Partially Wrapped Fukaya Category

• Partially Wrapped Fukaya Category is an A∞-category defined on Liouville manifolds with stops that restrict Hamiltonian flows, resulting in filtered Floer complexes.
• It features powerful localization theorems where stop removal yields the wrapped Fukaya category via precise chain-level constructions and moduli space analyses.
• The structure supports dualities and mirror symmetry, offering a framework for computing symplectic invariants and linking symplectic topology with algebraic geometry.

The partially wrapped Fukaya category is an $A_\infty$-category associated to a Liouville manifold (or more generally, to other open symplectic spaces such as Weinstein sectors, orbifolds, or pinched surfaces) equipped with a collection of geometric structures called “stops”. The notion generalizes the wrapped Fukaya category by “halting” certain Hamiltonian wrappings near specified hypersurfaces or subsets at infinity, yielding powerful localization theorems, new structural features, and deep connections to algebraic geometry, representation theory, and topological field theory.

1. Definition and Foundational Construction

The partially wrapped Fukaya category $\mathcal{W}(M, \{\sigma\})$ is defined for a Liouville domain $M$ with a collection of disjoint stops $\{\sigma\}$, where each stop is typically a Liouville hypersurface embedded into the contact boundary $\partial_\infty M$. A stop $\sigma$ restricts the allowed Hamiltonian flows for objects and morphisms: in forming the partially wrapped Floer cochain complex

$CW^*_\sigma(L_0, L_1) \subset CW^*(L_0, L_1)$

one counts only Hamiltonian chords and Floer trajectories that do not “hit” the stop—i.e., which have intersection number zero with $\sigma$ (Sylvan, 2016). In this way, wrapping is “partially” permitted: Lagrangians may be wrapped except across directions shielded by stops. The resulting category is a full $A_\infty$-subcategory of the wrapped Fukaya category, with morphisms generated by filtered Floer complexes according to the stops.

The $A_\infty$-structure is defined through counts of solutions to parametrized perturbed Cauchy-Riemann equations, subject to boundary and asymptotic conditions determined by the stops, along with universal and consistent choices of Floer data.

2. Algebraic Properties and Stop Removal Theorem

For a stop $\sigma$ that is strongly nondegenerate (i.e., where the fiber $F$ of the stop has nonvanishing symplectic cohomology and appropriate action-level conditions on the Hochschild fundamental cycle), the partially wrapped Fukaya category $\mathcal{W}_\sigma(M)$ satisfies a localization property: $$\mathcal{W}(M) \simeq \mathcal{W}_\sigma(M) \big/\mathcal{B}_\sigma$$ where $\mathcal{W}(M)$ is the wrapped Fukaya category (with no stop restriction) and $\mathcal{B}_\sigma$ is the (full) subcategory generated by Lagrangians localized near the stop (Sylvan, 2016, Ganatra et al., 2018). This formalizes the geometric notion that “removing a stop” yields the fully wrapped category as a quotient, mirroring, on the mirror symmetry side, the operation of removing a divisor.

The stop removal formula is proved through elaborate chain-level constructions involving moduli spaces of holomorphic curves (including discs with two negative punctures and arbitrary many positive punctures), shuffle products, and homotopy units. Key mappings, such as retraction homotopies

$$R_y = \mathrm{id} + \mu^1 \Delta_y + \Delta_y \mu^1,$$

witness homotopy equivalences after localization.

3. Local-to-Global Structures, Cosheaf Property, and Sectors

Partially wrapped Fukaya categories satisfy a cosheaf (or sectorial descent) property under Weinstein sectorial coverings: $$\operatorname{hocolim}_{\emptyset \neq I \subset \{1,\dots,n\}} \mathcal{W}(X_I) \xrightarrow{\sim} \mathcal{W}(X)$$ for sectorial covers $X = X_1 \cup \cdots \cup X_n$, with each $X_I$ a Weinstein sector (Ganatra et al., 2018). This realizes $\mathcal{W}(X)$ as a homotopy colimit over the local categories of the sectors—a categorical incarnation of Kontsevich’s skeleton cosheaf proposal.

On orbifold surfaces, cosheaf models have been explicitly constructed: one employs “ribbon complexes” built from admissible dissections, assigning local $A_\infty$-categories to each cell, and the global partially wrapped Fukaya category is recovered as global sections of this cosheaf (Barmeier et al., 23 Jul 2024). Equivalence with orbit categories arising from smooth double covers under group action is established, underpinning local-to-global principles and Morita invariance under elementary moves of the dissection.

