Quivers with potentials and actions of finite abelian groups

Let $G$ be a finite abelian group acting on a path algebra $kQ$ by permuting the vertices and preserving the arrowspans. Let $W$ be a potential on the quiver $Q$ which is fixed by the action. We study the skew group dg algebra $\Gamma_{Q, W}G$ of the Ginzburg dg algebra of $(Q, W)$. It is known that $\Gamma_{Q, W}G$ is Morita equivalent to another Ginzburg dg algebra $\Gamma_{Q_G, W_G}$, whose quiver $Q_G$ was constructed by Demonet. In this article we give an explicit construction of the potential $W_G$ as a linear combination of cycles in $Q_G$, and write the Morita equivalence explicitly. As a corollary, we obtain functors between the cluster categories corresponding to the two quivers with potentials.