Derived Wrapped Fukaya Categories

• Derived wrapped Fukaya categories are pretriangulated, idempotent-completed A∞-categories associated with Liouville/Weinstein manifolds, constructed via wrapping procedures that encode both local and global symplectic topology.
• They employ Hamiltonian Floer theory, quiver models, and microlocal sheaf techniques to establish equivalences with derived module categories, thereby advancing studies in homological mirror symmetry.
• Explicit examples on punctured surfaces, cotangent bundles, and mapping tori illustrate how geometric generators and local-to-global gluing capture intricate symplectic and algebraic structures.

A derived wrapped Fukaya category is a pretriangulated, idempotent-completed $A_\infty$-category associated to an exact symplectic (typically Liouville or Weinstein) manifold, with the "wrapping" procedure encoding both local and global symplectic topology, and admits enhancements and equivalences that are central to homological mirror symmetry, microlocal sheaf theory, and the categorical paper of symplectic/resolutive invariants. This article provides a comprehensive technical overview of derived wrapped Fukaya categories, their construction, algebraic and sheaf-theoretic models, local-to-global gluing, algebraic and geometric generators, and their roles in mirror symmetry and beyond.

1. Construction of (Partially) Wrapped Fukaya Categories and Derived Envelopes

Let $(M,\lambda_M)$ be a Liouville domain or its completion $\hat M$, equipped with a symplectic form $\omega=d\lambda_M$ and convex boundary. The wrapped Fukaya category $\mathcal{W}(M)$ is an $A_\infty$-category whose objects are exact, graded, spin Lagrangian submanifolds conical at infinity, with morphism complexes $CW^*(L_0,L_1)=\varinjlim CF^*(\phi_H^1(L_0),L_1)$ defined via Hamiltonian Floer theory and the direct limit over wrapping Hamiltonians. The higher products $\mu^k$ count holomorphic polygons with boundary on Lagrangians, standard $A_\infty$-relations, and action-filtration finiteness.

The partially wrapped Fukaya category $\mathcal{W}_\sigma(M)$ is defined when a stop $\sigma$ (a Liouville hypersurface or more generally a collection thereof) is included: morphisms are generated by Hamiltonian chords/disks avoiding the stop, and all $A_\infty$-operations are defined so as not to cross the stop. The stop-quotient process yields the fully wrapped category as $$\mathcal{W}(M) \simeq \mathcal{W}_\sigma(M)/\mathcal{B}_\sigma$$, with $\mathcal{B}_\sigma$ generated by stop-localized objects (Sylvan, 2016).

The derived wrapped Fukaya category $D^\pi\mathcal{W}(M)$ is obtained as the idempotent-completed pretriangulated envelope (category of twisted complexes with direct summands) of $\mathcal{W}(M)$, i.e., $$D^\pi \mathcal{W}(M)= H^0(\operatorname{Tw}^+\mathcal{W}(M))$$ (Katzarkov et al., 2017, Sylvan, 2016). The same applies to the partially wrapped case: $D^\pi\mathcal{W}_\sigma(M)$.

2. Algebraic and Quiver Models: Generation, Koszul Duality, and Gentle Algebras

For punctured surfaces (or stacky/nodal curves), there is an explicit equivalence between derived wrapped/partially wrapped Fukaya categories and derived categories of modules over quiver algebras (Lekili et al., 2017, Dedda, 2022). Generators are given by Lagrangian arcs or circles, and the endomorphism algebra is a path algebra of a quiver with relations explicitly determined by the surface decomposition or nodal combinatorics.

For surfaces $(\Sigma,\Lambda)$ with stops and a line field $\eta$, the partially wrapped Fukaya category $W(\Sigma,\Lambda;\eta)$ is split-generated by a set of non-intersecting graded arcs, and all higher $A_\infty$-operations beyond $m_2$ vanish. $W(\Sigma,\Lambda)$ is triangle equivalent to the perfect derived category $\operatorname{Perf}(A)$ of a homologically smooth, proper graded gentle algebra $A$ (Chang et al., 2022, Opper, 13 Oct 2025, Barmeier et al., 23 Jul 2024). For orbifold surfaces or surfaces with stacky points, the category is the perfect derived category of a graded skew-gentle algebra (Barmeier et al., 23 Jul 2024).

In cotangent bundle cases, Abouzaid established that the derived wrapped Fukaya category $D^\pi\mathcal{W}(T^*M)$ is equivalent to the triangulated category of modules over the dg-algebra of chains on the based loop space, $C_{-*}(\Omega M)$ (Abouzaid, 2010). This realizes the split-generation by a single cotangent fiber and relates the $A_\infty$-structure to string topology.

3. Sheaf-Theoretic and Microlocal Models

A key development is the identification of (derived) wrapped Fukaya categories of cotangent bundles with compact objects in the derived category of sheaves with prescribed microlocal support. For a closed isotropic $\Lambda \subset S^*M$,

$$\operatorname{Perf}(\mathcal{W}(T^*M,\Lambda))^{op} \simeq Sh^c_\Lambda(M)$$

where $Sh^c_\Lambda(M)$ denotes compact objects in the unbounded derived category of $k$-sheaves on $M$ with microsupport at infinity in $\Lambda$ (Ganatra et al., 2018). The entire local-to-global formalism extends to Weinstein sectors, with direct generation by conormals or cocores and functoriality under gluing and stop-removal.

Stops correspond, under the microlocal theory, to conditions on microsupport, establishing a parallel between geometric and sheaf-theoretic quotient operations.

