# Fukaya category

In symplectic topology, a discipline within mathematics, a Fukaya category of a symplectic manifold $(M,\omega )$ is a category ${\mathcal {F}}(M)$ whose objects are Lagrangian submanifolds of $M$ , and morphisms are Floer chain groups: $\mathrm {Hom} (L_{0},L_{1})=FC(L_{0},L_{1})$ . Its finer structure can be described in the language of quasi categories as an A-category.

They are named after Kenji Fukaya who introduced the $A_{\infty }$ language first in the context of Morse homology, and exist in a number of variants. As Fukaya categories are A-categories, they have associated derived categories, which are the subject of the celebrated homological mirror symmetry conjecture of Maxim Kontsevich. This conjecture has been computationally verified for a number of comparatively simple examples.

## Formal Definition

Let $(X,\omega )$  be a symplectic manifold. For each pair of Lagrangian submanifolds $L_{0},L_{1}\subset X$ , suppose they intersect transversely, then define the Floer cochain complex $CF^{*}(L_{0},L_{1})$  which is a module generated by intersection points $L_{0}\cap L_{1}$ . The Floer cochain complex is viewed as the set of morphisms from $L_{0}$  to $L_{1}$ . The Fukaya category is an $A_{\infty }$  category, meaning that besides ordinary compositions, there are higher composition maps

$\mu _{d}:CF^{*}(L_{d-1},L_{d})\otimes CF^{*}(L_{d-2},L_{d-1})\otimes \cdots \otimes CF^{*}(L_{1},L_{2})\otimes CF^{*}(L_{0},L_{1})\to CF^{*}(L_{0},L_{d}).$

It is defined as follows. Choose a compatible almost complex structure $J$  on the symplectic manifold $(X,\omega )$ . For generators $p_{d-1d},\ldots ,p_{01}$  of the cochain complexes on the left, and any generator $q_{0d}$  of the cochain complex on the right, the moduli space of $J$ -holomorphic polygons with $d+1$  faces with each face mapped into $L_{0},L_{1},\ldots ,L_{d}$  has a count

$n(p_{d-1d},\ldots ,p_{01};q_{0d})$

in the coefficient ring. Then define

$\mu _{d}(p_{d-1d},\ldots ,p_{01})=\sum _{q_{0d}\in L_{0}\cap L_{d}}n(p_{d-1d},\ldots ,p_{01})\cdot q_{0d}\in CF^{*}(L_{0},L_{d})$

and extend $\mu _{d}$  in a multilinear way.

The sequence of higher compositions $\mu _{1},\mu _{2},\ldots ,$  satisfy the $A_{\infty }$  relation because the boundaries of various moduli spaces of holomorphic polygons correspond to configurations of degenerate polygons.