We study the unwrapped Fukaya category of Lagrangian branes ending on a Legendrian knot. Our knots live at contact infinity in the cotangent bundle of a surface, the Fukaya category of which is equivalent to the category of constructible sheaves on the surface itself. Consequently, our category can be described as constructible sheaves with singular support controlled by the front projection of the knot. We use a theorem of Guillermou-Kashiwara-Schapira to show that the resulting category is invariant under Legendrian isotopies, and conjecture it is equivalent to the representation category of the Chekanov-Eliashberg differential graded algebra. We also find two connections to topological knot theory. First, drawing a positive braid closure on the annulus, the moduli space of rank-n objects maps to the space of local systems on a circle. The second page of the spectral sequence associated to the weight filtration on the pushforward of the constant sheaf is the (colored-by-n) triply-graded Khovanov-Rozansky homology. Second, drawing a positive braid closure in the plane, the number of points of our moduli spaces over a finite field with q elements recovers the lowest coefficient in 'a' of the HOMFLY polynomial of the braid closure.Comment: 92 pages, final journal version, Inventiones Mathematicae (2016

We prove a theorem relating torus-equivariant coherent sheaves on toric varieties to polyhedrallyconstructible sheaves on a vector space. At the level of K-theory, the theorem recovers Morelli's description of the K-theory of a smooth projective toric variety [M]. Specifically, let X be a proper toric variety of dimension n and let M R = Lie(T ∨ R ) ∼ = R n be the Lie algebra of the compact dual (real) torus T ∨ R ∼ = U (1) n . Then there is a corresponding conical Lagrangian Λ ⊂ T * M R and an equivalence of triangulated dg categories Perf T (X) ∼ = Shcc(M R ; Λ), where Perf T (X) is the triangulated dg category of perfect complexes of torus-equivariant coherent sheaves on X and Shcc(M R ; Λ) is the triangulated dg category of complex of sheaves on M R with compactly supported, constructible cohomology whose singular support lies in Λ. This equivalence is monoidal-it intertwines the tensor product of coherent sheaves on X with the convolution product of constructible sheaves on M R .

Many interesting spaces -including all positroid strata and wild character varieties -are moduli of constructible sheaves on a surface with microsupport in a Legendrian link. We show that the existence of cluster structures on these spaces may be deduced in a uniform, systematic fashion by constructing and taking the sheaf quantizations of a set of exact Lagrangian fillings in correspondence with isotopy representatives whose front projections have crossings with alternating orientations. It follows in turn that results in cluster algebra may be used to construct and distinguish exact Lagrangian fillings of Legendrian links in the standard contact three space.

Mathematical treatments of mirror symmetry can take different forms. We will focus on the one most suited to our purpose, the homological mirror symmetry between a projective toric variety and its Hori-Vafa Landau-Ginzburg mirror.1 Several possible versions of this Fukaya category arise in the literature -see Section 2.2.3, 6.3 and References [45, 2, 26] -though we mainly work with the one defined in [42]. Road, Evanston, IL 60208

For a Whitney stratification S of a space X (or, more generally, a topological stratification in the sense of Goresky and MacPherson) we introduce the notion of an S-constructible stack of categories on X. The motivating example is the stack of S-constructible perverse sheaves. We introduce a 2-category EP 2 (X, S), called the exit-path 2-category, which is a natural stratified version of the fundamental 2-groupoid. Our main result is that the 2-category of S-constructible stacks on X is equivalent to the 2-category of 2-functors 2Funct(EP 2 (X, S), Cat) from the exit-path 2-category to the 2-category of small categories.