equivalences in/of $(\infty,1)$-categories
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
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 – 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.
Fukaya categories have first been introduced in
The definitive reference is the book
See also
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.