Category of a symplectic manifold
In symplectic topology, a Fukaya category of a symplectic manifold
is a category
whose objects are Lagrangian submanifolds of
, and morphisms are Lagrangian Floer chain groups:
. Its finer structure can be described as an A∞-category.
They are named after Kenji Fukaya who introduced the
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
be a symplectic manifold. For each pair of Lagrangian submanifolds
that intersect transversely, one defines the Floer cochain complex
which is a module generated by intersection points
. The Floer cochain complex is viewed as the set of morphisms from
to
. The Fukaya category is an
category, meaning that besides ordinary compositions, there are higher composition maps
![{\displaystyle \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}).}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
It is defined as follows. Choose a compatible almost complex structure
on the symplectic manifold
. For generators
and
of the cochain complexes, the moduli space of
-holomorphic polygons with
faces with each face mapped into
has a count
![{\displaystyle n(p_{d-1,d},\ldots ,p_{0,1};q_{0,d})}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
in the coefficient ring. Then define
![{\displaystyle \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})}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
and extend
in a multilinear way.
The sequence of higher compositions
satisfy the
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
- ^ Kenji Fukaya, Morse homotopy,
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.
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; Oh, Yong-Geun; Ohta, Hiroshi; Ono, Kaoru (2009), Lagrangian intersection Floer theory: anomaly and obstruction. Part I, AMS/IP Studies in Advanced Mathematics, vol. 46, American Mathematical Society, Providence, RI; International Press, Somerville, MA, ISBN 978-0-8218-4836-4, MR 2553465
- Fukaya, Kenji; Oh, Yong-Geun; Ohta, Hiroshi; Ono, Kaoru (2009), Lagrangian intersection Floer theory: anomaly and obstruction. Part II, AMS/IP Studies in Advanced Mathematics, vol. 46, American Mathematical Society, Providence, RI; International Press, Somerville, MA, ISBN 978-0-8218-4837-1, MR 2548482
External links
- The thread on MathOverflow 'Is the Fukaya category "defined"?'