Wrapped Fukaya Categories in Symplectic Geometry

Updated 24 January 2026

Wrapped Fukaya categories are homologically enriched A∞ invariants for non-compact symplectic manifolds, capturing the dynamics of exact Lagrangians under Hamiltonian wrapping.
They incorporate stops and partial wrapping to precisely control behavior at infinity, using Hamiltonian chords and holomorphic disc counts for robust algebraic models.
These categories underpin deep connections to mirror symmetry, sheaf theory, and representation theory, enabling categorical localizations and advanced computations in symplectic topology.

Wrapped Fukaya categories are homologically enriched, $A_\infty$ -structured invariants of non-compact symplectic manifolds—especially Liouville or Weinstein manifolds—encoding the dynamics of exact Lagrangian submanifolds under Hamiltonian wrapping at infinity. They generalize the compact Fukaya category by incorporating “wrapping” in the non-compact domain, and their formal theory supports deep connections to homological mirror symmetry, symplectic duality, and categorical representation theory. They admit a robust formalism for stops and partial wrapping, produce algebraic models for symplectic invariants, relate directly to sheaf-theoretic microlocal categories, and underlie a range of computations in symplectic topology and algebraic geometry.

1. Liouville Geometry and the Foundation of Wrapping

A Liouville domain $M$ is a compact $(2n)$ -manifold with boundary and a one-form $\lambda$ such that the Liouville vector field $Z$ (defined by $\iota_Z \omega = \lambda$ , $\omega = d\lambda$ ) is outward-pointing along $\partial M$ . $M$ is completed to a non-compact Liouville manifold $\widehat{M}$ by attaching the positive symplectization $[1,\infty)\times\partial M$ with $\lambda=r \alpha$ , where $\alpha = \lambda|_{\partial M}$ (Auroux, 2013).

The objects of the wrapped Fukaya category $\mathcal{W}(M)$ are exact, graded (possibly immersed) Lagrangians conical at infinity: each $L \subset \widehat{M}$ is invariant under the Liouville flow for $r\gg 1$ and $\lambda|_L = d f$ for a compactly supported $f$ . The wrapped Floer complex $CW^*(L_0,L_1)$ is generated by time-1 Hamiltonian chords of a Hamiltonian $H$ growing quadratically at infinity (e.g., $H(r,y)=r^2$ for $r\gg1$ ). The $A_\infty$ -structure is furnished by counts of rigid, perturbed $J$ -holomorphic discs with boundary on the Lagrangians, and the wrapped category is defined via a direct limit over cofinal sequences of wrapping Hamiltonians (Auroux, 2013, Ganatra, 2013).

2. Partially Wrapped Categories, Stops, and Stopped Hamiltonians

To control the behavior at infinity more finely, stops (Liouville hypersurfaces or isotropic submanifolds in the contact boundary) are introduced. For $M$ with a collection of disjoint stops ( $\sigma$ ), the partially wrapped Fukaya category $\mathcal{W}(M,\sigma)$ restricts to exact Lagrangians conical at infinity and disjoint from the stop locus. The morphisms $CW^*_\sigma(L_0,L_1)$ are defined by filtering out those chords intersecting the stopped region.

An appropriate Hamiltonian $H$ is chosen so that its flow avoids pushing chords through the stop, ensuring the filtered complex is well-defined. The $A_\infty$ structure respects this filtration, as disk counts intersect the divisor or stop locus positively and preserve the non-intersection condition (Sylvan, 2016). Continuation functors between choices of Floer data yield quasi-equivalences of the resulting categories.

3. Structural Properties, Generation, and Categorical Localization

The algebraic structure of (partially) wrapped Fukaya categories is robust: cohomological unitality, existence of exact triangles reflecting geometric Lagrangian moves, and split-generation by finite collections of cocore or core Lagrangians are standard. In Weinstein settings, the wrapped category is generated by the cocores of index- $n$ critical points of the Liouville vector field (Chantraine et al., 2017). The open–closed map from Hochschild homology to symplectic cohomology is an isomorphism when the category is non-degenerate (i.e., the open–closed map hits the unit) (Ganatra, 2013).

The stop removal formula provides a categorical analog of divisor removal: if $\sigma$ is a strongly non-degenerate stop, then

$\mathcal{W}(M) \cong \operatorname{Coker}(\mathcal{W}(F) \to \mathcal{W}_\sigma(M))$

with the quotient $\mathcal{W}_\sigma(M)/\mathcal{B} \to \mathcal{W}(M)$ fully faithful, where $\mathcal{B}$ is the subcategory supported near the stop (Sylvan, 2016). This operation mirrors localization in the mirror Landau–Ginzburg model (removing a divisor corresponds to "removing a stop").

4. Examples and Mirror Symmetry Applications

A. Landau–Ginzburg Models: An exact Landau–Ginzburg model $W: M\to \mathbb{C}$ produces a stop $\sigma_W$ at a large-value fiber $W^{-1}(p)$ , and $\mathcal{W}_{\sigma_W}(M)$ is conjecturally equivalent to the Fukaya–Seidel category $\operatorname{Fuk}(M,W)$ . Removing the stop (i.e., setting $W=0$ ) recovers the fully wrapped category (Sylvan, 2016).

