Fukaya category


In symplectic topology, a discipline within mathematics, a Fukaya category of a symplectic manifold is a category whose objects are Lagrangian submanifolds of, and morphisms are Floer chain groups:. Its finer structure can be described in the language of quasi categories as an A∞-category.
They are named after Kenji Fukaya who introduced the language first in the context of Morse homology, and exist in a number of variants. As Fukaya categories are A∞-categories, they have associated derived categories, which are the subject of the celebrated homological mirror symmetry conjecture of Maxim Kontsevich. This conjecture has been computationally verified for a number of comparatively simple examples.

Formal Definition

Let be a symplectic manifold. For each pair of Lagrangian submanifolds, suppose they intersect transversely, then define the Floer cochain complex which is a module generated by intersection points. The Floer cochain complex is viewed as the set of morphisms from to. The Fukaya category is an category, meaning that besides ordinary compositions, there are higher composition maps
It is defined as follows. Choose a compatible almost complex structure on the symplectic manifold. For generators of the cochain complexes on the left, and any generator of the cochain complex on the right, the moduli space of -holomorphic polygons with faces with each face mapped into has a count
in the coefficient ring. Then define
and extend in a multilinear way.
The sequence of higher compositions satisfy the relation because the boundaries of various moduli spaces of holomorphic polygons correspond to configurations of degenerate polygons.