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Lagrangian cobordism functor in microlocal sheaf theory II
Symplectic Geometry · Submitted 9 Aug 2023
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim For an exact Lagrangian cobordism $L$ between Legendrians in $J^1(M)$ from $\Lambda_-$ to $\Lambda_+$ whose Legendrian lift is $\widetilde{L}$, we prove that sheaves in $Sh_{\widetilde{L}}(M \times \mathbb{R} \times \mathbb{R}_{>0})$ are equivalent to sheaves at the negative end $Sh_{\Lambda_-}(M \times \mathbb{R})$ together with the data of local systems $Loc({L})$ by studying sheaf quantizations for general noncompact Lagrangians.
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The abstract, sentence by sentence
For an exact Lagrangian cobordism $L$ between Legendrians in $J^1(M)$ from $\Lambda_-$ to $\Lambda_+$ whose Legendrian lift is $\widetilde{L}$, we prove that sheaves in $Sh_{\widetilde{L}}(M \times \mathbb{R} \times \mathbb{R}_{>0})$ are equivalent to sheaves at the negative end $Sh_{\Lambda_-}(M \times \mathbb{R})$ together with the data of local systems $Loc({L})$ by studying sheaf quantizations for general noncompact Lagrangians. Thus we interpret the Lagrangian cobordism functor between $Sh_{\Lambda_\pm}(M \times \mathbb{R})$ as a correspondence parametrized by $Loc({L})$. This enables one to consider generalizations to immersed Lagrangian cobordisms. We also prove that the Lagrangian cobordism functor is action decreasing and recover results on the lengths of embedded Lagrangian cobordisms. Finally, using the construction of Courte-Ekholm, we obtain a family of Legendrians with sheaf categories Morita equivalent to chains of based loop spaces of the Lagrangian fillings.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | For an exact Lagrangian cobordism $L$ between Legendrians in $J^1(M)$ from $\Lambda_-$ to $\Lambda_+$ whose Legendrian lift is $\widetilde{L}$, we prove that sheaves in $Sh_{\widetilde{L}}(M \times \mathbb{R} \times \mathbb{R}_{>0})$ are equivalent to sheaves at the negative end $Sh_{\Lambda_-}(M \times \mathbb{R})$ together with the data of local systems $Loc({L})$ by studying sheaf quantizations for general noncompact Lagrangians. | result 0.91 | 0.95 | 0.84 | asserted 1.00 |
| s1 | Thus we interpret the Lagrangian cobordism functor between $Sh_{\Lambda_\pm}(M \times \mathbb{R})$ as a correspondence parametrized by $Loc({L})$. | interpretation 1.00 | 0.86 | 0.27 | asserted 0.96 |
| s2 | This enables one to consider generalizations to immersed Lagrangian cobordisms. | interpretation 0.67 | 0.31 | 0.04 | hedged possibility 0.80 |
| s3 | We also prove that the Lagrangian cobordism functor is action decreasing and recover results on the lengths of embedded Lagrangian cobordisms. | result 0.98 | 0.95 | 0.08 | asserted 1.00 |
| s4 | Finally, using the construction of Courte-Ekholm, we obtain a family of Legendrians with sheaf categories Morita equivalent to chains of based loop spaces of the Lagrangian fillings. | result 0.92 | 0.89 | 0.08 | asserted 1.00 |
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Original abstract
For an exact Lagrangian cobordism $L$ between Legendrians in $J^1(M)$ from $\Lambda_-$ to $\Lambda_+$ whose Legendrian lift is $\widetilde{L}$, we prove that sheaves in $Sh_{\widetilde{L}}(M \times \mathbb{R} \times \mathbb{R}_{>0})$ are equivalent to sheaves at the negative end $Sh_{\Lambda_-}(M \times \mathbb{R})$ together with the data of local systems $Loc({L})$ by studying sheaf quantizations for general noncompact Lagrangians. Thus we interpret the Lagrangian cobordism functor between $Sh_{\Lambda_\pm}(M \times \mathbb{R})$ as a correspondence parametrized by $Loc({L})$. This enables one to consider generalizations to immersed Lagrangian cobordisms. We also prove that the Lagrangian cobordism functor is action decreasing and recover results on the lengths of embedded Lagrangian cobordisms. Finally, using the construction of Courte-Ekholm, we obtain a family of Legendrians with sheaf categories Morita equivalent to chains of based loop spaces of the Lagrangian fillings.
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How the literature uses it
2 arXiv papers cite this in 5 sentences
Papers citing it · 2 sentences without a confident label
Builds on it 1 paper
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For a conical Lagrangian $Λ⊂T^*_τ> 0(M ×_t)$, we can define [equation] When $Λ/ _>0$ is compact, Guillermou [ref] defined a doubling functor from the Kashiwara–Schapira stack $_Λ(Λ)$ to a simple sheaf $F_dbl$, whose microsupport [equation] In the previous work of the third author [ref] , we generalized the theorem to the case where $Λ/ _>0$ is non-compact under a suitable assumption, that there exists a tubular neighborhood of the Lagrangian submanifold of positive radius with respect to some Riemannian metric adapted to the symplectic form.
Lagrangian cobordism and shadow distance in Tamarkin category · Guillermou doubling functor · 0.97
Uses its method or tool 1 paper
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In the similar construction for microlocal sheaves [ref] , following [ref] and [ref] , we separate the Legendrian fronts of $T_-ϵ(L)$ and $T_ϵ(L)$ by applying a contact isotopy that sends $T_-ϵ(L) ∪T_ϵ(L)$ by $T_-h(L) ∪T_h(L)$ for $r ≫0$ using a cut-off of the Reeb flow.
Existence of generating families on Lagrangian cobordisms · Homotopy lifting of generating families · 0.88
What it cites · 40 sentences without a confident label
Compares with 2 papers
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Therefore, we will only give computations of Lagrangian handle attachments (the reader may compare with [ref] ).
Lagrangian cobordism functor in microlocal sheaf theory I · Computation of Lagrangian handle attachments · 0.83
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We remark that, though less general than the setting first paper [ref] , the setting in this paper is more intuitive and straightforward from the geometric perspective, and is the one considered in recent works on Legendrian weaves [ref] .
Algebraic Weaves and Braid Varieties · Context of Lagrangian cobordism · 0.92
Cites as support 1 paper
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For instance, we now know a number of Legendrian links with different Lagrangian fillings such that the images of local systems on the different fillings have overlaps (for example, when the moduli space of rank 1 local systems on the fillings induced different cluster charts on the moduli space of microlocal rank 1 sheaves [ref] ).
Augmentations, Fillings, and Clusters · Construction of Legendrians with simple sheaf categories · 0.97
Cites as background 8 papers
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Following the convention of [ref] , sheaf quantization is a functor from Lagrangian submanifolds $L ⊂T^*M$ with local systems to sheaves with singular support on the Legendrian lift, i.e. [equation] For conical Lagrangian cobordisms, we do not have a sheaf quantization functor in general.
Quantization of conic Lagrangian submanifolds of cotangent bundles · Sheaf Quantization of Lagrangian Cobordisms · 0.91
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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.
Brane structures in microlocal sheaf theory · Sheaf Quantization of Lagrangian Cobordisms · 0.85
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The doubling construction goes back to Guillermou [ref] (and Jin-Treumann [ref] ), who defined an unconditional doubling functor for a compact exact Lagrangian (or an exact Lagrangian with positive conical ends) [equation] and is also formulated in a different way in Nadler-Shende [ref] .
Sheaf quantization in Weinstein symplectic manifolds · Doubling Functor for Noncompact Legendrians · 0.85
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