Background
Basic concepts
equivalences in/of $(\infty,1)$-categories
Universal constructions
Local presentation
Theorems
Extra stuff, structure, properties
Models
The Fukaya category (named after Fukaya 1993) 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 is roughly defined as the vector space spanned 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
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 elusive.
When defined, the Fukaya category is an $A_\infty$-category which consitutes one side of the duality of homological mirror symmetry.
In string theory, the Fukaya category of a symplectic manifold $X$ represents the category of D-branes in the A-model – the A-branes – with target space $X$. 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.
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:
Monographs:
Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Karo Ono, Lagrangian intersection Floer theory - anomaly and obstruction, Studies in Advanced Mathematocs 46, AMS (2009) [ISBN:978-0-8218-5253-8]
Paul Seidel, Fukaya categories and Picard-Lefschetz theory, EMS (2008) [doi:10.4171/063]
Introduction:
and with an eye towards mirror symmetry:
On the relation to Lagrangian cobordism:
Relation to Yukawa couplings in intersecting D-brane models:
Last revised on June 2, 2024 at 22:24:24. See the history of this page for a list of all contributions to it.