Geometric crystals and Cluster ensembles in Kac-Moody setting

For a Kac-Moody group $G$, double Bruhat cells $G^{u,e}$ ($u$ is a Weyl group element) have positive geometric crystal structures. In arXiv:1210.2533, it is shown that there exist birational maps between `cluster tori’ ${\mathcal{X}}_{\mathrm{\Sigma }}$ (resp. ${\mathcal{A}}_{\mathrm{\Sigma }}$) and $G_{\mathrm{A}\mathrm{d}}^{u,e}$ (resp. $G^{u,e}$), and they are extended to regular maps from cluster $\mathcal{X}$ (resp. $\mathcal{A}$) -varieties to $G_{\mathrm{A}\mathrm{d}}^{u,e}$ (resp. $G^{u,e}$). The aim of this article is to construct certain positive geometric crystal structures on the cluster tori ${\mathcal{X}}_{\mathrm{\Sigma }}$ and ${\mathcal{A}}_{\mathrm{\Sigma }}$ by presenting their explicit formulae. In particular, the geometric crystal structures on the tori ${\mathcal{A}}_{\mathrm{\Sigma }}$ are obtained by applying the twist map. As a corollary, we see the sets of ${\mathbb{Z}}^T$-valued points of the cluster varieties have plural structures of crystals.