Quivers, Floer Cohomology, and Braid Group Actions

1a. Generalities. This paper investigates the connection between symplectic geometry and those parts of representation theory which revolve around the notion of categorification. The existence of such a connection, in an abstract sense, follows from simple general ideas. The difficult thing is to make it explicit. On the symplectic side, the tools needed for a systematic study of this question are not yet fully available. Therefore we concentrate on a single example, which is just complicated enough to indicate the depth of the relationship. The results can be understood by themselves, but a glimpse of the big picture certainly helps to explain them, and that is what the present section is for. Let Q be a category. An action of a group G on Q is a family (Fg)g∈G of functors from Q to itself, such that Fe ∼= idQ and Fg1Fg2 ∼= Fg1g2 for all g1, g2 ∈ G; here ∼= denotes isomorphism of functors. We will not distinguish between two actions (Fg) and (F̃g) such that Fg ∼= F̃g for all g. A particularly nice situation is when Q is triangulated and the Fg are exact functors. Then the action induces a linear representation of G on the Grothendieck group K(Q). The inverse process, in which one lifts a given linear representation to a group action on a triangulated category, is called categorification (of group representations). The connection with symplectic geometry is based on an idea of Donaldson. He proposed (in talks circa 1994) to associate to a compact symplectic manifold (M, ø) a category Lag(M,ω) whose objects are Lagrangian submanifolds L ⊂M , and whose morphisms are the Floer cohomology groups HF (L0, L1). The composition of morphisms would be given by products HF (L1, L2)×HF (L0, L1) −→ HF (L0, L2), which are defined, for example, in [39]. Let Symp(M,ω) be the group of symplectic automorphisms of M . Any φ ∈ Symp(M,ω) determines a family of isomorphisms HF (L0, L1) ∼= HF (φL0, φL1) for L0, L1 ∈ Ob(Lag(M,ω)) which are compatible with the products. In other words, φ induces an equivalence Fφ from Lag(M,ω) to itself. This is just a consequence of the fact that Lag(M,ω) is