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

Part of: Partial differential equations Homotopy theory

Published online by Cambridge University Press:  13 September 2024

Xin Jin Open the ORCID record for Xin Jin [Opens in a new window]
Xin Jin*
Affiliation:
Math Department, Boston College, Chestnut Hill, MA 02467, USA xin.jin@bc.edu
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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

microlocal sheaf theorysheaves of spectraLagrangian submanifoldsJ-homomorphismcorrespondences

MSC classification

Primary: 35A27: Microlocal methods; methods of sheaf theory and homological algebra in PDE
Secondary: 55P42: Stable homotopy theory, spectra

Information

Type
Research Article
Information
Compositio Mathematica , Volume 160 , Issue 9 , September 2024 , pp. 2005 - 2099
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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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