Cluster algebras arising from cluster tubes

We study the cluster algebras arising from cluster tubes with rank bigger than $1$. Cluster tubes are $2-$Calabi-Yau triangulated categories which contain no cluster tilting objects, but maximal rigid objects. Fix a certain maximal rigid object $T$ in the cluster tube $\mathcal{C}_n$ of rank $n$. For any indecomposable rigid object $M$ in $\mathcal{C}_n$, we define an analogous $X_M$ of Caldero-Chapton's formula (or Palu's cluster character formula) by using the geometric information of $M$. We show that $X_M, X_{M'}$ satisfy the mutation formula when $M,M'$ form an exchange pair, and that $X_{?}: M\mapsto X_M$ gives a bijection from the set of indecomposable rigid objects in $\mathcal{C}_n$ to the set of cluster variables of cluster algebra of type $C_{n-1}$, which induces a bijection between the set of basic maximal rigid objects in $\mathcal{C}_n$ and the set of clusters. This strengths a surprising result proved recently by Buan-Marsh-Vatne that the combinatorics of maximal rigid objects in the cluster tube $\mathcal{C}_n$ encode the combinatorics of the cluster algebra of type $B_{n-1}$ since the combinatorics of cluster algebras of type $B_{n-1}$ or of type $C_{n-1}$ are the same by a result of Fomin and Zelevinsky. As a consequence, we give a categorification of cluster algebras of type $C$.