Fukaya Category

In symplectic topology, 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: $\mathrm (L_0, L_1) = FC (L_0,L_1)$. 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, 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 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

:$\mu_d: CF^* (L_, L_d) \otimes CF^* (L_, L_)\otimes \cdots \otimes CF^*( L_1, L_2) \otimes CF^* (L_0, L_1) \to CF^* ( L_0, L_d).$

It is defined as follows. Choose a compatible almost complex structure $J$ on the symplectic manifold $(X, \omega)$. For generators $p_, \ldots, p_$ of the cochain complexes on the left, and any generator $q_$ 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

:$n(p_, \ldots, p_; q_)$

in the coefficient ring. Then define

:$\mu_d ( p_, \ldots, p_ ) = \sum_ n(p_, \ldots, p_) \cdot q_ \in CF^*(L_0, L_d)$

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.

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

References

Bibliography

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

*Paul Seidel, ''Fukaya categories and Picard-Lefschetz theory.'' Zurich lectures in Advanced Mathematics
*
*{{Citation, last1=Fukaya, first1=Kenji, authorlink1= Kenji Fukaya
, last2=Oh, first2=Yong-Geun, authorlink2=Oh Yong-Geun
, last3=Ohta, first3=Hiroshi
, last4=Ono, first4= Kaoru
, title=Lagrangian intersection Floer theory: anomaly and obstruction. Part II
, series=AMS/IP Studies in Advanced Mathematics
, volume=46
, publisher=American Mathematical Society, Providence, RI; International Press, Somerville, MA
, year=2009
, issue=2, isbn=978-0-8218-4837-1
, mr=2548482

External links

*Th
thread
on MathOverflow 'Is the Fukaya category "defined"?'

Symplectic geometry
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