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A Hamiltonian ∐n BO(n)-action, stratified Morse theory and the J-homomorphism
Published online by Cambridge University Press: 13 September 2024
Abstract
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.
Keywords
MSC classification
Secondary:
55P42: Stable homotopy theory, spectra
- Type
- Research Article
- Information
- Copyright
- © The Author(s), 2024. The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence
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