RECORD
A Hamiltonian $\coprod\limits_n BO(n)$-action, stratified Morse theory and the $J$-homomorphism
Symplectic Geometry · Submitted 18 Feb 2019
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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 $U/O\rightarrow B\mathrm{Pic}(\mathbf{S})$.
The compiled record
The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
Open the compiled record
The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | contribution 0.83 | 0.94 | 0.13 | asserted 1.00 |
| s1 | We employ the category of correspondences developed in [GaRo] to give an enrichment of stratified Morse theory by the $J$-homomorphism. | method 0.58 | 0.91 | 0.10 | asserted 1.00 |
| s2 | 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 $U/O\rightarrow B\mathrm{Pic}(\mathbf{S})$. | interpretation 0.47 | 0.27 | 0.64 | asserted 1.00 |
| s3 | 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. | contribution 0.92 | 0.94 | 0.06 | asserted 0.82 |
What was examined, and what was not
- arXiv listing abstract v8
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
The paper
Original 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.
Human review
How the literature uses it
3 arXiv papers cite this in 30 sentences
Papers citing it · 20 sentences without a confident label
Builds on it 1 paper
-
This is an upgraded version of the results that we have obtained in [ref] about constructions of (commutative) algebra and their modules in $()$, and it is frequently (sometimes implicitly) used in the arguments throughout Section [ref].
Microlocal sheaf categories and the $J$-homomorphism · Organization · 0.97
Uses its method or tool 1 paper
-
The construction will heavily rely on the main results in [ref] and Appendix [ref].
Microlocal sheaf categories and the $J$-homomorphism · A canonical functor $F_L: _L(U/O)^op→(Δ^1, )$ · 0.83
Cites it as background 2 papers
-
See [ref] for the role of $T^$ (and its associative version $T$) in the definition of a Cartesian fibration $^×, (_*)^op→N(_*)^op$ that exhibiting the Cartesian symmetric monoidal (and plain monoidal) structure of $$.
Microlocal sheaf categories and the $J$-homomorphism · Step 1: the case for $∙=n$ · 0.84
-
The twistings are related to the Maslov obstruction and similar homotopical considerations, and are studied in [ref] .
Sheaf quantization in Weinstein symplectic manifolds · Microlocal sheaves · 0.81
What it cites · 15 sentences without a confident label
Builds on 1 paper
-
Our construction of a 2-coCartesian fibration for $()$ is a slight modification of it in the $(∞,2)$-setting.] $()^⊗, Δ^op→N(Δ)^op$ (resp. $()^⊗, _*→N(_*)$) for the (symmetric) monoidal structure on $()$ inherited from a (symmetric) monoidal structure $^⊗$ as follows (See [ref] for a similar construction in the $(∞,1)$-categorical setting).
Enhanced six operations and base change theorem for higher Artin stacks · (Symmetric) monoidal structure on $()$ · 0.89
Uses the method or tool of 1 paper
-
The standard way to do this is to find a sheaf over prescribed coefficient with singular support in a conic Lagrangian lifting of the graph of the Hamiltonian action, called sheaf quantization (cf. [ref] , [ref] ).
Microlocal Category · Quantization of the Hamiltonian $∐_n BO(n)$-action and a natural “module" · 0.85
Cites as background 1 paper
-
Here we take the coefficient to be the universal one, the sphere spectrum $$, so sheaves are taking values in the stable $∞$-category of spectra $$ (see [ref] for basic properties of microlocal sheaves of spectra).
Brane structures in microlocal sheaf theory · Quantization of the Hamiltonian $∐_n BO(n)$-action and a natural “module" · 0.83
Discussion
Know something about this paper? Leave a comment, or highlight any passage above to comment on it.