In symplectic topology, a Fukaya category of a symplectic manifold
, ω )
whose objects are Lagrangian submanifolds of
, and morphisms are Lagrangian Floer chain groups:
Its finer structure can be described as an A∞-category.
They are named after Kenji Fukaya who introduced the
language first in the context of Morse homology,[1] 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.
[2] This conjecture has now been computationally verified for a number of examples.
be a symplectic manifold.
For each pair of Lagrangian submanifolds
that intersect transversely, one defines the Floer cochain complex
which is a module generated by intersection points
The Floer cochain complex is viewed as the set of morphisms from
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
of the cochain complexes, the moduli space of
-holomorphic polygons with
faces with each face mapped into
has a count in the coefficient ring.
Then define and extend
The sequence of higher compositions
relations because the boundaries of various moduli spaces of holomorphic polygons correspond to configurations of degenerate polygons.
This definition of Fukaya category for a general (compact) symplectic manifold has never been rigorously given.
The main challenge is the transversality issue, which is essential in defining the counting of holomorphic disks.