We describe local mirror symmetry from a mathematical point of view and make several A-model calculations using the e-print archive: http://xxx.lanl.gov/hep-th/9903053 496 T.-M. CHIANG, A. KLEMM, S.-T. YAU, AND E. ZASLOW mirror principle (localization). Our results agree with B-model computations from solutions of Picard-Fuchs differential equations constructed form the local geometry near a Fano surface within a Calabi-Yau manifold. We interpret the GromovWitten-type numbers from an enumerative point of view. We also describe the geometry of singular surfaces and show how the local invariants of singular surfaces agree with the smooth cases when they occur as complete intersections.

We describe jm isomorphism of categories conjectured by Kontsevich. If M and M are mirror pairs then the conjectural equivalence is between the derived category of coherent sheaves on M and a suitable version of Fukaya's category of Lagrangian submanifolds^on M. We prove this equivalence when M is an elliptic curve and M is its dual curve, exhibiting the dictionary in detail.

Let X X be a compact real analytic manifold, and let T ∗ X T^*X be its cotangent bundle. Let S h ( X ) Sh(X) be the triangulated dg category of bounded, constructible complexes of sheaves on X X . In this paper, we develop a Fukaya A ∞ A_\infty -category F u k ( T ∗ X ) Fuk(T^*X) whose objects are exact, not necessarily compact Lagrangian branes in the cotangent bundle. We write T w F u k ( T ∗ X ) Tw Fuk(T^*X) for the A ∞ A_\infty -triangulated envelope of F u k ( T ∗ X ) Fuk(T^*X) consisting of twisted complexes of Lagrangian branes. Our main result is that S h ( X ) Sh(X) quasi-embeds into T w F u k ( T ∗ X ) Tw Fuk(T^*X) as an A ∞ A_\infty -category. Taking cohomology gives an embedding of the corresponding derived categories.

We exhibit a transformation taking special Lagrangian submanifolds of a Calabi-Yau together with local systems to vector bundles over the mirror manifold with connections obeying deformed HermitianYang-Mills equations. That is, the transformation relates supersymmetric A-and B-cycles. In this paper, we assume that the mirror pair are dual torus fibrations with flat tori and that the A-cycle is a section.We also show that this transformation preserves the (holomorphic) Chern-Simons functional for all connections. Furthermore, on corresponding moduli spaces of super symmetric cycles it identifies the graded tangent spaces and the holomorphic m-forms. In particular, we verify Vafa's mirror conjecture with bundles in this special case. e-print archive: http://xxx.lanl.gov/math.DG/0005118

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