Counting using Hall Algebras III. Quivers with Potentials

Abstract

For a quiver with potential, we can associate a vanishing cycle to each representation space. If there is a nice torus action on the potential, the vanishing cycles can be expressed in terms of truncated Jacobian algebras. We study how these vanishing cycles change under the mutation of Derksen-Weyman-Zelevinsky. The wall-crossing formula leads to a categorification of quantum cluster algebras under some assumption. This is a special case of A. Efimov's result, but our approach is more concrete and down-to-earth. We also obtain a counting formula relating the representation Grassmannians under sink-source reflections.