Exchange graphs and Ext quivers

We study the oriented exchange graph EG∘(ΓNQ) of reachable hearts in the finite-dimensional derived category D(ΓNQ) of the CY-N Ginzburg algebra ΓNQΓNQ associated to an acyclic quiver Q . We show that any such heart is induced from some heart in the bounded derived category D(Q)D(Q) via some ‘Lagrangian immersion’ L:D(Q)→D(ΓNQ). We build on this to show that the quotient of EG∘(ΓNQ) by the Seidel–Thomas braid group is the exchange graph CEGN−1(Q)CEGN−1(Q) of cluster tilting sets in the (higher) cluster category CN−1(Q)CN−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 D(ΓNQ).