Exchange graphs and Ext quivers

We study the oriented exchange graph $\textrm{EG}^\circ(\Gamma_{N}\,Q)$ 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 $\textrm{EG}^\circ(\Gamma_{N}\,Q)$ by the Seidel-Thomas braid group is the exchange graph $\textrm{CEG}_{N-1}(Q)$ 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 $\mathcal{D}(\Gamma_{N}\,Q)$.

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