nLab Fukaya category



Symplectic geometry

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

Stable Homotopy theory



The Fukaya category (named after Fukaya 1993) 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 is roughly defined as the vector space spanned 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}) \longrightarrow 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 elusive.

When defined, the Fukaya category is an A A_\infty -category which consitutes one side of the duality of homological mirror symmetry.

In string theory

A-Model topological string

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 and see at homological mirror symmetry.

Yukawa couplings in intersecting D-brane models

In intersecting D-brane models Yukawa couplings are encoded by worldsheet instantons of open strings stretching between the intersecting D-branes (see Marchesano 03, Section 7.5). Mathematically this is encoded by derived hom-spaces in a Fukaya category (see Marchesano 03, Section 7.5).

table grabbed from Marchesano 03


Fukaya categories are named after:

  • Kenji Fukaya, Morse homotopy, A A_\infty-category, and Floer homologies, Proceedings of GARC Workshop on Geometry and Topology ‘93 (Seoul, 1993) [pdf]



  • Denis Auroux, A beginner’s introduction to Fukaya categories, lectures at Summer School on Contact and Symplectic Topology, Université de Nantes (June 2011) [arXiv:1301.7056]

and with an eye towards mirror symmetry:

  • Felipe Espreafico G. Ramos, Mirror Symmetry and Fukaya Categories (2020) [pdf, pdf]

On the relation to Lagrangian cobordism:

Relation to Yukawa couplings in intersecting D-brane models:

Last revised on April 1, 2024 at 07:53:26. See the history of this page for a list of all contributions to it.