Cambrian lattices are fractionally Calabi-Yau via 2-cluster combinatorics

Reading constructed a Cambrian lattice $C_{\mathrm{\Gamma }}$ for each oriented finite type Coxeter diagram $\mathrm{\Gamma }$. We show that the derived category of representations of $C_{\mathrm{\Gamma }}$ is fractionally Calabi-Yau for any $\mathrm{\Gamma }$, confirming a conjecture of Chapoton. This extends a result of Rognerud for Cambrian lattices of type $A$ with linear orientation, better known as Tamari lattices. If $\mathrm{\Gamma }$ is crystallographic, then $C_{\mathrm{\Gamma }}$ is given by the lattice of torsion classes of any hereditary algebra $\mathrm{\Lambda }$ of type $\mathrm{\Gamma }$. In this case we introduce and study a class of intervals in $C_{\mathrm{\Gamma }}$ whose combinatorics matches the combinatorics of $2$-cluster tilting objects in the 2-cluster category of $\mathrm{\Lambda }$. This allows us to compute the Calabi-Yau dimension of $C_{\mathrm{\Gamma }}$.