Quantization of conic Lagrangian submanifolds of cotangent bundles
@article{Guillermou2012QuantizationOC, title={Quantization of conic Lagrangian submanifolds of cotangent bundles}, author={St'ephane Guillermou}, journal={arXiv: Symplectic Geometry}, year={2012} }
Let $M$ be a manifold and $\Lambda$ a compact exact connected Lagrangian submanifold of $T^*M$. We can associate with $\Lambda$ a conic Lagrangian submanifold $\Lambda'$ of $T^*(M\times R)$. We prove that there exists a canonical sheaf $F$ on $M\times R$ whose microsupport is $\Lambda'$ outside the zero section. We deduce the already known results that the Maslov class of $\Lambda$ is $0$ and that the projection from $\Lambda$ to $M$ induces isomorphisms between the homotopy groups.
34 Citations
Brane structures in microlocal sheaf theory
- Mathematics
- 2017
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…
Generating families and constructible sheaves
- Mathematics
- 2015
Let $\Lambda$ be a Legendrian in the jet space of some manifold $X$. To a generating family presentation of $\Lambda$, we associate a constructible sheaf on $X \times \mathbb{R}$ whose singular…
The singular support of sheaves is $\gamma$-coisotropic
- Mathematics
- 2022
We prove that the singular support of an element in the derived category of sheaves is $\gamma$-coisotropic, a notion defined in [Vit22]. We prove that this implies that it is involutive in the sense…
A Hamiltonian $\coprod\limits_n BO(n)$-action, stratified Morse theory and the $J$-homomorphism
- Mathematics
- 2019
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…
Microlocal sheaf categories and the $J$-homomorphism
- Mathematics
- 2020
Let $X$ be a smooth manifold and $\mathbf{k}$ be a commutative (or at least $\mathbb{E}_2$) ring spectrum. Given a smooth exact Lagrangian $L\hookrightarrow T^*X$, the microlocal sheaf theory…
Kasteleyn operators from mirror symmetry
- MathematicsSelecta Mathematica
- 2019
Given a consistent bipartite graph $\Gamma$ in $T^2$ with a complex-valued edge weighting $\mathcal{E}$ we show the following two constructions are the same. The first is to form the Kasteleyn…
Twisted generating functions and the nearby Lagrangian conjecture.
- Mathematics
- 2020
We prove that, for closed exact embedded Lagrangian submanifolds of cotangent bundles, the homomorphism of homotopy groups induced by the stable Lagrangian Gauss map vanishes. In particular, we prove…
Lagrangian cobordism functor in microlocal sheaf theory
- Mathematics
- 2021
. Given a Lagrangian cobordism L of Legendrian submanifolds from Λ − to Λ + , we construct a functor Φ ∗ L : Sh c Λ + ( M ) → Sh c Λ − ( M ) ⊗ C −∗ (Ω ∗ Λ − ) C −∗ (Ω ∗ L ) between sheaf categories…
Microlocal Category for Weinstein Manifolds via the h-Principle
- MathematicsPublications of the Research Institute for Mathematical Sciences
- 2021
On a Weinstein manifold, we define a constructible co/sheaf of categories on the skeleton. The construction works with arbitrary coefficients, and depends only on the homotopy class of a section of…
Compact Exact Lagrangian Intersections in Cotangent Bundles via Sheaf Quantization
- MathematicsPublications of the Research Institute for Mathematical Sciences
- 2019
We show that the cardinality of the transverse intersection of two compact exact Lagrangian submanifolds in a cotangent bundle is bounded from below by the dimension of the Hom space of sheaf…
References
SHOWING 1-10 OF 11 REFERENCES
Constructible sheaves and the Fukaya category
- Mathematics
- 2006
Let $X$ be a compact real analytic manifold, and let $T^*X$ be its cotangent bundle. Let $Sh(X)$ be the triangulated dg category of bounded, constructible complexes of sheaves on $X$. In this paper,…
Exact Lagrangian submanifolds in simply-connected cotangent bundles
- Mathematics
- 2008
We consider exact Lagrangian submanifolds in cotangent bundles. Under certain additional restrictions (triviality of the fundamental group of the cotangent bundle, and of the Maslov class and second…
Sheaf quantization of Hamiltonian isotopies and applications to non displaceability problems
- Mathematics
- 2010
Let I be an open interval, M be a real manifold, T*M its cotangent bundle and \Phi={\phi_t}, t in I, a homogeneous Hamiltonian isotopy of T*M defined outside the zero-section. Let \Lambda be the…
Nearby Lagrangians with vanishing Maslov class are homotopy equivalent
- Mathematics
- 2010
We prove that the inclusion of every closed exact Lagrangian with vanishing Maslov class in a cotangent bundle is a homotopy equivalence. We start by adapting an idea of Fukaya-Seidel-Smith to prove…
Microlocal branes are constructible sheaves
- Mathematics
- 2006
Let X be a compact real analytic manifold, and let T* X be its cotangent bundle. In a recent paper with Zaslow (J Am Math Soc 22:233–286, 2009), we showed that the dg category Shc(X) of constructible…
On triangulated orbit categories
- Mathematics
- 2005
We show that the category of orbits of the bounded derived category of a hereditary category under a well-behaved autoequivalence is canonically triangulated. This answers a question by A. Buan, R.…
Microlocal theory of sheaves and Tamarkin's non displaceability theorem
- Mathematics
- 2011
This paper is an attempt to better understand Tamarkin's approach of classical non-displaceability theorems of symplectic geometry, based on the microlocal theory of sheaves, a theory whose main…
Microlocal condition for non-displaceablility
- Mathematics
- 2008
We formulate a sufficient condition for non-displaceability (by Hamiltonian symplectomorphisms which are identity outside of a compact) of a pair of subsets in a cotangent bundle. This condition is…
An introduction to symplectic topology
- Mathematics
- 1991
Proposition 1.4. (1) Any symplectic vector space has even dimension (2) Any isotropic subspace is contained in a Lagrangian subspace and Lagrangians have dimension equal to half the dimension of the…
Homogénéisation symplectique et Applications de la théorie des faisceaux à la topologie symplectique
- Philosophy
- 2012
Dans une premiere partie, nous developperons la theorie de l'homogeneisation symplectique ainsi que ses applications a la theorie de Mather et a la rigidite symplectique. Les invariants spectraux…