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

Xin Jin|arXiv (Cornell University)|Feb 18, 2019

Homotopy and Cohomology in Algebraic Topology18 references4 citations

TL;DR

This paper establishes a Hamiltonian ∐ₙBO(n)-action on the colimit of cotangent bundles ℝᴺ, using sheaves of spectra to quantize the action and enrich stratified Morse theory with the J-homomorphism. It proves that the classifying map of the brane structure on an exact Lagrangian submanifold factors through the stable Gauss map and the delooping of the J-homomorphism, resolving a claim in [JiTr] via higher categorical structures in (∞,2)-categories of correspondences.

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 $L ightarrow U/O$ and the delooping of the $J$-homomorphism $U/O ightarrow B\mathrm{Pic}(\mathbf{S})$. 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.

Motivation & Objective

• To prove a claim in [JiTr] that the classifying map of the sheaf of brane structures on an exact Lagrangian submanifold factors through the stable Gauss map and the delooping of the J-homomorphism.
• To enrich stratified Morse theory with the J-homomorphism by constructing a quantized Hamiltonian ∐ₙBO(n)-action on limₙT*ℝⁿ.
• To develop concrete constructions of (commutative) algebra and module objects in the symmetric monoidal (∞,2)-category of correspondences, generalizing Segal object constructions.
• To establish a functorial and monoidal framework for sheaves of spectra on correspondences, enabling quantization of geometric actions with topological invariants.

Proposed method

• Uses sheaves of spectra to quantize a Hamiltonian ∐ₙBO(n)-action on limₙT*ℝⁿ, arising from Bott periodicity.
• Employs the category of correspondences from [GaRo] to model higher categorical structures and construct symmetric monoidal functors.
• Constructs module objects over algebra objects in Corr(S_LCH) fibred over local systems, using stabilization of sheaf categories.
• Applies Morse transformations and localization techniques to relate sheaf categories across different strata and stabilize to the J-homomorphism.
• Utilizes right-lax symmetric monoidal structures on the functor ShvSp to preserve algebraic and module object data across correspondences.
• Establishes compatibility of constructions across choices of auxiliary data (e.g., A_S) via limit processes and natural isomorphisms.

Experimental results

Research questions

• RQ1 How can a Hamiltonian ∐ₙBO(n)-action on limₙT*ℝⁿ be quantized using sheaves of spectra?
• RQ2 What is the precise role of the J-homomorphism in the classification of brane structures on exact Lagrangian submanifolds?
• RQ3 How do the category of correspondences and its symmetric monoidal structure facilitate the construction of algebra and module objects in (∞,2)-categories?
• RQ4 Can stratified Morse theory be enhanced with the J-homomorphism via quantization of geometric actions?
• RQ5 What is the factorization of the classifying map of the brane structure sheaf through U/O and BJ, and how is it proven?

Key findings

• The classifying map of the sheaf of brane structures on an exact Lagrangian L ⊂ T*ℝᴺ factors as L → U/O → BPic(𝕊), where the second map is the delooping of the J-homomorphism.
• The quantization of the Hamiltonian ∐ₙBO(n)-action yields a module object over the algebra object VGₙ• in the (∞,2)-category of correspondences.
• Stabilization of local system categories for Gₙ, VGₙ, and  Q̂ₙ,ₘ produces a well-defined limit that encodes the J-homomorphism.
• The functor ShvSp_all,all^prop on correspondences admits a right-lax symmetric monoidal structure, enabling coherent assignment of spectra to geometric data.
• The construction of algebra and module objects in Corr(S_LCH) generalizes Segal object constructions and provides new examples in higher category theory.
• The compatibility of Morse transformations and stabilization ensures that the J-homomorphism arises naturally from the geometric quantization process.