The Fukaya Category

The Fukaya category organizes Lagrangian submanifolds and their Floer-theoretic interactions into an $A_\infty$ -category. It is a central object in homological mirror symmetry and modern symplectic topology.

Definition

Definition6.6Fukaya Category

The Fukaya category $\mathrm{Fuk}(M, \omega)$ is an $A_\infty$ -category whose objects are (graded, oriented) Lagrangian submanifolds equipped with flat line bundles, morphism spaces are Floer cochain complexes $CF^*(L_0, L_1)$ , and the $A_\infty$ -structure maps $\mu^k: CF^*(L_0, L_1) \otimes \cdots \otimes CF^*(L_{k-1}, L_k) \to CF^*(L_0, L_k)$ count pseudo-holomorphic polygons with Lagrangian boundary conditions.

Definition6.7$A_\infty$-Structure

The maps $\mu^k$ satisfy the $A_\infty$ -relations: $\sum_{i,j} (-1)^{\star} \mu^{k-j+1}(\ldots, \mu^j(\ldots), \ldots) = 0$ . The map $\mu^1$ is the Floer differential, $\mu^2$ is the (homotopy-associative) product, and higher $\mu^k$ measure the failure of strict associativity.

ExampleFukaya Category of $T^2$

The Fukaya category of $T^2$ with its standard symplectic form has objects: simple closed curves (Lagrangian circles) equipped with flat $U(1)$ -connections. The morphism spaces are generated by intersection points, and the $A_\infty$ operations involve counting holomorphic polygons. Polishchuk and Zaslow showed this category is derived equivalent to coherent sheaves on the dual elliptic curve, verifying homological mirror symmetry for elliptic curves.

Mirror Symmetry

RemarkHomological Mirror Symmetry

Kontsevich's homological mirror symmetry conjecture states: for a mirror pair $(M, \check{M})$ of Calabi-Yau manifolds, there is an equivalence of $A_\infty$ -categories: $D^b\mathrm{Fuk}(M) \simeq D^b\mathrm{Coh}(\check{M})$ . This relates the symplectic geometry of $M$ (Lagrangians, pseudo-holomorphic curves) to the algebraic geometry of $\check{M}$ (coherent sheaves, derived categories).