Constructible sheaves and the Fukaya category

Abstract

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.

Boundaries, mirror symmetry, and symplectic duality in 3d N = 4 $$ \mathcal{N}=4 $$ gauge theory

“…As explained in appendix C.2 and summarized in figure 32, this twist defines an A-model to M H in complex structure I H ζ=1 . Kapustin and Witten [30] defined a functor (generalized by Gukov and Witten [53]) When M H is a cotangent bundle, Nadler and Zaslow proved that the functor I D provides an equivalence of categories [129]. This statement is expected to be true more generally, and we will assume here that it holds for the fully resolved Higgs and Coulomb branches of 3d N = 4 gauge theories.…”

“…The twists at (0, 1) ∈ CP 1 × CP 1 and at (1, 0) effectively lead to massive A-models on the original 3d Higgs and Coulomb branches, respectively. By a result of Nadler and Zaslow [129] (originating in work of Kapustin and Witten [30]), the categories of branes in these theories are equivalent to the derived module categories O H and O C when M H and M C are cotangent bundles. We expect this equivalence to hold for more general M H and M C as well.…”

Ribbon graphs and mirror symmetry

“…It was proven in [B, FLTZ] that constructible sheaves on S 1 with this stratification are equivalent to coherent sheaves on P 1 . Since this category of constructible sheaves is equivalent to a Fukaya category on T * (S 1 ), by [NZ,N1], this equivalence is a form of or mirror symmetry.…”

Microlocal Euler classes and Hochschild homology

“…Note that thanks to Nadler and Zaslow [NZ09], the category D b R-c (k M ) is equivalent to the Fukaya category of the symplectic manifold T * M, and this is another argument to treat sheaves from a microlocal point of view. Acknowledgments The second named author warmly thanks Stéphane Guillermou for helpful discussions.…”