Wrapped Fukaya Category

• Wrapped Fukaya category is an A∞-category that models Floer cohomology for non-compact Lagrangians in Liouville/Weinstein manifolds using Hamiltonian wrapping techniques.
• It employs direct limit constructions over Hamiltonians and counts holomorphic strips and polygons, ensuring invariance under different auxiliary choices.
• This framework bridges symplectic topology, mirror symmetry, and representation theory through explicit algebraic models, sectorial descent, and microlocal sheaf theory.

The wrapped Fukaya category is a central object in symplectic topology and homological mirror symmetry, encoding the Floer-theoretic data of non-compact Lagrangian submanifolds in Liouville or Weinstein manifolds, with “wrapping” dynamics that probe the geometry both at infinity and in the interior. It is constructed as a cohomologically unital $A_\infty$-category, and its structure, generation, invariance, and duality properties are the subject of extensive research. Wrapped Fukaya categories also admit various generalizations and localizations, such as the partially wrapped and Rabinowitz versions, and they act as bridges between geometry, topology, and representation theory.

1. Definitions and Fundamental Construction

Let $(X,\omega=d\lambda)$ be a Liouville (or more generally, Weinstein) manifold, with a cylindrical end modeled by $[1,\infty)_r\times Y$, $\lambda=r\,\alpha$ and Liouville vector field $Z=r\partial_r$ (Auroux, 2013, Ganatra et al., 2018, Morimura, 28 Dec 2025). An exact Lagrangian $L\subset X$ is cylindrical at infinity if $L\cap\{r\gg 1\}= [1,\infty)\times\Lambda$ for some Legendrian $\Lambda\subset Y$ and $\lambda|_L$ is exact, with primitive vanishing at infinity.

The wrapped Floer cochain complex between $L_0,L_1$ is generated by intersections between a Hamiltonian "wrapped" pushforward of $L_0$ and $L_1$, together with chords at infinity: $$CW^*(L_0,L_1;H) = \bigoplus_{x \in X_H(L_0,L_1)} \mathbb{K} \cdot x$$ where $X_H(L_0,L_1)$ are time-1 Hamiltonian chords. The differential counts inhomogeneous holomorphic strips. Higher $A_\infty$-operations $\mu^k$ count holomorphic polygons, and $\mu^1$ recovers the Floer differential.

The wrapped Fukaya category $\mathcal{W}(X)$ is the $A_\infty$-category with objects the (branes) admissible exact Lagrangians, morphisms given by $CW^*$-complexes, and higher products by holomorphic polygon counts (Auroux, 2013, Gao, 2017, Morimura, 28 Dec 2025).

Direct limit presentation: The wrapped complex is often constructed as a direct limit over cofinal families of Hamiltonians (with increasing slopes at infinity), using continuation maps to define

$$CW^*(L_0, L_1) = \varinjlim_{H_i} CF^*(L_0, L_1; H_i)$$

(Auroux, 2013, Gao, 2017, Morimura, 28 Dec 2025).

2. Generation, Invariance, and Duality

A crucial feature is the split-generation property: in cotangent bundles, a single cotangent fiber $T^*_qM$ split-generates the wrapped Fukaya category $\mathcal{W}(T^*M)$, inducing a quasi-equivalence with modules over the Pontryagin algebra $C_{-*}(\Omega_qM)$ (Abouzaid, 2010). More generally, for Weinstein manifolds, the cocores of index-$n$ critical handles split-generate $\mathcal{W}(W)$ (Chantraine et al., 2017); in sectorial settings, the cocores and linking disks to stops are generators (Ganatra et al., 2018). These generational results also underpin the proof that the open-closed map from Hochschild homology $HH_*(\mathcal{W}(X))$ to symplectic cohomology $SH^*(X)$ is an isomorphism in nondegenerate cases (Ganatra, 2013, Chantraine et al., 2017, Ritter et al., 2012).

Invariance results: The quasi-equivalence class of $\mathcal{W}(X)$ and the $A_\infty$ structure are independent of auxiliary choices, such as cylindrical adjustments of metrics, Hamiltonians, or almost complex structures, up to isotopy or Lipschitz equivalence (Bae et al., 2019, Morimura, 28 Dec 2025). This is implemented by continuation functors, which become formal inverses in the $\infty$-categorical localization view.

“Wrapped Fukaya category as localization”: The construction via abstract wrapped Floer setups and their localization along continuation morphisms yields an explicit model for $\mathcal{W}(X)$ as the $\infty$-categorical localization of the raw Floer (pre-)category at the set of continuation maps (Morimura, 28 Dec 2025).

3. Sectorial, Plumbing, and Local-to-Global Computations

Sectorial descent: If $X$ is covered by Weinstein sectors (or plumbing pieces), then $\mathcal{W}(X)$ is computed by homotopy colimit over the diagram of local wrapped Fukaya categories and their overlaps (Ganatra et al., 2018, Karabas et al., 2021, Karabas et al., 17 May 2024): $\mathcal{W}(X) \simeq \hocolim \left( \mathcal{W}(X_1) \leftarrow \mathcal{W}(X_1 \cap X_2) \rightarrow \mathcal{W}(X_2) \right)$

Plumbings: For plumbings of cotangent bundles along a quiver $Q$, the wrapped Fukaya category is explicitly equivalent to a dg quiver category, with objects indexed by the vertices and morphisms/relations controlled by the plumbing graph, gradings, and based loop algebra data (Karabas et al., 17 May 2024, Bae et al., 2022). For $n \geq 3$, $W(P)$ is the Ginzburg dg algebra of the graded quiver, and for surfaces, the multiplicative preprojective algebra is obtained.

