Fukaya category

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In symplectic topology, a discipline within mathematics, a Fukaya category of a symplectic manifold is a category whose objects are Lagrangian submanifolds of , and morphisms are Floer chain groups: . 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 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 been computationally verified for a number of comparatively simple examples.

Formal Definition[edit]

Let be a symplectic manifold. For each pair of Lagrangian submanifolds , suppose they intersect transversely, then define 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 of the cochain complexes on the left, and any generator of the cochain complex on the right, 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 relation because the boundaries of various moduli spaces of holomorphic polygons correspond to configurations of degenerate polygons.

References[edit]

  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.
  • 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, 46.1, 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, 46.2, American Mathematical Society, Providence, RI; International Press, Somerville, MA, ISBN 978-0-8218-4837-1, MR 2548482
  • The thread on MathOverflow 'Is the Fukaya category "defined"?'