4. Local-to-Global Properties and Gluing via Homotopy Colimits

Gluing formulas for wrapped Fukaya categories exploit their presentation as homotopy colimits of semifree dg/A$_\infty$ categories. For Weinstein manifolds with sectorial covers $W=W_1\cup W_2$, one has

$$WFuk(W) \simeq \operatorname{hocolim}(WFuk(W_1) \leftarrow WFuk(F) \rightarrow WFuk(W_2))$$

where $F$ is the common sectorial boundary (Karabas et al., 2021). The explicit semifree dg-category models allow for inductive computations and facilitate the computation of wrapped Fukaya categories for complex manifolds (e.g., cotangent bundles of lens spaces, plumbings, etc.).

In the case of surfaces and orbifold surfaces, explicit cosheaf-of-categories structures and sectorial descent formalize the local-to-global reconstruction and Morita invariance (Barmeier et al., 23 Jul 2024).

5. Mirror Symmetry and HMS-Equivalences

Derived wrapped Fukaya categories are central to non-compact homological mirror symmetry (HMS). For punctured surfaces or stacky/nodal curves, there are explicit HMS equivalences of the form

$$D^b(\mathcal{A}_C) \simeq \mathcal{W}(X,\Lambda),\quad D^b(\mathrm{Coh}\,C) \simeq \mathcal{W}(X),\quad \mathrm{Perf}(C) \simeq \mathcal{F}(X)$$

where $\mathcal{A}_C$ is an Auslander order, $C$ is a stacky nodal curve, and $X$ a punctured surface with stops (Lekili et al., 2017).

For cotangent bundles and skeleta of simplicial type, the derived category of the partially wrapped Fukaya category is equivalent (up to split-closure) to the derived category of coherent sheaves/coherent sheaves on toric (possibly non-complete) mirrors (Katzarkov et al., 2017).

Maurer–Cartan, formality, and deformation-theory results establish robust invariance and derived Picard group structures for these categories, tying autoequivalences to mapping class groups and local system data (Opper, 13 Oct 2025).

Rabinowitz wrapped Fukaya categories, as constructed in (Ganatra et al., 2022), provide categorical invariants for formal punctured neighborhoods of infinity, and their relationship (via mapping cone construction) with the usual derived wrapped Fukaya category encodes the failure of Poincaré duality in the open setting, and is reflected in the categorical structure underlying mirror symmetry with respect to compactifications, singular mirrors, or Landau–Ginzburg models.

6. Applications, Examples, and Further Developments

Representative cases include:

• Surfaces and Quivers: For the once-punctured disk with $m$ boundary stops, $D^b(\Bbbk A_{m-1}) \simeq \mathcal{W}(0;m)$ (Lekili et al., 2017). For annuli, tori, and in higher genus, the equivalences involve gentle/skew-gentle path algebras, with explicit generation by Lagrangian arcs and computation of endomorphism algebras (Chang et al., 2022, Barmeier et al., 23 Jul 2024).
• Mapping tori and dynamical invariants: The wrapped Fukaya category of a symplectic mapping torus is derived equivalent to a twisted tensor product of the original category with a nodal curve, distinguishing symplectic mapping tori that are diffeomorphic and share contact and symplectic cohomology invariants (Kartal, 2019).
• SYZ fibrations and open Calabi–Yau: The wrapped Fukaya categories of tame semi-toric SYZ fibrations realize analytic or algebraic mirrors as Jacobian rings, with the closed–open map giving explicit isomorphisms (Groman, 2018).
• Recollement and category cuts: Cutting along arcs in surfaces induces recollements of partially wrapped Fukaya categories, corresponding to derived recollements of gentle algebras (Chang et al., 2022).

Recent results establish that (skew-)gentle algebras arising from surface models are closed under derived equivalence, and that derived invariants of surfaces (with line fields and stops) classify the derived wrapped Fukaya categories up to equivalence (Opper, 13 Oct 2025, Barmeier et al., 23 Jul 2024).

7. Structural Features and Current Directions

Derived wrapped Fukaya categories exhibit the following structural properties:

• Generation and split-generation: Explicit geometric generators are given by cocores, conormals, or Lagrangian arcs. The split-generation criteria are linked to open–closed maps and Calabi–Yau structures (Abouzaid, 2010, Ganatra et al., 2018).
• Idempotent and triangulated envelopes: Pass to derived envelopes by adjoining mapping cones and idempotent splittings; all constructions are compatible with higher $A_\infty$-structure.
• Functoriality and autoequivalences: Mapping class group actions, deformation and Hochschild invariants, and exponentials of Hochschild cohomology provide comprehensive descriptions of autoequivalence groups in both characteristic zero and positive characteristic settings (Opper, 13 Oct 2025).
• Gluing and cosheafification: The categories admit descriptions as (co)sheaves on skeleta with gluing given by homotopy colimits; local models (e.g., type $A$, $D$ disks, annuli, orbifold points) control the entire global theory.

Ongoing research continues to generalize these frameworks to higher-dimensional Weinstein sectors, arboreal skeleta, and situations where stops, orbifold points, and stacky structures proliferate, with applications to singularity theory and the categorical formalism for mirror symmetry.

References:

(Abouzaid, 2010, Sylvan, 2016, Lekili et al., 2017, Katzarkov et al., 2017, Ganatra et al., 2018, Kartal, 2019, Groman, 2018, Gao, 2017, Karabas et al., 2021, Ganatra et al., 2022, Chang et al., 2022, Opper, 13 Oct 2025, Dedda, 2022, Barmeier et al., 23 Jul 2024).