Fukaya category explained

(X, \omega) is a category l F (X) whose objects are Lagrangian submanifolds of X , and morphisms are Lagrangian Floer chain groups: 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,^[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 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)} ⊗ CF ^{*} (L _{d-2} , L _{d-1} ) ⊗ … ⊗ CF ^{*(} L _{1,} L _{2)} ⊗ CF ^{*} (L _{0,} L _{1)} \to CF ^{*} ( L _{0,} L _{d).} J on the symplectic manifold (X, \omega) . For generators p _{d-1,} \in CF ^{*(L} _{d -1} ,L _{d),} \ldots, p _{0,} \in CF ^{*(L} _{0 ,L} _{1)} and q _{0,} \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 countn(p _{d-1,} , \ldots, p _{0,} ; q _{0,} )

in the coefficient ring. Then define

\mu _{d} ( p _{d-1,} , \ldots, p _{0,} ) = \sum _{q _{0,} \in L _{0} \cap L _{d}} n(p _{d-1,} , \ldots, p _{0,} ) ⋅ q _{0,} \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

External links

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

Notes and References

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.