Constructible sheaves and the Fukaya category

By D. Nadler, Eric Zaslow. 2006

266 citations recorded by the source corpus.

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, we develop a Fukaya $A_\infty$-category $Fuk(T^*X)$ whose objects are exact, not necessarily compact Lagrangian branes in the cotangent bundle. We write $Tw Fuk(T^*X)$ for the $A_\infty$-triangulated envelope of $Fuk(T^*X)$ consisting of twisted complexes of Lagrangian branes. Our main result is that $Sh(X)$ quasi-embeds into $Tw Fuk(T^*X)$ as an $A_\infty$-category. Taking cohomology gives an embedding of the corresponding derived categories.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/04714f7067470e62535fd52f4bb9f42fade4ac61

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Citations in the corpus

T-Duality and Homological Mirror Symmetry of Toric Varieties (2008, methodology): “Inverting equations, we can write L as a graph over an interval over length |a + b| in R, which corresponds to a constructible sheaf by [27].”
Quantization of conic Lagrangian submanifolds of cotangent bundles (2012, background): “We also point out that the link between the microlocal theory of sheaves and the symplectic geometry is studied in another way in [10, 9].”
Derived equivalences of gentle algebras via Fukaya categories (2018, background): “In the presence of unmarked boundary components, there is a dual construction to the construction of partially wrapped Fukaya categories, W(Σ,Λ; η), namely, the infinitesimal wrapped Fukaya categories F(Σ,Λ; η), originally introduced for general symplectic manifolds in [22] and studied in the case o”
Branes and Quantization (2008, methodology): “matters. For a related approach to the A-model, see [ 13 ].”
Exact Lagrangian submanifolds in simply-connected cotangent bundles (2007, unknown): “ar X iv :m at h/ 07 01 78 3v 2 [ m at h. SG ] 1”
A Beginner’s Introduction to Fukaya Categories (2013, background): “Theorem 4.6 (Fukaya-Seidel-Smith [21], Nadler-Zaslow [31], Abouzaid [7], Kragh [26]).”
Microlocal theory of sheaves and Tamarkin's non displaceability theorem (2011, background): “(Note however that the use of the microlocal theory of sheaves also appeared in a related context in [7, 9, 8].”
Symmetries and stabilization for sheaves of vanishing cycles (2012, background): “Nadler and Zaslow [39, 40] show that if X is a real analytic manifold (for instance, a complex manifold), then the derived category D c(X) of constructible sheaves on X is equivalent to a certain derived Fukaya category DFuk(T X) of exact Lagrangians in T X .”
Derived Categories (2016, background): “Algebraic analysis, including differential, microdifferential and DQ modules (see [74], [134], [77]) and microlocal sheaf theory (see [75]), with its application to symplectic topology (see [149], [110]).”
Stability conditions, torsion theories and tilting (2009, unknown): “Nadler and Zaslow [9] construct an equivalence”
On partially wrapped Fukaya categories (2016, methodology): “meant to be equivalent to the dg-categories Coh(X) and Perf(X), respectively. The equivalence proceeds by combining the coherent-constructible correspondence [13] and the Nadler{Zaslow correspondence [32]. This approach is explored by Fang{Liu{Treumann{ Zaslow in [14]. In good cases, each of these m”