nLab
Fukaya category

Context

Symplectic geometry

(,1)(\infty,1)-Category theory

Stable Homotopy theory

Contents

Idea

The Fukaya category of a symplectic manifold XX is an A-∞ category having Lagrangian submanifolds of XX as objects. When two Lagrangian submanifolds L 1L_1 and L 2L_2 of XX meet transversally, their hom-space in the Fukaya category can be roughly defined as the free vector space generated by the intersection points xL 1L 2x\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,,L k+1L_1, L_2,\dots,L_{k+1} intersect transversally one has a clear geometric intuition of the multiplication

m k:Hom(L 1,L 2)Hom(L k,L k+1)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 XX whose boundaries lie on the given Lagrangian submanifolds, but when intersections are nontransverse, the definition of m km_k becomes more evasive.

In quantum field theory and string theory

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

The assignment that sends a symplectic manifold to its Fukaya category extends to a functor on a variant of the symplectic category with Lagrangian correspondences as morphisms. This is supposed to be the FQFT incarnation of Donaldson theory. See at Lagrangian correspondences and category-valued TFT for more on this.

References

Fukaya categories have first been introduced in

  • Kenji Fukaya, Morse homotopy, A 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

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.

Discussion of the relation to Lagrangian cobordism is in

Last revised on November 17, 2014 at 17:27:59. See the history of this page for a list of all contributions to it.