Constructible sheaves and the Fukaya category

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_∞ -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_∞ -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