Microlocal branes are constructible sheaves

By D. Nadler. 2006

202 citations recorded by the source corpus.

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 sheaves on X quasi-embeds into the triangulated envelope F(T* X) of the Fukaya category of T* X. We prove here that the quasi-embedding is in fact a quasi-equivalence. When X is a complex manifold, one may interpret this as a topological analogue of the identification of Lagrangian branes in T* X and regular holonomic $${{\mathcal D}_X}$$ -modules developed by Kapustin (A-branes and noncommutative geometry, arXiv:hep-th/0502212) and Kapustin and Witten (Commun Number Theory Phys 1(1):1–236, 2007) from a physical perspective. As a concrete application, we show that compact connected exact Lagrangians in T* X (with some modest homological assumptions) are equivalent in the Fukaya category to the zero section. In particular, this determines their (complex) cohomology ring and homology class in T* X, and provides a homological bound on their number of intersection points. An independent characterization of compact branes in T* X has recently been obtained by Fukaya et al. (Invent Math 172(1):1–27, 2008).

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/39425997a3022a8ab00a551ce9e352f66507acc0

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

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”
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 .”
The Coherent-Constructible Correspondence for Toric Deligne-Mumford Stacks (2009, methodology): “We also have functoriality involving Fukaya categories [38].”
Nearby Lagrangians with vanishing Maslov class are homotopy equivalent (2010, methodology): “As the foundations of the wrapped Fukaya category were not developed at the time, the proof was not complete because the other two models for the Fukaya category of a cotangent bundle were not technically adequate to prove the result: (1) the Lefschetz pencil approach to studying Fukaya categories, ”
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].”
Sheaf quantization of Hamiltonian isotopies and applications to non displaceability problems (2010, background): “al nondisplaceability theorem conjectured by Arnold. (Tamarkin’s ideas have also been exposed and developed in [9].) Note that the use of sheaf theory in symplectic topology already appeared in [15], [19], [20] and [21]. In this paper, we will also ﬁnd a new proof of the non-displaceability theorem ”
Microlocal Morse theory of wrapped Fukaya categories (2018, background): “Because we treat the wrapped category, our result has broader implications than the original [37, 33].”
Knot contact homology (2011, background): “1 is its application to an interesting general question in symplectic topology: to what extent do symplectic- and contact-geometric objects naturally associated to objects in smooth topology remember the underlying smooth topology? More specifically, how much of the smooth topology is encoded by hol”
T-Duality and Homological Mirror Symmetry of Toric Varieties (2008, methodology): “The symplectomorphism of inversion on the fibers intertwines with Verdier duality of constructible sheaves under microlocalization [25].”
Global topology of the Hitchin system (2011, result): “The mathematical content of [KW] maybe phrased that the combination of this latter Nadler-Zaslow type equivalence with the homological mirror symmetry in (5.12) leads to the proposed Geometric Langlands program of [La1, BeD].”
Homological mirror symmetry for the 4-torus (2009, unknown): “Fukaya category. Nadler’s paper [ Na06 ], which concerns a certain Fukaya category of Lagrangian submanifolds of the cotangent bundle, has highlighted the importance of the notion of resolutions of the diagonal in proofs of such a result.”
A categorification of Morelli’s theorem (2010, background): “In [15, 17] a “microlocalization” theorem is proved, showing the dg category of constructible sheaves on a real analytic manifold is quasi-equivalent to the unwrapped Fukaya category of Lagrangian submanifolds of the cotangent bundle.”
Constructible sheaves and the Fukaya category (2006, background): “Further arguments show that this is in fact a quasi-equivalence [28].”