Partially Wrapped Fukaya Categories

• Partially wrapped Fukaya categories are A∞-categories defined on Liouville domains with stop data that restricts allowable Floer trajectories.
• The stop removal process constructs a quotient of the fully wrapped category, providing a categorical localization technique in symplectic geometry.
• Applications include resolutions of singularities, mirror symmetry for Landau–Ginzburg models, and microlocal sheaf-theoretic realizations.

A partially wrapped Fukaya category is an $A_\infty$-category associated to a Liouville domain equipped with additional "stop data"—that is, a collection of Liouville hypersurfaces (the stops) in the contact boundary. This structure interpolates between the compact Fukaya category (with no wrapping allowed at infinity) and the fully wrapped Fukaya category (with all Reeb chords permitted), producing a flexible invariant that encodes both symplectic and algebraic aspects of non-compact symplectic manifolds, Landau–Ginzburg models, mirror symmetry, and categorical localization phenomena.

1. Stops and the Definition of Partial Wrapping

A stop in a Liouville domain $M$ is a closed Liouville hypersurface $\sigma \subset \partial M$ that itself carries an induced Liouville structure. Stops are designed to block the trajectories of the Reeb vector field: wrapped Floer theory is modified so that only those Hamiltonian chords (corresponding to intersections of Lagrangians or Reeb chords) that avoid crossing the stop are counted as morphisms. More precisely, for Lagrangians $L_0, L_1$, the morphism complex $CW_*^{\mathrm{pw}}(L_0, L_1)$ in the partially wrapped category is the subcomplex of the usual wrapped Floer complex generated by intersection points and chords $\gamma$ with intersection number $n_\sigma(\gamma) = 0$ with the stop.

The $A_\infty$-structure maps $\mu^d$ are defined by moduli spaces of holomorphic disks with boundary punctures and consistent Floer data, as in the wrapped or compact Fukaya category, but with the added restriction that only configurations compatible with the stop filtration (involving zero intersection with the stop) contribute. The algebraic structure is thus "filtered" by intersection number with the stop, and moduli spaces are equipped with conformal rescaling functions, perturbation data, and sign-twisting parameters to ensure well-definedness of the differential and higher products (Sylvan, 2016).

2. Algebraic Framework and Stop Removal

The key technical device in the theory is the "stop removal" procedure. If $W$ denotes the wrapped Fukaya category and $W(\sigma)$ the partially wrapped category with stop $\sigma$, then under a strong nondegeneracy assumption (nonvanishing symplectic cohomology and minimal-action Hochschild cycle for the stop), there is a homotopy retraction of chain complexes: $$R_y = \operatorname{id} + \mu^1 \Delta_y + \Delta_y \mu^1,$$ where $\Delta_y$ is a chain-homotopy operator built from holomorphic disks interacting with the stop and $y$ is a Hochschild cycle representing the stop. This retraction identity implies that the quotient of $W$ by the subcategory of objects supported near the stop $\mathcal{B}_\sigma$ is quasi-isomorphic to the partially wrapped category $W(\sigma)$, the morphisms and $A_\infty$-operations restricting to the subcomplex annihilated by $n_\sigma$ (Sylvan, 2016).

This construction realizes "removing a divisor" on the symplectic side and is the mirror of categorical localization via removing a hypersurface on the B-model. The stop removal results in a fully faithful functor $W/\mathcal{B}_\sigma \to W(\sigma)$ and, at the cohomological level, isomorphisms of Hochschild (co)homology and symplectic cohomology respecting ring and module structures.

3. Moduli Spaces, Open–Closed Maps, and Two-Pointed Models

The construction of all $A_\infty$-operations and the open–closed (OC) and closed–open (CO) maps relies on a family of moduli spaces of punctured holomorphic disks:

• Stasheff Associahedra $\mathcal{R}^d$: moduli of stable disks with one negative (output) and $d$ positive (inputs) boundary punctures define the $A_\infty$-structure $\mu^d$ via counts of disk maps with sign-twisting data computed by the incremental sign vector $(1,2,\ldots,d)$.
• Open–Closed and Closed–Open Domains $\mathcal{R}^1_d$: disks with one interior negative (or positive) puncture and $d$ boundary inputs yield the chain-level open–closed map

$$\mathrm{OC}_d : (CW^*(L_{d-1},L_d)\otimes\ldots\otimes CW^*(L_0,L_1)) \to CH^*(M).$$

Summing over $d$ recovers the standard OC map from Hochschild chains of the (partially) wrapped Fukaya category to symplectic cohomology $SH^*(M)$; similarly for CO.