B. Pairs-of-pants and toric mirrors: In $M=(\mathbb{C}^*)^n$ with $W=\sum z_i$ , the stop recovers the pair-of-pants skeleton; the partially wrapped category mirrors categories of coherent sheaves on the complement of an anticanonical divisor (Sylvan, 2016).

C. Hypertoric and hyperplane arrangements: For the complement of a polarized hyperplane arrangement $(M(\mathbb{V}),\xi)$ , the partially wrapped category is generated by Lagrangians from bounded chambers and is quasi-isomorphic to an equivariant hypertoric convolution algebra, confirming mirror symmetry predictions (Lee et al., 2024, Côté et al., 2024).

D. Surface categories: Partially wrapped Fukaya categories of surfaces with stops are formal and Morita equivalent to perfect derived categories of gentle or skew-gentle algebras (Barmeier et al., 2024, Chang et al., 2022, Barmeier et al., 18 Dec 2025).

E. Microlocal–sheaf correspondence: For cotangent bundles $(T^*M, \Lambda)$ , $\mathcal{W}(T^*M,\Lambda)$ is quasi-equivalent to the category of compact objects in the derived category of sheaves on $M$ with microsupport in $\Lambda$ (Ganatra et al., 2018).

5. Advanced Formulations: Rabinowitz, Categorical Punctures, and Calabi–Yau Structures

The Rabinowitz wrapped Fukaya category $\operatorname{RW}(X)$ , defined via a mapping cone of Floer complexes for positive and negative Hamiltonians, captures failures of Poincaré duality and models the categorical formal punctured neighborhood of infinity (Efimov's $\widehat{\mathcal{W}(X)}_\infty$ ). For Weinstein $X$ , there is a quasi-equivalence $\operatorname{RW}(X)\cong\widehat{\mathcal{W}(X)}_\infty$ , confirming conjectures on categorical compactifications (Ganatra et al., 2022).

The wrapped Fukaya category for a non-degenerate Liouville/Weinstein manifold is homologically smooth and admits a non-compact Calabi–Yau structure: the diagonal bimodule is isomorphic to the inverse dualizing bimodule up to shift induced by degree (Ganatra, 2013). The open–closed and closed–open string maps provide isomorphisms between Hochschild (co)homology and symplectic cohomology.

6. Functoriality, Cobordisms, and Sectorial Descent

Wrapped (=partially wrapped) Fukaya categories are functorial under Lagrangian correspondences and local-to-global under sectorial descent. Künneth functors yield fully faithful injections for product sectors, and explicit sectorial covers produce practical computation tools for categories via homotopy colimit diagrams (Gao, 2017, Ganatra et al., 2018, Karabas et al., 2021). Exact Lagrangian cobordisms induce equivalences or exact triangles in the derived Fukaya category, with higher symmetry captured by the functoriality to module categories and stable $\infty$ -categorical structures (Tanaka, 2016).

Stop removal is always realized as a categorical localization: removing a mostly Legendrian stop amounts to quotienting by the associated linking disks (Ganatra et al., 2018). Recollement and ladder constructions relate cuts and surgeries in the symplectic geometry directly to algebraic recollements and exceptional sequences in algebraic models (Chang et al., 2022).

7. Deformation Theory and Algebraic Models

For surfaces and their orbifold variants, partially wrapped Fukaya categories are classified algebraically—gentle, skew-gentle, and related associative (possibly graded) algebras arise from formal or sectorial dissections. $A_\infty$ -deformations correspond geometrically to partial compactifications (removal of stops or compactifying boundary components as orbifold points), as controlled by Hochschild cohomology: all deformations are "geometric," with obstruction theory resolved by weak duals and unbounded twisted complexes (Barmeier et al., 2024, Barmeier et al., 18 Dec 2025).

Formality results (vanishing of higher $A_\infty$ -products) and Koszul duality align the wrapped category with known algebraic and representation-theoretic categories, such as hypertoric category $\mathcal{O}$ (Côté et al., 2024), and the compact derived category of singularities in mirror symmetry situations (Ganatra et al., 2022).

References:

Partially wrapped Fukaya categories—definition and stop removal: (Sylvan, 2016)
Sectorial descent and generation: (Ganatra et al., 2018, Chantraine et al., 2017)
Rabinowitz and formal punctured neighborhoods: (Ganatra et al., 2022)
Gentle/skew-gentle algebraic models and surface theory: (Barmeier et al., 2024, Chang et al., 2022, Barmeier et al., 18 Dec 2025)
Microlocal Morse (sheaf/Fukaya) theory: (Ganatra et al., 2018)
Mirror symmetry and hypertoric convolution: (Lee et al., 2024, Côté et al., 2024)
Wrapped category as colimit and gluing: (Karabas et al., 2021)

For comprehensive coverage of technical frameworks, deformation theory, sectorial analysis, and deep links to representation theory and mirror symmetry, wrapped and partially wrapped Fukaya categories form the backbone of current categorical approaches to symplectic geometry.