Quivers, Floer cohomology, and braid group actions |
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Authors: | Mikhail Khovanov Paul Seidel |
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Affiliation: | Department of Mathematics, University of California at Davis, Davis, California 95616-8633 ; Department of Mathematics, Ecole Polytechnique, F-91128 Palaiseau, France -- and -- School of Mathematics, Institute for Advanced Study, Einstein Drive, Princeton, New Jersey 08540 |
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Abstract: | We consider the derived categories of modules over a certain family ( ) of graded rings, and Floer cohomology of Lagrangian intersections in the symplectic manifolds which are the Milnor fibres of simple singularities of type We show that each of these two rather different objects encodes the topology of curves on an -punctured disc. We prove that the braid group acts faithfully on the derived category of -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 -modules. The full embedding would connect the two occurrences of the braid group, thus explaining the similarity between them. |
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