• Two-pointed Models: To relate these open–closed/closed–open maps to bimodule and diagonal resolutions, one introduces the moduli spaces of pairs of disks $\mathcal{R}_{k,\ell}$, glued along a common output and stratified by input coincidences. The highest strata correspond to shuffle permutations, and lower strata to "tricolored" associahedra. This enables an alternate chain model for the OC and CO maps (denoted $\tensor[_{2}]{OC}{}$ and $\tensor[_{2}]{CO}{}$), which are shown to be homotopy equivalent to the standard maps after quasi-isomorphism between two-pointed and ordinary Hochschild complexes (Ganatra, 2013).

The analysis of boundary strata and degeneration of the moduli spaces guarantees that these operations are chain maps and are compatible, up to explicit homotopies, with ring and module structures. Unstable operations (e.g., strips, units) provide homotopies between $\mu^2(–,e_L)$ and the identity, confirming unitality and the correct homological structure.

4. Relation to Mirror Symmetry and Landau–Ginzburg Models

The construction is motivated in part by homological mirror symmetry. In an exact Landau–Ginzburg model $(M, W)$, the "stop" corresponds to a Liouville hypersurface associated to the divisor or fiber at infinity of the superpotential $W$. The partially wrapped Fukaya category that disallows chords crossing this stop is mirror to the derived category of coherent sheaves on the anticanonical divisor.

This relationship is formalized by the stop removal formula, which mirrors the localization sequence associated with removal of a hypersurface in the B-model. Moreover, in the toric Fano setting, the superpotential dictates the stop at infinity, and the resulting partially wrapped Fukaya category realizes the mirror of the Fukaya–Seidel category of $(M,W)$ (Sylvan, 2016).

5. Functoriality, Continuation, and Invariance

Partially wrapped Fukaya categories are robust under choices of Hamiltonians, almost complex structures, and Floer data. Chain-equivalence of categories associated to different Floer data is established via "continuation functors" constructed without cascades—instead, direct counts of glued holomorphic curves parameterized by the interpolating Floer datum provide chain homotopies. The continuation transform is given by

$F = \operatorname{Id} + \mu^1 h + h \mu^1,$

which, on cohomology, becomes the identity (Sylvan, 2016). This directly implies invariance of the $A_\infty$-structure under auxiliary data.

6. Applications and Extensions

Partially wrapped Fukaya categories play a central role as building blocks for the computation and structure theory of Fukaya categories on noncompact and singular symplectic manifolds. Key applications include:

• Resolutions of Singularities: The partially wrapped Fukaya category of a punctured surface with stops provides a categorical (smooth and proper) resolution of the compact Fukaya category, paralleling how the derived category of modules over the Auslander order resolves the category of perfect complexes on nodal stacky curves (Lekili et al., 2017).
• Cutting and Gluing Formulas: Using stops and sectorial decompositions, one can reconstruct global categories from local data via homotopy colimits and cosheaf-theoretic descent. The process is governed by the localization property—enlarging the stop (i.e., removing part of the boundary) corresponds to localizing at the subcategory generated by linking disks or cocores (Ganatra et al., 2018).
• Mirror Symmetry for Toric and Noncomplete Varieties: In the context of toric geometry, partially wrapped Fukaya categories in cotangent bundles mirror equivariant/non-equivariant derived categories of coherent sheaves on toric targets, even in non-complete (quasi-affine) cases—a result unattainable via Landau–Ginzburg models alone (Katzarkov et al., 2017).
• Microlocal Sheaf-Theoretic Realization: The partially wrapped Fukaya category of $T^*M$ stopped at an isotropic subset $\Lambda \subset S^*M$ is equivalent to the compact objects in the derived category of sheaves with microsupport in $\Lambda$ (Ganatra et al., 2018). This opens analysis of these categories via microlocal methods and Morse theory.

7. Structural Properties and Categorification

From an algebraic perspective, the category encodes a localization sequence that formally resembles the Gysin sequence in sheaf theory or the Orlov blowup formula, with exact triangles, generation by cocores and linking disks, and explicit formulas for Künneth theorems and pushforward functors. Under certain conditions (e.g., stops given by stopping hypersurfaces associated to Lefschetz fibrations), the Fukaya–Seidel category emerges as a partially wrapped category generated by thimbles—the cocores of the attaching handles (Ganatra et al., 2018).

This framework supports a categorical approach to mirror symmetry, birational geometry, and microlocal theory, situating partially wrapped Fukaya categories as a fundamental and computable invariant interpolating between compact and fully wrapped (non-compact) settings.