4. Generators, Exact Triangles, and Structural Theorems

The presence of stops dramatically alters generation and exact triangle phenomena. For Weinstein manifolds or sectors with a (mostly Legendrian) stop, it is proved that the partially wrapped Fukaya category is generated by the cocores of critical Weinstein handles and by Lagrangian linking disks (“meridians”) associated to the stop (Ganatra et al., 2018). An exact triangle records the algebraic effect of “wrapping through a stop”: $$L^{(w)} \rightarrow L \rightarrow D_p \rightarrow [1]$$ where $L^{(w)}$ is the image of $L$ under a positive wrapping across the Legendrian stop at a point $p$, and $D_p$ is the linking disk at $p$. This exact triangle is fundamental to both computations and the stop removal localization.

On orbifold surfaces, formal generators from “formal” or “DG” dissections yield endomorphism algebras which are graded (skew-)gentle, and the entire category is described as the perfect derived category of such an algebra, with derived invariance under elementary moves (Barmeier et al., 23 Jul 2024). In classical surface cases, formal generators, full exceptional sequences, and recollement sequences are analyzed via cuts along arcs and corresponding gentle algebra reductions (Chang et al., 2022).

5. Dualities, Open–Closed Maps, and Functorialities

Partially wrapped Fukaya categories admit deep structures relating Hochschild (co)homology and symplectic invariants. Under a non-degeneracy condition, natural maps

$$\operatorname{HH}_*(\mathcal{W}) \to SH^*(M) \to \operatorname{HH}^*(\mathcal{W})$$

are isomorphisms compatible with ring and module structures, with the duality encoded in a non-compact Calabi-Yau structure (Ganatra, 2013). The construction of open–closed and closed–open maps utilizes new moduli spaces of pairs of discs, stratified by point identifications and resolved via gluing arguments, yielding higher $A_\infty$-operations and ensuring required algebraic and module/ring compatibilities.

Continuation functors between different partially wrapped Fukaya categories (e.g., under changes of stops or Hamiltonians) are constructed through chain-level moduli space counts without cascades, streamlining technical arguments and ensuring invariance up to quasi-equivalence (Sylvan, 2016). Orlov and Viterbo transfer functors (and their spherical/localization structures) enable the comparison and localization of categories under change of domain or stop (Sylvan, 2019).

6. Mirror Symmetry, Algebraic Models, and Derived Equivalences

The partially wrapped Fukaya category appears as the mirror counterpart to categorical constructions on the commutative algebraic side. For instance, the algebraic “stop removal” formula corresponds to removing a divisor, yielding agreement between the partially wrapped category of a Landau–Ginzburg model and the mirror side’s derived category (Sylvan, 2016).

In the context of punctured surfaces and stacky curves, explicit mirror equivalences are established: the derived category of modules over Auslander orders on a stacky nodal curve matches the partially wrapped Fukaya category of a punctured surface with stops, both providing categorical resolutions of singular or incomplete data (Lekili et al., 2017).

For toric and hyperkähler situations, local and global models for partially wrapped Fukaya categories produce graded gentle and skew–gentle (or, under group actions, skew-group) algebras, with derived equivalence realized through orbit category techniques, ribbon complexes, or deformation quantization (Katzarkov et al., 2017, Barmeier et al., 23 Jul 2024, Amiot et al., 24 May 2024, Côté et al., 3 Jun 2024). The explicit construction and classification of generators and tilting objects facilitate the computation of morphism spaces and Koszul duality structure.

7. Theoretical Implications and Applications

The partially wrapped Fukaya category serves as a bridge between open and closed string invariants, categorical localization in topology and geometry, and structural theorems in representation and mirror symmetry. It provides a framework for constructing and analyzing:

• Generation, cosheaf descent, and recollement of Fukaya categories (Ganatra et al., 2018, Chang et al., 2022).
• Algebraic models for categories of symplectic mapping tori and locally trivial fibrations via twisted tensor products (Kartal, 2019).
• Relative Calabi-Yau structures, formality, and explicit calculation of symplectic cohomology and Hochschild invariants in various geometric settings (Ganatra, 2013, Côté et al., 3 Jun 2024).
• Connections to orbifold and perverse sheaf categories via microlocal Morse theory and mirror symmetry (Ganatra et al., 2018, Côté et al., 3 Jun 2024).

The explicit description of these categories via cosheaf, algebraic, and combinatorial models—together with deep functoriality, localization, and duality properties—has established partially wrapped Fukaya categories as central objects in symplectic topology, higher category theory, and homological mirror symmetry.