Exchange graphs and Ext quivers

TL;DR

This paper explores the structure of exchange graphs of hearts in derived categories related to CY-N Ginzburg algebras, revealing their connection to cluster categories and providing new interpretations of cluster tilting sets.

Contribution

It demonstrates that hearts in the derived category of a CY-N Ginzburg algebra originate from hearts in the derived category of an acyclic quiver, linking exchange graphs to higher cluster categories.

Findings

Hearts are induced from hearts in the derived category of the quiver.

The quotient of the exchange graph by the braid group relates to higher cluster categories.

Provides a new interpretation of Buan-Thomas' coloured quiver in terms of Ext quivers.

Abstract

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 $\mathcal{D}(\Gamma_{N}\,Q)$.