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Brane structures in microlocal sheaf theory
Symplectic Geometry · Submitted 13 Apr 2017
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Its fiber is the $\infty$-category of spectra, and we construct a Hamiltonian invariant, fully faithful functor from $\Gamma(L,\mathrm{Brane}_L)$ to the $\infty$-category of sheaves of spectra on $M$ with singular support in $\Lambda$.
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The abstract, sentence by sentence
Let $L$ be an exact Lagrangian submanifold of a cotangent bundle $T^* M$, asymptotic to a Legendrian submanifold $\Lambda \subset T^{\infty} M$. We study a locally constant sheaf of $\infty$-categories on $L$, called the sheaf of brane structures or $\mathrm{Brane}_L$. Its fiber is the $\infty$-category of spectra, and we construct a Hamiltonian invariant, fully faithful functor from $\Gamma(L,\mathrm{Brane}_L)$ to the $\infty$-category of sheaves of spectra on $M$ with singular support in $\Lambda$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Let $L$ be an exact Lagrangian submanifold of a cotangent bundle $T^* M$, asymptotic to a Legendrian submanifold $\Lambda \subset T^{\infty} M$. | background 1.00 | 0.09 | 0.02 | asserted 0.86 |
| s1 | We study a locally constant sheaf of $\infty$-categories on $L$, called the sheaf of brane structures or $\mathrm{Brane}_L$. | contribution 1.00 | 0.89 | 0.07 | asserted 1.00 |
| s2 | Its fiber is the $\infty$-category of spectra, and we construct a Hamiltonian invariant, fully faithful functor from $\Gamma(L,\mathrm{Brane}_L)$ to the $\infty$-category of sheaves of spectra on $M$ with singular support in $\Lambda$. | result 0.56 | 0.93 | 0.75 | asserted 1.00 |
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The paper
Original abstract
Let $L$ be an exact Lagrangian submanifold of a cotangent bundle $T^* M$, asymptotic to a Legendrian submanifold $\Lambda \subset T^{\infty} M$. We study a locally constant sheaf of $\infty$-categories on $L$, called the sheaf of brane structures or $\mathrm{Brane}_L$. Its fiber is the $\infty$-category of spectra, and we construct a Hamiltonian invariant, fully faithful functor from $\Gamma(L,\mathrm{Brane}_L)$ to the $\infty$-category of sheaves of spectra on $M$ with singular support in $\Lambda$.
Human review
How the literature uses it
20 arXiv papers cite this in 76 sentences
Papers citing it · 50 sentences without a confident label
Builds on it 1 paper
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For a simple object $∈^Y(M)$ with $()= L$ is a smooth Lagrangian submanifold, we will construct the Novikovized microlocalization (a generalization of [ref] ).
Sheaf quantization from exact WKB analysis · Sheaf quantization associated to brane data · 0.89
Uses its method or tool 2 papers
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Here we adapt the proof of Jin-Treumann [ref] .
Estimating Reeb chords using microlocal sheaf theory · Two Sheaf Categories · 0.88
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Then we can cut out small open balls $Ω_α$ in $⊂L$ by neighborhoods of the form $B_α×I_α⊂V×_t$, where $B_α⊂V$ and $I_α⊂_t$ are open balls and intervals respectively (cf. [ref] ).
Microlocal sheaf categories and the $J$-homomorphism · An alternative construction of $μ_L$ · 0.85
Compares with it 2 papers
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The two statements above are slightly different from the results presented in [ref] .
Conjugate Fillings and Legendrian Weaves · Existence of sheaf quantization for Lagrangian fillings · 0.88
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In this section we basically show that for costandard Lagrangian branes, our fully faithful functor [equation] coincides with the functor Jin-Treumann constructed [ref] .
Lagrangian cobordism functor in microlocal sheaf theory I · Comparison with the Filling Functor · 1.00
Cites it as support 1 paper
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By [ref] , or [ref] , the two strata $_1(_1)$ and $_1(_2)$ are non-empty.
Microlocal Theory of Legendrian Links and Cluster Algebras · Erasing $Y$-trees on weaves and vanishing loci of face merodromies. · 0.99
Cites it as background 14 papers
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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).
A Hamiltonian $\coprod\limits_n BO(n)$-action, stratified Morse theory and the $J$-homomorphism · Quantization of the Hamiltonian $∐_n BO(n)$-action and a natural “module" · 0.83
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Let $T_u: J^1(M ×ℝ_>0) →J^1(M ×ℝ_>0)$ be the Reeb flow.Guillermou and Jin-Treumann [ref] considered the sheaf quantization of closed exact Lagrangians or exact Lagrangians with positive conical ends.
Lagrangian cobordism functor in microlocal sheaf theory II · Sheaf Quantization of Lagrangian Cobordisms · 0.85
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Sheaf quantization was invented by Tamarkin [ref] and developed in [ref] , [ref] , [ref] .
Sheaf quantization from exact WKB analysis · Sheaf quantization associated to brane data · 1.00
What it cites · 12 sentences without a confident label
Builds on 1 paper
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Our main result generalizes [ref] on compact Lagrangians and our proof uses some of the same techniques.
Quantization of conic Lagrangian submanifolds of cotangent bundles · Precedents · 0.98
Uses the method or tool of 1 paper
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The most basic of these techniques, probing the symplectic geometry of $T^* M$ by studying sheaves on $M ×$, was pioneered by Tamarkin [ref] and has many other recent applications.
Microlocal condition for non-displaceablility · Precedents · 0.97
Cites as background 3 papers
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(In [ref] one of us has called it the “Kashiwara-Schapira sheaf;” in [ref] it is called the “Kashiwara-Schapira stack.”) The foundational text [ref] of the theory is significantly older than the foundational text of $∞$-category theory [ref] , which we require to make sense of the Proposition above.
Polytopes and Skeleta · Coefficients · 0.93
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(In [ref] one of us has called it the “Kashiwara-Schapira sheaf;” in [ref] it is called the “Kashiwara-Schapira stack.”) The foundational text [ref] of the theory is significantly older than the foundational text of $∞$-category theory [ref] , which we require to make sense of the Proposition above.
Quantization of conic Lagrangian submanifolds of cotangent bundles · Coefficients · 0.93
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[ref] leaves open the problem of identifying this coefficient category.
A stable infinity-category of Lagrangian cobordisms · Lagrangian cobordisms · 0.81
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