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Quivers, Floer cohomology, and braid group actions

Authors: Mikhail Khovanov and Paul Seidel
Journal: J. Amer. Math. Soc. 15 (2002), 203-271
MSC (2000): Primary 18G10, 53D40, 20F36
Published electronically: September 24, 2001
MathSciNet review: 1862802
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Abstract: We consider the derived categories of modules over a certain family $A_m$ ($m \geq 1$) of graded rings, and Floer cohomology of Lagrangian intersections in the symplectic manifolds which are the Milnor fibres of simple singularities of type $A_m.$ We show that each of these two rather different objects encodes the topology of curves on an $(m+1)$-punctured disc. We prove that the braid group $B_{m+1}$ acts faithfully on the derived category of $A_m$-modules, and that it injects into the symplectic mapping class group of the Milnor fibers. The philosophy behind our results is as follows. Using Floer cohomology, one should be able to associate to the Milnor fibre a triangulated category (its construction has not been carried out in detail yet). This triangulated category should contain a full subcategory which is equivalent, up to a slight difference in the grading, to the derived category of $A_m$-modules. The full embedding would connect the two occurrences of the braid group, thus explaining the similarity between them.

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Additional Information

Mikhail Khovanov
Affiliation: Department of Mathematics, University of California at Davis, Davis, California 95616-8633

Paul Seidel
Affiliation: Department of Mathematics, Ecole Polytechnique, F-91128 Palaiseau, France – and – School of Mathematics, Institute for Advanced Study, Einstein Drive, Princeton, New Jersey 08540

Received by editor(s): July 6, 2000
Received by editor(s) in revised form: April 3, 2001
Published electronically: September 24, 2001
Additional Notes: The first author was supported by NSF grants DMS 96-27351 and DMS 97-29992 and, later on, by the University of California at Davis. The second author was supported by NSF grant DMS-9304580 and by the Institut Universitaire de France.
Article copyright: © Copyright 2001 American Mathematical Society