We will explain how the cluster-tilting subcategories of a cluster category give rise to A- and X-cluster structures in the sense of Fock-Goncharov. In particular, as T varies through the cluster-tilting subcategories, the Grothendieck groups $K_0(T)$ and $K_0(\mathrm{fd}\, T)$ admit families of homomorphisms tropicalizing the birational maps used to build the corresponding A- and X-cluster varieties.

Using our framework, we construct an X-cluster character for any cluster category and relate it to the A-cluster character. We also show the existence of a canonical quantization of any Hom-finite exact cluster category, extending examples originally due to Geiß-Leclerc-Schröer.

This is joint work with Matthew Pressland (Glasgow).