  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.
