }{\mathbf{R}}^N$ is given by the composition of the stable Gauss map $L\to U/O$ and the delooping of the $J$-homomorphism $U/O\to B\mathrm{Pic}(\mathbf{S})$. 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.

arXiv comment96 pages; Compositio Math. (to appear)