Fukaya category

In symplectic topology, a Fukaya category of a symplectic manifold $(X,\omega )$ is a category ${\mathcal {F}}(X)$ whose objects are Lagrangian submanifolds of $X$, and morphisms are Lagrangian Floer chain groups: $\mathrm {Hom} (L_{0},L_{1})=CF(L_{0},L_{1})$. Its finer structure can be described as an A_∞-category.

They are named after Kenji Fukaya who introduced the $A_{\infty }$ 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 $(X,\omega )$ 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.

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; Oh, Yong-Geun; Ohta, Hiroshi; Ono, Kaoru (2009), Lagrangian intersection Floer theory: anomaly and obstruction. Part I, AMS/IP Studies in Advanced Mathematics, vol. 46, American Mathematical Society, Providence, RI; International Press, Somerville, MA, ISBN 978-0-8218-4836-4, MR 2553465
• Fukaya, Kenji; Oh, Yong-Geun; Ohta, Hiroshi; Ono, Kaoru (2009), Lagrangian intersection Floer theory: anomaly and obstruction. Part II, AMS/IP Studies in Advanced Mathematics, vol. 46, American Mathematical Society, Providence, RI; International Press, Somerville, MA, ISBN 978-0-8218-4837-1, MR 2548482

External links

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