[Paper Review] Microlocal branes are constructible sheaves

David Nadler|arXiv (Cornell University)|Dec 14, 2006

Homotopy and Cohomology in Algebraic Topology17 references14 citations

TL;DR

This paper establishes a quasi-equivalence between the dg category of constructible sheaves on a compact real analytic manifold $X$ and the triangulated envelope of the Fukaya category of its cotangent bundle $T^*X$, proving that microlocal branes—Lagrangian branes in $T^*X$—are categorically equivalent to constructible sheaves. The key result is that the microlocalization functor $\mu_X$ is not just fully faithful but also essentially surjective, categorically identifying the two structures and extending the characteristic cycle isomorphism to a derived level.

ABSTRACT

Let $X$ be a real analytic manifold, and let $T^*X$ be its cotangent bundle. In a recent paper with E. Zaslow \cite{NZ}, we showed that the dg category $Sh_c(X)$ 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 complex, one may interpret this as a topological analogue of the identification of Lagrangian branes in $T^*X$ and holonomic $D_X$-modules developed by Kapustin and Kapustin-Witten 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-Seidel-Smith.

Motivation & Objective

• To establish a quasi-equivalence between the dg category of constructible sheaves on a compact real analytic manifold $X$ and the triangulated envelope of the Fukaya category of $T^*X$.
• To extend the microlocalization functor $\mu_X$ from a fully faithful embedding to a quasi-equivalence, thereby categorically identifying constructible sheaves with microlocal branes.
• To provide a homological characterization of compact exact Lagrangians in $T^*X$, showing they are equivalent to the zero section in the Fukaya category under mild homological assumptions.
• To demonstrate that $A_\infty$-calculations in the Fukaya category are independent of the choice of compatible almost complex structure and taming perturbations, ensuring invariance of the construction.

Proposed method

• Constructs a microlocalization $A_\infty$-functor $\mu_X: Sh_c(X) \to F(T^*X)$, which embeds the dg category of constructible sheaves into the triangulated envelope of the Fukaya category of $T^*X$.
• Proves that the induced cohomology functor $H(\mu_X): D_c(X) \to DF(T^*X)$ is not only fully faithful but also essentially surjective, establishing a quasi-equivalence.
• Uses non-characteristic isotopies and continuation maps to relate Floer-theoretic calculations among branes to sheaf-theoretic constructions, particularly via the diagonal brane and product branes.
• Applies a priori diameter bounds on $J$-holomorphic disks to show invariance of $A_\infty$-operations under changes of compatible almost complex structures $J$, via homotopy arguments.
• Demonstrates that taming perturbations in the brane structure do not affect $A_\infty$-operations by constructing a $[0,1]$-family of almost complex structures that interpolate between perturbations.
• Relies on compactness of moduli spaces of pseudoholomorphic polygons and the tameness condition to ensure finitely determined, invariant calculations.

Experimental results

Research questions

• RQ1 Is the microlocalization functor $\mu_X$ a quasi-equivalence between $Sh_c(X)$ and $F(T^*X)$, or merely a fully faithful embedding?
• RQ2 Can the characteristic cycle isomorphism between constructible functions and conical Lagrangian cycles be lifted to a derived, categorical equivalence?
• RQ3 Are compact connected exact Lagrangians in $T^*X$ equivalent to the zero section in the Fukaya category under standard homological assumptions?
• RQ4 Does the $A_\infty$-structure of the Fukaya category of $T^*X$ remain invariant under changes of compatible almost complex structures and taming perturbations?
• RQ5 Can the triangulated envelope of the Fukaya category be fully described by constructible sheaves via microlocalization?

Key findings

• The microlocalization functor $\mu_X$ induces a quasi-equivalence between the dg category of constructible sheaves $Sh_c(X)$ and the triangulated envelope $F(T^*X)$ of the Fukaya category of $T^*X$, proving that microlocal branes are categorically equivalent to constructible sheaves.
• The cohomology-level functor $H(\mu_X): D_c(X) \to DF(T^*X)$ is an equivalence of triangulated categories, not just fully faithful.
• Compact connected exact Lagrangians in $T^*X$ are isomorphic to the zero section in $DF(T^*X)$, implying they have the same cohomology ring and homology class.
• The $A_\infty$-operations in the Fukaya category are invariant under changes of compatible almost complex structures and taming perturbations, ensuring the construction is well-defined and robust.
• The characteristic cycle homomorphism $CC$ fits into a commutative diagram with the Grothendieck group $K_0$ and the microlocalization functor, confirming consistency with classical results.
• The triangulated envelope $F(T^*X)$ is generated by the image of $\mu_X$, so every object in $DF(T^*X)$ arises from a constructible sheaf via microlocalization.