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Fukaya category

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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

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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

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

in the coefficient ring. Then define

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

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References

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  1. ^ Kenji Fukaya, Morse homotopy, 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.

Bibliography

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  • 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
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