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

Xin Jin

Abstract

We use sheaves of spectra to quantize a Hamiltonian $\coprod\limits_n BO(n)$-action on $\varinjlim\limits_{N}T^*\mathbf{R}^N$ 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 $L\subset T^*\mathbf{R}^N$ 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 in [GaRo], which might be of interest by its own.

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

Abstract

We use sheaves of spectra to quantize a Hamiltonian -action on 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 -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 is given by the composition of the stable Gauss map and the delooping of the -homomorphism . 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 -category of correspondences, generalizing the construction out of Segal objects in [GaRo], which might be of interest by its own.

Paper Structure

This paper contains 55 sections, 73 theorems, 376 equations, 3 figures.

Table of Contents

  1. Introduction
  2. A Hamiltonian $\coprod\limits_n BO(n)$-action and Bott periodicity
  3. Quantization of the Hamiltonian $\coprod\limits_n BO(n)$-action and a natural "module"
  4. Stratified Morse theory
  5. The $J$-homomorphism
  6. Statement of the main results
  7. Expected strategy of the proof of Claim \ref{['claim: 1']}
  8. The category of correspondences and the functor of taking sheaves of spectra
  9. Applications of the category of correspondences to quantizations
  10. Related results and future work
  11. Acknowledgement
  12. Background and results on the $(\infty,2)$-categories of correspondences
  13. The definition of the $(\infty,2)$-categories of correspondences
  14. (Symmetric) monoidal structure on $\mathbf{Corr}(\mathcal{C})$
  15. A construction of Cartesian fibration $\mathcal{C}^{\times,\Delta}\rightarrow N(\Delta)$ (resp. $\mathcal{C}^{\times,(\mathrm{Fin}_*)^{op}}\rightarrow N(\mathrm{Fin}_*)^{op}$) for a Cartesian monoidal structure
  16. ...and 40 more sections

Key Result

Proposition 1.2

The functor is symmetric monoidal, and the local system $(\pi_{VG})_*\varpi_{VG}$ is the commutative algebra object in $\mathrm{Loc}(G;\mathrm{Sp})$ classified by $J$.

Figures (3)

  • Figure 1: A picture of $\Gamma_{[n],s,e}$.
  • Figure 2: A picture of $S_{[n],e_+}$.
  • Figure 3: An illustration of $S_0, H=H_0, (x_0,\xi_0)=(0,0;0, -dt)$, the curves $\partial B_{\rho(\epsilon)}((0,\epsilon))$ (the dashed circles) and the smoothings $H_\epsilon$ (the dashed curves that have tangency with $H$). The shaded region enclosed by the dashed $H_\epsilon$ and the solid $H_0$ is $U_\epsilon-U_0$.

Theorems & Definitions (142)

  • Claim 1.1
  • Proposition 1.2: (See Proposition \ref{['prop: equiv model of J']})
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Remark 1.7
  • Theorem 1.8
  • Proposition 2.1
  • Definition 2.2
  • ...and 132 more