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

Revised on November 17, 2014 17:27:59 by Urs Schreiber (217.155.201.6)