Partially wrapped and stopped categories: Introduction of stops (hypersurfaces at infinity cutting off wrapping) yields partially wrapped Fukaya categories, where Reeb/Hamiltonian wrapping is restricted to avoid the stop loci. These categories model microlocal sheaves via the Nadler–Zaslow correspondence and admit explicit algebraic presentations for surfaces via gentle algebras and derived recollements (Ganatra et al., 2018, Sylvan, 2016, Chang et al., 2022).

4. Mirror Symmetry, Algebraic Models, and Applications

Homological mirror symmetry: The wrapped Fukaya category $\mathcal{W}(X)$ and its variants have been identified with derived categories of coherent sheaves, singularities, or algebraic dg-categories on mirror spaces. For instance, in toric or SYZ settings, the wrapped Floer cohomology of a Lagrangian section is identified with the ring of functions on the Hori–Vafa mirror, and the derived category $\mathrm{D}^\pi W(M)$ is equivalent to $\mathrm{Perf}(\mathcal{O}(M^\vee))$ (Groman, 2018). The cluster category associated to a quiver $Q$ also appears as a quotient $\mathcal{W}(X_Q)/\mathcal{F}(X_Q)$, matching the Rabinowitz Fukaya category (Bae et al., 2022, Ganatra et al., 2022).

Microlocal sheaf theory: The partially wrapped category $W(T^*M,\Lambda)$ stopped at $\Lambda \subset S^*M$ matches the compact objects in the derived category of sheaves with microsupport in $\Lambda$ (Ganatra et al., 2018). This bridges Floer theory with constructible sheaf theory and allows the transfer of known sheaf-theoretic computations to wrapped Floer settings.

Recollement and algebraic presentations: For surfaces, the partially wrapped Fukaya categories are triangle equivalent to the perfect derived category of graded gentle algebras, providing a geometric-algebraic dictionary. The process of “cutting” surfaces induces recollement diagrams, and the existence of silting objects, simple-minded collections, and exceptional sequences characterizes algebraic generation properties (Chang et al., 2022, Barmeier et al., 18 Dec 2025).

5. Advanced Structures: Rabinowitz, Deformation, and Cobordism Enhancements

Rabinowitz Fukaya category: The Rabinowitz category $RW(X)$ captures the failure of wrapped Floer cohomology to satisfy Poincaré duality and is defined as the cone of the continuation map between $CF^*(-H)$ and $CF^*(H)$. $RW(X)$ is shown to coincide with the categorical formal punctured neighborhood of infinity of $W(X)$ and is identified with quotient categories (e.g., singularity or cluster categories) (Ganatra et al., 2022, Bae et al., 2022).

$A_\infty$-deformation theory: For partially wrapped Fukaya categories of surfaces, all $A_\infty$ deformations are realized geometrically as categories of orbifold surfaces via partial compactification. The solution to the curvature problem utilizes unbounded twisted complexes and the notion of a weak dual (Barmeier et al., 18 Dec 2025).

Cobordism pairings: There is a stable $\infty$-category $Lag(X)$ of noncompact Lagrangians and cobordisms which pairs with the wrapped Fukaya category, extending the duality and representing compact branes up to cobordism equivalence (Tanaka, 2016).

6. Technical Features: $A_\infty$ Structures, Moduli, and Compactness

Core technical features of wrapped Fukaya categories include:

• Moduli spaces of holomorphic polygons (decorated with Floer data—Hamiltonian, complex structure, weights) are central to the $A_\infty$ structure; operations $\mu^k$ count rigid configurations.
• Compactness, transversality, and monotonicity conditions on moduli space ensure well-behaved differentials and higher products, with potentially nontrivial bubbling in monotone or non-exact settings (Ritter et al., 2012, Groman, 2018).
• Homotopy colimits and semifree dg algebra presentations offer practical computational tools for glueing and explicit category computations (Karabas et al., 2021, Karabas et al., 17 May 2024).
• Exact triangles and Surgery exact triangles model categorical operations like Dehn twists and Lagrangian surgeries.

7. Outlook and Connections

Wrapped Fukaya categories serve as a unifying framework relating symplectic topology, categorical representation theory, algebraic geometry, mirror symmetry, and low-dimensional topology. Their explicit and axiomatic formulations, as well as their local-global and deformation-theoretic properties, enable both practical calculation and conceptual transfer to algebraic, microlocal, and dg categorical contexts. Recent advances refine their axiomatic basis (minimal localization models), expand their computational reach (plumbing and sectorial descent), and deepen their role in mirror symmetry and categorical topology (Morimura, 28 Dec 2025, Karabas et al., 17 May 2024, Bae et al., 2022, Barmeier et al., 18 Dec 2025, Ganatra et al., 2018, Ganatra et al., 2022, Chantraine et al., 2017, Ganatra, 2013, Groman, 2018, Chang et al., 2022, Gao, 2017).