Fukaya category

In symplectic topology, a Fukaya category of a symplectic manifold $(X,\omega )$ is a category ${ℱ}(X)$ whose objects are Lagrangian submanifolds of $X$, and morphisms are Lagrangian Floer chain groups: $\mathrm{H}\mathrm{o}\mathrm{m}(L_0,L_1)=CF(L_0,L_1)$. Its finer structure can be described as an A_∞-category. They are named after Kenji Fukaya who introduced the $A_{\mathrm{\infty }}$ 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 $C{F}^{*}(L_0,L_1)$ 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 $A_{\mathrm{\infty }}$ category, meaning that besides ordinary compositions, there are higher composition maps

${\mu }_d:C{F}^{*}(L_{d-1},L_d)\otimes C{F}^{*}(L_{d-2},L_{d-1})\otimes \cdots \otimes C{F}^{*}(L_1,L_2)\otimes C{F}^{*}(L_0,L_1)\to C{F}^{*}(L_0,L_d).$

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

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

in the coefficient ring. Then define

${\mu }_d(p_{d-1,d},\dots ,p_{0,1})=\sum _{q_{0,d}\in L_0\cap L_d}n(p_{d-1,d},\dots ,p_{0,1})\cdot q_{0,d}\in C{F}^{*}(L_0,L_d)$

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

The sequence of higher compositions ${\mu }_1,{\mu }_2,\dots ,$ satisfy the $A_{\mathrm{\infty }}$ 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 (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"?'