A Hamiltonian ∐ n BO(n)-action, stratified Morse theory and the J-homomorphism

We use sheaves of spectra to quantize a Hamiltonian ₙ BO (n) -action on ₍T^* RN 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 T^* RN is given by the composition of the stable Gauss map L U/O and the delooping of the J -homomorphism U/O B Pic (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 (, 2) -category of correspondences, generalizing the construction out of Segal objects of Gaitsgory and Rozenblyum, which might be of independent interest.