Ice quivers with potential arising from once-punctured polygons and Cohen-Macaulay modules

Given a tagged triangulation of a once-punctured polygon $P^{\ast }$ with $n$ vertices, we associate an ice quiver with potential such that the frozen part of the associated frozen Jacobian algebra has the structure of a Gorenstein $K[X]$-order $\mathrm{\Lambda }$. Then we show that the stable category of the category of Cohen-Macaulay $\mathrm{\Lambda }$-modules is equivalent to the cluster category $\mathcal{C}$ of type $D_n$. It gives a natural interpretation of the usual indexation of cluster tilting objects of $\mathcal{C}$ by tagged triangulations of $P^{\ast }$. Moreover, it extends naturally the triangulated categorification by $\mathcal{C}$ of the cluster algebra of type $D_n$ to an exact categorification by adding coefficients corresponding to the sides of $P$. Finally, we lift the previous equivalence of categories to an equivalence between the stable category of graded Cohen-Macaulay $\mathrm{\Lambda }$-modules and the bounded derived category of modules over a path algebra of type $D_n$.