Fukaya Category in Symplectic Topology

• Fukaya Category is an A∞-category of Lagrangian submanifolds defined via Floer cohomology and pseudo-holomorphic curve counts.
• Its higher operations are derived from counting holomorphic discs with marked points, ensuring associativity up to homotopy with applications like the Cardy relation.
• The framework supports generation criteria and mirror symmetry by connecting symplectic invariants with algebraic structures such as twisted complexes.

The Fukaya category is a central object in modern symplectic topology, mirror symmetry, and homological algebra. It is an $A_\infty$-category whose objects are Lagrangian submanifolds—possibly decorated with additional structure—inside a symplectic manifold. The morphisms are defined via Floer cochain complexes, with composition and higher-order compositions ($A_\infty$-operations) governed by counts of pseudo-holomorphic curves. The structure encodes both analytic and homological information about the ambient symplectic geometry. Technical foundations of the Fukaya category are achieved through detailed treatments of moduli spaces of holomorphic curves, energy and action estimates, orientations, and chain-level algebraic constructions, especially in the wrapped and relative settings and in the presence of symplectic or algebraic degenerations.

1. Algebraic and Analytic Foundations

The Fukaya category, both in its compact and wrapped (non-compact) incarnations, is constructed as a $\mathit{cohomologically}$-graded $A_\infty$-category. Objects are suitably nice Lagrangian submanifolds, typically required to be exact or monotone, and possibly equipped with orientation, spin structure, grading, and flat local systems. Morphism spaces $\operatorname{hom}(L_0, L_1)$ are generated by intersection points or Hamiltonian chords, with Floer differentials defined by counts of rigid pseudo-holomorphic strips solving a perturbed Cauchy-Riemann equation.

Higher $A_\infty$-products $\mu^k$ are defined by counting holomorphic discs with $k+1$ marked boundary points (cornered polygons), subject to appropriate boundary conditions on the sequence of Lagrangians. These compositions satisfy quadratic $A_\infty$-relations, encoding associativity up to controlled homotopy. For non-compact settings, as in the wrapped Fukaya category, additional Hamiltonian perturbations—often with quadratic growth at infinity—are carefully used to ensure transversality and compactness of moduli spaces; the roles of action and energy estimates are essential for avoiding non-compactness phenomena (Abouzaid, 2010, Auroux, 2013, Smith, 2014).

The $A_\infty$-structure is strictly unital and minimal (i.e., the identity morphisms are closed under $m_1$, and higher $m_k$ vanish on units for $k>2$) in a well-tuned analytic setup, with orientations and signs assigned via determinant bundles of linearized Cauchy-Riemann operators, tracked systematically through moduli space degenerations and boundary gluing. This underpins all subsequent algebraic constructions and functoriality.

2. Operations, Open–Closed Maps, and the Cardy Relation

Chain-level operations are defined using detailed Floer data for various moduli spaces: discs (governing $A_\infty$-structure), discs with two outputs (bimodule coproducts), and annuli and discs with interior punctures (open–closed and closed–open maps) (Abouzaid, 2010).

• The open–closed map $\mathrm{OC}$ from cyclic Hochschild complexes of the Fukaya category to symplectic cohomology is defined by counting rigid holomorphic disks with one interior puncture asymptotic to a closed orbit, matching the analytic structure near infinity with prescribed Hamiltonians and families of almost complex structures.
• The closed–open map $\mathrm{CO}$ goes in the reverse direction, relating symplectic cohomology to self–Floer cohomology of a given Lagrangian.
• Mapping these operations into concrete algebraic terms requires explicit handling of chain-level sign conventions (primarily Koszul rules), the role of time-shifting and cutoff one-forms, and the topological energy.

Critical is the Cardy relation, which equates two "routes" for maps from Hochschild homology to Floer cohomology. The proof constructs explicit chain-level homotopies between the two compositions, each involving moduli spaces of broken curves with corrected weights. This ensures the compatibility of open and closed string structures at the chain level, a foundational assertion in modern symplectic topology (Abouzaid, 2010).

3. Generation Criteria and Twisted Complexes

A key structural result is the split-generation criterion: if a collection of Lagrangian branes yields a chain representative of the identity in symplectic cohomology, every Lagrangian is split-generated by this collection in the idempotent closure (triangulated envelope) of the Fukaya category (Abouzaid, 2010). Algebraically, this is achieved by constructing a "universal twisted complex" from bar-type models for Yoneda modules, leveraging the Yoneda embedding and standard homological algebra.

In practice, this means that the Fukaya category is determined, up to idempotent completion, by a finite generating set whenever its fundamental class is realized in symplectic (quantum) cohomology. This result is pivotal for applications to mirror symmetry, where generators on the symplectic side correspond to bases of objects on the derived category of coherent sheaves.

4. Energy, Compactness, and Sign Data

The construction depends crucially on action and energy estimates for pseudo-holomorphic maps: quadratically growing Hamiltonians ensure that orbits or chords at infinity have negative action, precluding energy escape and controlling compactness for moduli spaces. The analysis employs sub-closed forms and global non-positivity conditions, excluding unwanted solutions (Abouzaid, 2010).

Orientation lines for chords or orbits are defined via determinant (index) bundles, and gluing arguments provide isomorphisms between the determinant lines at boundaries and nodes, allowing for coherent sign assignments across all operations. Throughout, the ℝ-action trivialization on strips or cylinders underpins the formation of chain-level invariants and module structures.

5. Explicit Models and Applications

Comprehensive chain-level models are constructed for all operations, including higher $A_\infty$-products, bimodule maps, and open–closed string maps (Abouzaid, 2010). These chain-level frameworks are validated by:

• Careful characterization and compactification of moduli spaces, including their Deligne-Mumford boundary stratifications.
• Explicit gluing and smoothing constructions, ensuring compatibility of perturbation data, signs, and orientation lines.
• Action estimates ensuring no nonconstant pseudo-holomorphic curve escapes to infinity or leads to energy-divergence.

Practical consequences include:

• The ability to check generation criteria for subcategories by determining whether the identity in symplectic cohomology maps into their Hochschild image.
• The foundation for categorifying quantum invariants and matching them to operations in homological mirror symmetry.
• The explicit realization of the Cardy relation, validating the algebraic consistency of open and closed string operations in the wrapped Fukaya category.

6. Summary Table: Core Structural Ingredients

Construction	Geometric Data	Algebraic Output
Moduli of Discs	Riemann surfaces, boundary labels	$A_\infty$-products via holomorphic counts
Discs with two outputs	Annuli; disc with double output	Coproduct/bimodule structure
One-int. puncture disc	Mixed boundary/interior puncture	Open–closed/closed–open maps
Energy & Compactness	Hamiltonians, one-forms, actions	Controls moduli compactness, excludes escapes
orientations/signs	Index bundles, ℝ-trivialization	Consistent chain-level operations
Universal twisted complexes	Bar construction, Yoneda modules	Generation, idempotent completion

This explicit, chain-level framework for the wrapped Fukaya category, with guarantee of analytic and algebraic coherence (including signs, orientations and homotopies), underpins current approaches to categorification in symplectic geometry, the proof of mirror symmetry conjectures, and generation criteria for collections of Lagrangian branes (Abouzaid, 2010).