The Fukaya category is a mathematical structure arising in symplectic geometry and homological mirror symmetry. It is an $A_\infty$-category whose objects are suitably decorated Lagrangian submanifolds of a symplectic manifold, and whose morphism spaces are Floer cochain complexes. The composition maps are defined using counts of pseudo‑holomorphic polygons, giving the category an $A_\infty$ structure rather than a strict associative one.
Definition and construction
Let $(M,\omega)$ be a closed (or suitably convex) symplectic manifold of dimension $2n$. An object of the (derived) Fukaya category $\mathcal{F}(M)$ consists of data $(L,\mathbf{b})$ where:
- $L\subset M$ is a compact, oriented, relatively spin Lagrangian submanifold (i.e., $\omega|_L=0$);
- $\mathbf{b}$ is a bounding cochain (a solution of the Maurer–Cartan equation) in the Floer complex of $L$, used to achieve unobstructedness when disc bubbling occurs.
For a pair of objects $(L_0,\mathbf{b}_0)$ and $(L_1,\mathbf{b}1)$, the morphism space is the Floer cochain complex $$ \operatorname{hom}{\mathcal{F}(M)}\big((L_0,\mathbf{b}_0),(L_1,\mathbf{b}_1)\big) = CF^*(L_0,L_1; \Lambda), $$ where $\Lambda$ denotes a suitable Novikov field that records symplectic area contributions.
The higher composition maps $$ \mu^k: \hom(L_{k-1},L_k) \otimes \cdots \otimes \hom(L_0,L_1) \to \hom(L_0,L_k)[2-k] $$ are defined by counting rigid pseudo‑holomorphic maps $$ u: (D^2, \partial D^2) \to (M, \cup_i L_i) $$ with $k+1$ marked boundary points mapping to the given Lagrangians. The counts are weighted by the exponential of the symplectic area, ensuring convergence in the Novikov ring. These maps satisfy the $A_\infty$ relations, which encode associativity up to higher homotopies.
Variants
Several versions exist, depending on additional structures:
- Exact Fukaya category: for exact symplectic manifolds (e.g., cotangent bundles), where the Novikov ring can be omitted.
- Wrapped Fukaya category: incorporates non‑compact Lagrangians with suitable behavior at infinity, relevant for Liouville manifolds.
- Partially wrapped and relative Fukaya categories: further modifications adapted to specific geometric contexts.
Properties
- $A_\infty$-structure: The composition maps satisfy the Stasheff identities, making $\mathcal{F}(M)$ an $A_\infty$-category rather than a strict dg-category.
- Homological invariance: Up to quasi‑equivalence, the Fukaya category depends only on the symplectic isotopy class of $(M,\omega)$ and not on auxiliary choices (almost complex structure, perturbations) after appropriate homotopy‑theoretic corrections.
- Duality: For compact $M$, the Fukaya category is conjecturally (and in many cases proved to be) Calabi‑Yau, i.e., it admits a non‑degenerate cyclic pairing compatible with the $A_\infty$ structure.
- Generation: In many settings, a finite collection of Lagrangians generates $\mathcal{F}(M)$ in the sense that any object can be obtained from them via twisted complexes.
Applications
- Homological mirror symmetry (HMS): Kontsevich’s conjecture asserts an equivalence of triangulated categories $$ D^\pi\mathcal{F}(M) ;\cong; D^b\operatorname{Coh}(X^\vee), $$ where $X^\vee$ is a mirror complex manifold and $D^\pi$ denotes the split‑closed derived category. This equivalence has been established for several classes of manifolds (e.g., elliptic curves, quartic K3 surfaces, certain toric varieties).
- Symplectic topology: The Fukaya category encodes deep symplectic invariants; for instance, non‑vanishing of certain objects yields obstructions to Lagrangian embeddings and to symplectic isotopies.
- Low‑dimensional topology: Wrapped Fukaya categories of cotangent bundles relate to categories of constructible sheaves and to the representation theory of braid groups.
Historical notes
The notion originated in the early 1990s through works of Kenji Fukaya, who introduced $A_\infty$-structures on Floer complexes of Lagrangians. The systematic development of the Fukaya category, including rigorous foundations of transversality and obstruction theory, was carried out in a series of papers by Fukaya–Oh–Ohta–Ono (often abbreviated as FOOO). Subsequent contributions by Seidel, Abouzaid, Nadler, and many others have expanded the theory to various geometric contexts.
References (selected)
- K. Fukaya, Y.-G. Oh, H. Ohta, K. Ono, Lagrangian Intersection Floer Theory – Anomaly and Obstruction, AMS (2009).
- P. Seidel, Fukaya Categories and Picard–Lefschetz Theory, European Math. Soc. (2008).
- M. Abouzaid, A Topological Model for the Fukaya Categories of Plumbings, J. Differential Geom. (2010).
- D. Auroux, Homological Mirror Symmetry and Symplectic Geometry, Proceedings of the International Congress of Mathematicians (2018).
The Fukaya category remains an active area of research, with ongoing work on its algebraic foundations, categorical equivalences, and connections to other branches of mathematics and theoretical physics.
