Compact Fukaya Category

Updated 29 August 2025

The compact Fukaya category is an A∞-category defined by closed, relatively spin Lagrangian submanifolds and Floer cochain complexes.
It constructs its structure via counts of pseudoholomorphic disks and rigorous moduli space techniques ensuring unitality and homotopy invariance.
It connects with wrapped and Rabinowitz categories, underpinning mirror symmetry, quantum cohomology, and modern microlocal sheaf-theoretic methods.

A compact Fukaya category encodes the symplectic topology of a compact or convex symplectic manifold via an $A_\infty$ -category whose objects are compact, relatively spin Lagrangian submanifolds (possibly equipped with additional data such as gradings and local systems) and whose morphisms are constructed from Floer cochain complexes arising from intersection orbits and pseudoholomorphic curve counts. In the context of Weinstein or Liouville manifolds, as well as algebraic varieties and mirror symmetry, the compact Fukaya category serves as a core symplectic invariant and interacts functorially with wrapped, partially wrapped, and Rabinowitz Fukaya categories. Its formal properties are deeply tied to the underlying moduli theory of $J$ -holomorphic curves and the categorical architecture imposed by 2-categorical and microlocal sheaf-theoretic structures.

1. Foundations and Construction

The compact Fukaya category, denoted $\mathcal{F}(X)$ for a symplectic manifold $X$ , defines its objects as closed, relatively spin, graded Lagrangian submanifolds. Morphisms between objects are the Floer cochain complexes $CF^*(L_0,L_1)$ , assembled using orientation lines and grading data, with higher $A_\infty$ -operations $m_k$ given by counts of rigid, pseudoholomorphic disks (or polygons) with boundary mapped cyclically to a sequence of Lagrangians. Perturbative transversality and compactness are achieved using stabilizing divisors, as in Cieliebak-Mohnke, to control the moduli spaces of holomorphic disks and guarantee strict unitality and well-definedness of the $A_\infty$ -structure (Charest et al., 2015).

Compactness of the relevant moduli spaces is ensured by Gromov compactness, energy bounds dependent on the symplectic area, and diameter constraints. In particular, in Weinstein manifolds, Liouville flows and conic topology guarantee that compact pieces can be glued into the global category, under control of the singular support of branes (Nadler, 2011). The Floer cohomology $HF^*(L_0,L_1)$ appears as the cohomology of $m_1$ (the Floer differential), and the higher operations satisfy the $A_\infty$ -relations via boundary strata identification of the moduli spaces.

Formality and minimal cyclic models are explicitly constructed in simple cases, such as the flat symplectic two-torus, where the entire $A_\infty$ -structure is determined by combinatorial counts of polygons and their derivatives, and higher operations are realized as derivatives of transversal structure constants (Kajiura, 2018).

2. Homotopy Invariance, Kuranishi Structures, and Virtual Fundamental Cycles

The independence of the Fukaya $A_\infty$ -structure from auxiliary choices—almost complex structure, perturbation data, or the stabilizing divisor—is essential. Given two sets of higher products $m = \{m_k\}$ and $m' = \{m'_k\}$ constructed with different data, there is an $A_\infty$ -quasi-isomorphism between the associated categories; hence, their weak Maurer-Cartan moduli spaces are gauge equivalent, and Floer cohomology groups are invariant up to canonical isomorphism (Charest et al., 2015).

The geometric underpinning of the entire construction is rooted in the theory of Kuranishi structures, now reinterpreted as a 2-category $\mathbf{Kur}$ with robust gluing, morphisms, and fiber products. Every Fukaya-Oh-Ohta-Ono (FOOO) Kuranishi space (the standard analytic setup for regularizing holomorphic curve moduli spaces) can be upgraded to a compact Kuranishi space in this sense, and the full 2-category is equivalent to the 2-category of derived orbifolds, $\mathbf{dOrb}$ (Joyce, 2015). The tangent-obstruction exact sequence

$0 \longrightarrow T_x X \longrightarrow T_v V \longrightarrow E|_v \longrightarrow O_x X \longrightarrow 0$

at a point $x \in X$ in a Kuranishi chart $(V,E,\Gamma,s,\psi)$ formalizes the virtual fundamental class construction essential for the algebraic structure on the compact Fukaya category.

3. Sheaf-Theoretic and Morse-Theoretic Structures

For Weinstein manifolds, the compact Fukaya category $\operatorname{Perf}F(M)$ (i.e., its derived category) admits a recollement description that formalizes its assembly from local data. The model is categorically Morse-theoretic: decompose $M$ into its unstable Weinstein cells $\{C_p\}$ , and reconstruct the category via a gluing monad $T$ whose entries $T_{q,p} = \mathfrak{i}_q^! \circ \mathfrak{i}_{p!}$ encode "Morse differential" data. The recollement diagram aligns with the classical theory of constructible sheaves via adjunctions and exact triangles, and the global category arises as the global sections of a sheaf on the conic topology: $\mathcal{F}_M^{\mathrm{pre}}(U) = \operatorname{Perf}F(M) / \operatorname{Null}(M, U)$ (Nadler, 2011).

