Algebraic and Symplectic Geometry Seminar

Let be a quiver with a potential given by successive mutations from a quiver with a potential . Then we have an equivalence of the derived categories of dg-modules over the Ginzburg dg-algebras satisfying the following condition: a simple module over the dg-algebra for is either concentrated on degree 0 or concentrated on degree 1 as a dg-module over the dg-algebra for . As an application of this equivalence, I will give a description of the space of stability conditions.

I will introduce the theory of cluster categories after Amiot and Plamondon. For a quiver with a potential, the cluster category is defined as the quotient of the category of perfect dg-modules by the category of dg-modules with finite dimensional cohomologies. We can show the existence of the equivalence in the first talk as an application of the cluster category. I will also propose a definition of a counting invariant for each element in the cluster category.