Fukaya category

Formal definition

Let (X, \omega) be a symplectic manifold. For each pair of Lagrangian submanifolds L_0, L_1 \subset X that intersect transversely, one defines the Floer cochain complex CF^*(L_0, L_1) which is a module generated by intersection points L_0 \cap L_1 . The Floer cochain complex is viewed as the set of morphisms from L_0 to L_1 . The Fukaya category is an A_\infty category, meaning that besides ordinary compositions, there are higher composition maps

: \mu_d: CF^* (L_{d-1}, L_d) \otimes CF^* (L_{d-2}, L_{d-1})\otimes \cdots \otimes CF^( L_1, L_2) \otimes CF^ (L_0, L_1) \to CF^* ( L_0, L_d).

It is defined as follows. Choose a compatible almost complex structure J on the symplectic manifold (X, \omega) . For generators p_{d-1, d} \in CF^(L_{d-1},L_d), \ldots, p_{0, 1} \in CF^(L_0,L_1) and q_{0, d} \in CF^*(L_0,L_d) of the cochain complexes, the moduli space of J -holomorphic polygons with d+ 1 faces with each face mapped into L_0, L_1, \ldots, L_d has a count

: n(p_{d-1, d}, \ldots, p_{0, 1}; q_{0, d})

in the coefficient ring. Then define

: \mu_d ( p_{d-1, d}, \ldots, p_{0, 1} ) = \sum_{q_{0, d} \in L_0 \cap L_d} n(p_{d-1, d}, \ldots, p_{0, 1}) \cdot q_{0, d} \in CF^*(L_0, L_d)

and extend \mu_d in a multilinear way.

The sequence of higher compositions \mu_1, \mu_2, \ldots, satisfy the A_\infty 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.

References

Bibliography

• Denis Auroux, A beginner's introduction to Fukaya categories.

• Paul Seidel, Fukaya categories and Picard-Lefschetz theory. Zurich lectures in Advanced Mathematics

• {{Citation|last1=Fukaya|first1=Kenji|authorlink1= Kenji Fukaya

• {{Citation|last1=Fukaya|first1=Kenji|authorlink1= Kenji Fukaya

References

1. Kenji Fukaya, ''Morse homotopy, $A\_\backslash infty$ category and Floer homologies'', MSRI preprint No. 020-94 (1993)
2. Kontsevich, Maxim, ''Homological algebra of mirror symmetry'', Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), 120–139, Birkhäuser, Basel, 1995.