These formal sheaf-theoretic and Morse-theoretic (monadic) gluings connect the Fukaya category to microlocal sheaf theory, enabling explicit computation, especially when the stalks in the conic topology can be related to quiver representations (e.g., the $A_{n-1}$ -quiver category for the local model of $\mathbb{C}$ with $n$ rays).

4. Algebraic Structures and String Maps

The algebraic structure of the compact Fukaya category is tightly linked with quantum cohomology, symplectic cohomology, and Hochschild (co)homology via open–closed (OC) and closed–open (CO) string maps (Ritter et al., 2012). The OC map

$\mathrm{OC}\colon HH_*(\mathcal{F}(X)) \longrightarrow QH^*(X)$

and the CO map

$\mathrm{CO}\colon QH^*(X) \longrightarrow HH^*(\mathcal{F}(X))$

are defined at the chain level by counts of rigid pseudoholomorphic curves with boundary and interior marked points and satisfy algebraic module and unital algebra compatibility. The compact Fukaya category $F(E)$ is included into the wrapped Fukaya category $W(E)$ via an acceleration functor $\mathcal{A}F$ , preserving OC/CO diagrams and eigenvalue splittings associated to $c_1(TE)$ . Algebraic criteria, such as Abouzaid's split-generation, guarantee that certain collections of Lagrangians (e.g., toric fibers with local systems in compact toric varieties) generate the category (Smith, 2018), provided the CO map is injective on Hochschild cohomology.

5. Examples and Computations

Explicit minimal and cyclic $A_\infty$ -structures have been constructed for the compact Fukaya category of $T^2$ , with structure constants realized as generalized theta functions and higher products determined by polygon counts and their derivatives (Kajiura, 2018). On higher-genus surfaces, topological variants (disregarding the area form and using admissibility conditions) allow for a combinatorial description using immersed curves, bigon and polygon counts, and twisted complexes (Azam et al., 2019). In toric varieties, summands of the Fukaya category are generated by explicit objects (toric fibers with local systems) tied to quantum cohomology eigenspaces (Smith, 2018).

Localization techniques produce "compact" Fukaya categories for singular hypersurfaces by localizing the nearby smooth fiber's Fukaya category at Seidel's natural transformation, linking to the derived Knörrer periodicity theorem and ensuring compatibility with mirror symmetry, particularly for large complex structure limit degenerations (Jeffs, 2020). In Landau–Ginzburg settings, cones over the quantum cap action associated with monodromy orbits yield corrected Fukaya categories tailored for mirror symmetry and variation operators (Cho et al., 2020).

6. Relationships with Wrapped, Partially Wrapped, and Rabinowitz Categories

The compact Fukaya category $\mathcal{F}(X)$ is canonically a full subcategory of the wrapped Fukaya category $\mathcal{W}(X)$ , and inclusion is realized via the acceleration functor. While wrapped theory encodes infinitely many generators arising from Reeb chords and is sensitive to the non-compactness of $X$ , the compact Fukaya category captures the "proper" information associated to closed Lagrangians. Koszul duality appears in settings such as plumbing spaces, where the compact category is thick inside the wrapped category and the quotient gives rise to cluster categories (and, in suitable settings, to the Rabinowitz Fukaya category) (Bae et al., 2022).

The Rabinowitz Fukaya category $RW(X)$ is built as the cone over the continuation map between Floer complexes for opposite Hamiltonians and vanishes when at least one Lagrangian is compact, thus encoding the "failure of Poincaré duality" of the wrapped theory and representing the categorical formal punctured neighborhood of infinity (Ganatra et al., 2022). This category is central to understanding the boundary behavior and the "nonproper part" of the Fukaya category in mirror symmetry.

7. Triangulated Persistence Structures and Metric Rigidity

The derived Fukaya category admits refinement to a triangulated persistence category (TPC) by tracking the natural energy (action) filtration on Floer complexes (Biran et al., 2023). Here, morphisms are persistence modules with shifts corresponding to energy levels, and exact (or nearly exact) triangles are endowed with weights measuring Floer action. This framework produces a fragmentation pseudo-metric on compact exact Lagrangians, quantifying the "cost" (in terms of energy) of connecting two Lagrangians via cone decompositions, and is tightly controlled by Floer-theoretic energy estimates and spectral invariants. The nonflatness of these weights encodes symplectic rigidity, and the resulting metric exhibits stability and non-squeezing properties intrinsic to the symplectic topology of the underlying space.

In summary, the compact Fukaya category is a homotopically robust, $A_\infty$ -structured invariant built from compact Lagrangians in a symplectic manifold, equipped with a rich algebraic and categorical framework reflecting both local and global symplectic topology, and deeply intertwined with mirror symmetry, quantum/closed string invariants, and modern microlocal and categorical tools. Its construction, invariance, and properties are governed by advanced virtual geometry techniques (Kuranishi, derived orbifold structures), sheaf-theoretic localization, and deep connections to representation theory, algebraic geometry, and topological field theory.