Abstract
We consider the derived categories of modules over a certain familyAmA_m(m≥1m \geq 1) of graded rings, and Floer cohomology of Lagrangian intersections in the symplectic manifolds which are the Milnor fibres of simple singularities of typeAm.A_m.We show that each of these two rather different objects encodes the topology of curves on an(m+1)(m+1)-punctured disc. We prove that the braid groupBm+1B_{m+1}acts faithfully on the derived category ofAmA_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 ofAmA_m-modules. The full embedding would connect the two occurrences of the braid group, thus explaining the similarity between them.
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