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In [[symplectic topology]], a discipline within mathematics, a '''Fukaya category''' of a [[symplectic manifold]] <math>(M, \omega)</math> is a  [[Category (mathematics)|category]] <math>\mathcal F (M)</math> whose objects are [[Lagrangian submanifold]]s of <math>M</math>, and [[morphism]]s are [[Floer homology|Floer chain groups]]: <math>\mathrm{Hom} (L_0, L_1) = FC (L_0,L_1)</math>. Its finer structure can be described in the language of [[quasi-category|quasi categories]] as an [[A∞-algebra|''A''<sub>∞</sub>-category]].
 
They are named after [[Kenji Fukaya]] who introduced the <math>A_\infty</math> language first in the context of [[Morse homology]], and exist in a number of variants. As Fukaya categories are [[A∞-algebra|''A''<sub>∞</sub>-categories]], they have associated [[derived categories]], which are the subject of a celebrated conjecture of [[Maxim Kontsevich]]: the [[homological mirror symmetry]]. This conjecture has been verified by computations for a variety of comparatively simple examples.
 
==References==
 
*P. Seidel, ''Fukaya categories and Picard-Lefschetz theory'', Zurich lectures in Advanced Mathematics
*Fukaya, Y-G. Oh, H. Ohta, K. Ono, ''Lagrangian Intersection Floer Theory'', Studies in Advanced Mathematics
*The [http://mathoverflow.net/questions/2905/is-the-fukaya-category-defined thread] on  [[MathOverflow]] 'Is the Fukaya category "defined"?'
 
[[Category:Symplectic geometry]]
 
{{differential-geometry-stub}}

Revision as of 15:20, 3 January 2014

In symplectic topology, a discipline within mathematics, a Fukaya category of a symplectic manifold (M,ω) is a category ℱ(M) whose objects are Lagrangian submanifolds of M, and morphisms are Floer chain groups: Hom(L0,L1)=FC(L0,L1). 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 A∞ language first in the context of Morse homology, and exist in a number of variants. As Fukaya categories are A∞-categories, they have associated derived categories, which are the subject of a celebrated conjecture of Maxim Kontsevich: the homological mirror symmetry. This conjecture has been verified by computations for a variety of comparatively simple examples.

References

  • P. Seidel, Fukaya categories and Picard-Lefschetz theory, Zurich lectures in Advanced Mathematics
  • Fukaya, Y-G. Oh, H. Ohta, K. Ono, Lagrangian Intersection Floer Theory, Studies in Advanced Mathematics
  • The thread on MathOverflow 'Is the Fukaya category "defined"?'

Template:Differential-geometry-stub

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