Exchange graphs and Ext quivers

We study the oriented exchange graph ${\text{EG}}^{\circ }({\mathrm{\Gamma }}_{N}Q)$ of reachable hearts in the finite-dimensional derived category $\mathcal{D}({\mathrm{\Gamma }}_{N}Q)$ of the CY-$N$ Ginzburg algebra ${\mathrm{\Gamma }}_{N}Q$ associated to an acyclic quiver $Q$. We show that any such heart is induced from some heart in the bounded derived category $\mathcal{D}(Q)$ via some `Lagrangian immersion’ $\mathcal{L}:\mathcal{D}(Q)\to \mathcal{D}({\mathrm{\Gamma }}_{N}Q)$. We build on this to show that the quotient of ${\text{EG}}^{\circ }({\mathrm{\Gamma }}_{N}Q)$ by the Seidel-Thomas braid group is the exchange graph ${\text{CEG}}_{N-1}(Q)$ of cluster tilting sets in the (higher) cluster category ${\mathcal{C}}_{N-1}(Q)$. As an application, we interpret Buan-Thomas’ coloured quiver for a cluster tilting set in terms of the Ext quiver of any corresponding heart in $\mathcal{D}({\mathrm{\Gamma }}_{N}Q)$.