Английская Википедия:Fukaya category
In symplectic topology, a Fukaya category of a symplectic manifold <math>(X, \omega)</math> is a category <math>\mathcal F (X)</math> whose objects are Lagrangian submanifolds of <math>X</math>, and morphisms are Lagrangian Floer chain groups: <math>\mathrm{Hom} (L_0, L_1) = CF (L_0,L_1)</math>. Its finer structure can be described as an A∞-category.
They are named after Kenji Fukaya who introduced the <math>A_\infty</math> 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 <math> (X, \omega) </math> be a symplectic manifold. For each pair of Lagrangian submanifolds <math> L_0, L_1 \subset X </math> that intersect transversely, one defines the Floer cochain complex <math> CF^*(L_0, L_1) </math> which is a module generated by intersection points <math> L_0 \cap L_1 </math>. The Floer cochain complex is viewed as the set of morphisms from <math> L_0 </math> to <math> L_1 </math>. The Fukaya category is an <math> A_\infty </math> category, meaning that besides ordinary compositions, there are higher composition maps
- <math> \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). </math>
It is defined as follows. Choose a compatible almost complex structure <math> J </math> on the symplectic manifold <math> (X, \omega) </math>. For generators <math> p_{d-1, d} \in CF^*(L_{d-1},L_d), \ldots, p_{0, 1} \in CF^*(L_0,L_1) </math> and <math> q_{0, d} \in CF^*(L_0,L_d) </math> of the cochain complexes, the moduli space of <math> J </math>-holomorphic polygons with <math> d+ 1 </math> faces with each face mapped into <math> L_0, L_1, \ldots, L_d </math> has a count
- <math> n(p_{d-1, d}, \ldots, p_{0, 1}; q_{0, d}) </math>
in the coefficient ring. Then define
- <math> \mu_d ( p_{d-1, d}, \ldots, p_{0, 1} ) = \sum_{q_{0, d} \in L_0 \cap L_d} n(p_{d-1, d}, \ldots, p_{0, 1}) \cdot q_{0, d} \in CF^*(L_0, L_d)</math>
and extend <math> \mu_d </math> in a multilinear way.
The sequence of higher compositions <math> \mu_1, \mu_2, \ldots, </math> satisfy the <math> A_\infty </math> 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
References
Bibliography
- Denis Auroux, A beginner's introduction to Fukaya categories.
- Paul Seidel, Fukaya categories and Picard-Lefschetz theory. Zurich lectures in Advanced Mathematics
- Шаблон:Citation
- Шаблон:Citation
External links
- The thread on MathOverflow 'Is the Fukaya category "defined"?'
- ↑ Kenji Fukaya, Morse homotopy, <math> A_\infty </math> 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.
