# Contents

## Idea

The Fukaya category of a symplectic manifold $X$ is an A-∞ category having Lagrangian submanifolds of $X$ as objects. When two Lagrangian submanifolds ${L}_{1}$ and ${L}_{2}$ of $X$ meet transversally, their hom-space in the Fukaya category can be roughly defined as the free vector space generated by the intersection points $x\in {L}_{1}\cap {L}_{2}$; one of the main difficulties in giving a rigorous definition of the Fukaya category in general relies precisely in the problem of correctly defining the hom-spaces for nontransversal intersections. As one could expect, the same difficulty carries on to the definition of the multilinear operations in the Fukaya category: when Lagrangians ${L}_{1},{L}_{2},\dots ,{L}_{k+1}$ intersect transversally one has a clear geometric intuition of the multiplication

${m}_{k}:\mathrm{Hom}\left({L}_{1},{L}_{2}\right)\otimes \cdots \otimes \mathrm{Hom}\left({L}_{k},{L}_{k+1}\right)\to \mathrm{Hom}\left({L}_{1},{L}_{k+1}\right)$m_k\colon Hom(L_1,L_2)\otimes\cdots\otimes Hom(L_k,L_{k+1})\to Hom(L_1,L_{k+1})

in terms of counting pseudo-holomorphic disks into $X$ whose boundaries lie on the given Lagrangian submanifolds, but when intersections are nontransverse, the definition of ${m}_{k}$ becomes more evasive.

In string theory, the Fukaya category of a symplectic manifold $X$ represents the category of D-branes in the A-model with target space $X$. For Landau-Ginzburg models, the category of D-branes for the A-model is described by Fukaya-Seidel categories.

## References

Fukaya categories have first been introduced in

• Kenji Fukaya, Morse homotopy, ${A}_{\infty }$-category, and Floer homologies. Proceedings of GARC Workshop on Geometry and Topology ‘93 (Seoul, 1993). (link)

The definitive reference is the book

• Fukaya-Oh-Ohta-Ono, Lagrangian intersection Floer theory - anomaly and obstruction