Fukaya category

In symplectic topology, a discipline within mathematics, a Fukaya category of a symplectic manifold $(M,\omega )$ is a category ${\mathcal {F}}(M)$ whose objects are Lagrangian submanifolds of $M$ , and morphisms are Floer chain groups: $\mathrm {Hom} (L_{0},L_{1})=FC(L_{0},L_{1})$ . 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 $A_{\infty }$ 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 been computationally verified for a number of comparatively simple examples.

Formal Definition

Let $(X,\omega )$ be a symplectic manifold. For each pair of Lagrangian submanifolds $L_{0},L_{1}\subset X$ , suppose they intersect transversely, then define 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 $p_{d-1d},\ldots ,p_{01}$ of the cochain complexes on the left, and any generator $q_{0d}$ of the cochain complex on the right, 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

$n(p_{d-1d},\ldots ,p_{01};q_{0d})$

in the coefficient ring. Then define

$\mu _{d}(p_{d-1d},\ldots ,p_{01})=\sum _{q_{0d}\in L_{0}\cap L_{d}}n(p_{d-1d},\ldots ,p_{01})\cdot q_{0d}\in CF^{*}(L_{0},L_{d})$

and extend $\mu _{d}$ in a multilinear way.

The sequence of higher compositions $\mu _{1},\mu _{2},\ldots ,$ satisfy the $A_{\infty }$ relation because the boundaries of various moduli spaces of holomorphic polygons correspond to configurations of degenerate polygons.

References

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

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, 46.1, 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, 46.2, American Mathematical Society, Providence, RI; International Press, Somerville, MA, ISBN 978-0-8218-4837-1, MR 2548482
The thread on MathOverflow 'Is the Fukaya category "defined"?'

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[1] Kenji Fukaya, Morse homotopy, $A_{\infty }$ 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.