Quivers, Floer cohomology, and braid group actions

We consider the derived categories of modules over a certain family A m A_m ( m ≥ 1 m ≥ 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 . 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 group B m + 1 Bₘ₊₁ acts faithfully on the derived category of A m 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 A_m -modules. The full embedding would connect the two occurrences of the braid group, thus explaining the similarity between them.