Frontiers of Mathematics in China >
Lie bialgebra structure on cyclic cohomology of Fukaya categories
Received date: 14 Apr 2014
Accepted date: 24 Oct 2014
Published date: 24 Jun 2015
Copyright
Let M be an exact symplectic manifold with contact type boundary such that c1(M) = 0. Motivated by noncommutative symplectic geometry and string topology, we show that the cyclic cohomology of the Fukaya category of M has an involutive Lie bialgebra structure.
Key words： Fukaya category; cyclic cohomology; Lie bialgebra
Xiaojun CHEN , Hai-Long HER , Shanzhong SUN . Lie bialgebra structure on cyclic cohomology of Fukaya categories[J]. Frontiers of Mathematics in China, 2015 , 10(5) : 1057 -1085 . DOI: 10.1007/s11464-015-0440-8
| 1 |
Abouzaid M. A topological model for the Fukaya categories of plumbings. J Differential Geom, 2011, 87(1): 1-80
|
| 2 |
Bocklandt R, Le Bruyn L. Necklace Lie algebras and noncommutative symplectic geometry. Math Z, 2002, 240: 141-167
|
| 3 |
Chas M, Sullivan D. Closed string operators in topology leading to Lie bialgebras and higher string algebra. In: The Legacy of Niels Henrik Abel. Berlin: Springer, 2004, 771-784
|
| 4 |
Chen K-T. Iterated path integrals. Bull Amer Math Soc, 1977, 83: 831-879
|
| 5 |
Chen X, Eshmatov F, Gan W L. Quantization of the Lie bialgebra of string topology. Comm Math Phys, 2011, 301(1): 37-53
|
| 6 |
Costello K. Topological conformal field theories and Calabi-Yau categories. Adv Math, 2007, 210(1): 165-214
|
| 7 |
Fukaya K. Morse homotopy, A∞-category, and Floer homologies. In: Kim H J, ed. Proceedings of GARC Workshop on Geometry and Topology ’93 (Seoul, 1993). Lecture Notes, No 18. Seoul Nat Univ, Seoul, 1993, 1-102
|
| 8 |
Fukaya K. Floer homology and mirror symmetry. II. Minimal surfaces, geometric analysis and symplectic geometry (Baltimore, MD, 1999). Adv Stud Pure Math, Vol 34. Tokyo: Math Soc Japan, 2002, 31-127
|
| 9 |
Fukaya K. Cyclic symmetry and adic convergence in Lagrangian Floer theory. Kyoto J Math, 2010, 50(3): 521-590
|
| 10 |
Fukaya K, Oh Y-G, Ohta H, Ono K. Lagrangian Intersection Floer Theory: Anomaly and Obstruction. Part I and II. AMS/IP Studies in Advanced Mathematics, Vol 46. Providence: Amer Math Soc/Internation Press, 2009
|
| 11 |
Fukaya K, Seidel P, Smith I. Exact Lagrangian submanifolds in simply-connected cotangent bundles. Invent Math, 2008, 172(1): 1-27
|
| 12 |
Fukaya K, Seidel P, Smith I. The symplectic geometry of cotangent bundles from a categorical viewpoint. In: Homological Mirror Symmetry. Lecture Notes in Phys, Vol 757. Berlin: Springer, 2009, 1-26
|
| 13 |
Getzler E, Jones J D S. A∞-algebras and the cyclic bar complex. Illinois J Math, 1989, 34: 256-283
|
| 14 |
Getzler E, Jones J D S, Petrack S. Differential forms on loop space and the cyclic bar complex. Topology, 1991, 30: 339-371
|
| 15 |
Ginzburg V. Noncommutative symplectic geometry, quiver varieties and operads. Math Res Lett, 2001, 8: 377-400
|
| 16 |
Hamilton A. Noncommutative geometry and compactifications of the moduli space of curves. arXiv: 0801.0904
|
| 17 |
Jones J D S. Cyclic homology and equivariant homology. Invent Math, 1987, 87: 403-423
|
| 18 |
Kontsevich M. Formal (non)commutative symplectic geometry. In: The Gelfand Mathematical Seminars 1990-1992. Boston; Birkhäuser, 1993, 173-187
|
| 19 |
Kontsevich M. Feynman diagrams and low-dimensional topology. In: 1-st European Congress of Mathematics, 1992, Paris, Vol II. Progr Math, Vol 120. Basel: Birkhäuser, 1994, 97-121
|
| 20 |
Kontsevich M. Homological algebra of mirror symmetry. In: Proceedings of the International Congress of Mathematicians, Zürich 1994, Vol I. Basel: Birkhäuser, 1995, 120-139
|
| 21 |
Kontsevich M, Soibelman Y. Notes on A∞ algebras, A∞ categories and noncommutative geometry. In: Homological Mirror Symmetry. Lecture Notes in Phys, Vol 757. Berlin: Springer, 2009, 153-219
|
| 22 |
Loday J-L. Cyclic Homology. 2nd ed. Grundlehren Math Wiss, Vol 301. Berlin: Springer-Verlag, 1998
|
| 23 |
McDuff D, Salamon D. Introduction to Symplectic Topology. 2nd ed. Oxford Mathematical Monographs. New York: The Clarendon Press, Oxford University Press, 1998
|
| 24 |
Nadler D. Microlocal branes are constructible sheaves. Selecta Math (N S), 2009, 15(4): 563-619
|
| 25 |
Piunikhin S, Salamon D, Schwarz M. Symplectic Floer-Donaldson theory and quantum cohomology. In: Thomas C B, ed. Contact and Symplectic Geometry. Cambridge: Cambridge Univ Press, 1996, 171-200
|
| 26 |
Schedler T. A Hopf algebra quantizing a necklace Lie algebra canonically associated to a quiver. Int Math Res Not, 2005, 12: 725-760
|
| 27 |
Seidel P. Fukaya Categories and Picard-Lefschetz Theory. Zürich Lectures in Advanced Mathematics. Zürich: European Mathematical Society, 2008
|
/
| 〈 |
|
〉 |