Fukaya category

In symplectic topology, a Fukaya category of a symplectic manifold [math]\displaystyle{ (X, \omega) }[/math] is a category [math]\displaystyle{ \mathcal F (X) }[/math] whose objects are Lagrangian submanifolds of [math]\displaystyle{ X }[/math], and morphisms are Lagrangian Floer chain groups: [math]\displaystyle{ \mathrm{Hom} (L_0, L_1) = CF (L_0,L_1) }[/math]. Its finer structure can be described as an A_∞-category. They are named after Kenji Fukaya who introduced the [math]\displaystyle{ A_\infty }[/math] 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 now been computationally verified for a number of examples.

Formal definition

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

[math]\displaystyle{ \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). }[/math]

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

[math]\displaystyle{ n(p_{d-1, d}, \ldots, p_{0, 1}; q_{0, d}) }[/math]

in the coefficient ring. Then define

[math]\displaystyle{ \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) }[/math]

and extend [math]\displaystyle{ \mu_d }[/math] in a multilinear way.

The sequence of higher compositions [math]\displaystyle{ \mu_1, \mu_2, \ldots, }[/math] satisfy the [math]\displaystyle{ A_\infty }[/math] 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.

See also

• Homotopy associative algebra

Bibliography

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

• Paul Seidel, Fukaya categories and Picard-Lefschetz theory. Zurich lectures in Advanced Mathematics
• Fukaya, Kenji (2009), Lagrangian intersection Floer theory: anomaly and obstruction. Part I, AMS/IP Studies in Advanced Mathematics, 46, American Mathematical Society, Providence, RI; International Press, Somerville, MA, ISBN 978-0-8218-4836-4 
• Fukaya, Kenji (2009), Lagrangian intersection Floer theory: anomaly and obstruction. Part II, AMS/IP Studies in Advanced Mathematics, 46, American Mathematical Society, Providence, RI; International Press, Somerville, MA, ISBN 978-0-8218-4837-1 

External links

• The thread on MathOverflow 'Is the Fukaya category "defined"?'