Exchange graphs and Ext quivers

Abstract: We study the oriented exchange graph $\textrm{EG}<sup>\circ(\Gamma_{N}\,Q)$ of reachable hearts in the finite-dimensional derived category $\mathcal{D}(\Gamma_{N}\,Q)$ of the CY-$N$ Ginzburg algebra $\Gamma_{N}Q$ 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' $$\mathcal{L}:\mathcal{D}(Q)\to\mathcal{D}(\Gamma_{N}\,Q)$$. We build on this to show that the quotient of $\textrm{EG}<sup>\circ(\Gamma_{N}\,Q)$ by the Seidel-Thomas braid group is the exchange graph $\textrm{CEG}<em>{N-1}(Q)$ of cluster tilting sets in the (higher) cluster category $\mathcal{C}</em>{N-1}(Q)$. 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 $\mathcal{D}(\Gamma_{N}\,Q)$.