nLab
Fukaya category

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

$$m_k:\mathrm{Hom}(L_1,L_2)\otimes \cdots \otimes \mathrm{Hom}(L_k,L_{k+1})\to \mathrm{Hom}(L_1,L_{k+1})$$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.

Related concepts

• Weinstein symplectic category

References

Fukaya categories have first been introduced in

• Kenji Fukaya, Morse homotopy, $A_{\mathrm{\infty }}$-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

See also

• Paul Seidel, Fukaya categories and Picard-Lefschetz theory.

A beginners introduction is given by Denis Auroux, see link, a text (available as ArXiv 1301.7056), based on a series of lectures given at a Summer School on Contact and Symplectic Topology at Université de Nantes in June 2011.