Fukaya category

 ( Symplectic Geometry ) 

In symplectic topology, a Fukaya category of a symplectic manifold $(X,\; \backslash omega)$ is a category $\backslash mathcal\; F\; (X)$ whose objects are Lagrangian submanifolds of $X$, and are Lagrangian Floer homology: $\backslash mathrm\{Hom\}\; (L\_0,\; L\_1)\; =\; CF\; (L\_0,L\_1)$. Its finer structure can be described as an A_∞-category.

They are named after Kenji Fukaya who introduced the $A\_\backslash infty$ language first in the context of Morse homology, Kenji Fukaya, Morse homotopy, $A\_\backslash infty$ category and Floer homologies, MSRI preprint No. 020-94 (1993) 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.Kontsevich, Maxim, Homological algebra of mirror symmetry, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), 120–139, Birkhäuser, Basel, 1995. This conjecture has now been computationally verified for a number of examples.

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Formal definition Let $(X,\; \backslash omega)$ be a symplectic manifold. For each pair of Lagrangian submanifolds $L\_0,\; L\_1\; \backslash subset\; X$ that intersect transversely, one defines the Floer cochain complex $CF^*(L\_0,\; L\_1)$ which is a module generated by intersection points $L\_0\; \backslash cap\; L\_1$. The Floer cochain complex is viewed as the set of morphisms from $L\_0$ to $L\_1$. The Fukaya category is an $A\_\backslash infty$ category, meaning that besides ordinary compositions, there are higher composition maps

$\backslash mu\_d:\; CF^*\; (L\_\{d-1\},\; L\_d)\; \backslash otimes\; CF^*\; (L\_\{d-2\},\; L\_\{d-1\})\backslash otimes\; \backslash cdots\; \backslash otimes\; CF^*(\; L\_1,\; L\_2)\; \backslash otimes\; CF^*\; (L\_0,\; L\_1)\; \backslash to\; CF^*\; (\; L\_0,\; L\_d).$

It is defined as follows. Choose a compatible almost complex structure $J$ on the symplectic manifold $(X,\; \backslash omega)$. For generators $p\_\{d-1,\; d\}\; \backslash in\; CF^*(L\_\{d-1\},L\_d),\; \backslash ldots,\; p\_\{0,\; 1\}\; \backslash in\; CF^*(L\_0,L\_1)$ and $q\_\{0,\; d\}\; \backslash in\; CF^*(L\_0,L\_d)$ of the cochain complexes, the moduli space of $J$-holomorphic polygons with $d+\; 1$ faces with each face mapped into $L\_0,\; L\_1,\; \backslash ldots,\; L\_d$ has a count

$n(p\_\{d-1,\; d\},\; \backslash ldots,\; p\_\{0,\; 1\};\; q\_\{0,\; d\})$

in the coefficient ring. Then define

$\backslash mu\_d\; (\; p\_\{d-1,\; d\},\; \backslash ldots,\; p\_\{0,\; 1\}\; )\; =\; \backslash sum\_\{q\_\{0,\; d\}\; \backslash in\; L\_0\; \backslash cap\; L\_d\}\; n(p\_\{d-1,\; d\},\; \backslash ldots,\; p\_\{0,\; 1\})\; \backslash cdot\; q\_\{0,\; d\}\; \backslash in\; CF^*(L\_0,\; L\_d)$

and extend $\backslash mu\_d$ in a multilinear way.

The sequence of higher compositions $\backslash mu\_1,\; \backslash mu\_2,\; \backslash ldots,$ satisfy the $A\_\backslash infty$ 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.

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See also

• Homotopy associative algebra

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Bibliography

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

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

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External links

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

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