A Hamiltonian ∐_<i>n</i><i>BO</i>(<i>n</i>)-action, stratified Morse theory and the <i>J</i>-homomorphism

We use sheaves of spectra to quantize a Hamiltonian $\coprod _n BO(n)$ -action on $\varinjlim _{N}T^*\mathbf {R}^N$ that naturally arises from Bott periodicity. We employ the category of correspondences developed by Gaitsgory and Rozenblyum [ A study in derived algebraic geometry, vol. I. Correspondences and duality , Mathematical Surveys and Monographs, vol. 221 (American Mathematical Society, 2017)] to give an enrichment of stratified Morse theory by the $J$ -homomorphism. This provides a key step in the work of Jin [ Microlocal sheaf categories and the $J$ -homomorphism , Preprint (2020), arXiv:2004.14270v4] on the proof of a claim of Jin and Treumann [ Brane structures in microlocal sheaf theory , J. Topol. 17 (2024), e12325]: 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 of Gaitsgory and Rozenblyum, which might be of independent interest.