Total stability functions for type $\mathbb{A}$ quivers

Abstract

For a quiver $Q$ of Dynkin type ${\mathbb{A}}_n$, we give a set of $n-1$ inequalities which are necessary and sufficient for a linear stability condition (a.k.a. central charge) $Z:K_0(Q)\to \mathbb{C}$ to make all indecomposable representations stable. We furthermore show that these are a minimal set of inequalities defining the space $\mathcal{T}\mathcal{S}(Q)$ of total stability conditions, considered as an open subset of ${\mathbb{R}}^{Q_0}\times ({\mathbb{R}}_{>0}{)}^{Q_0}$. We then use these inequalities to show that each fiber of the projection of $\mathcal{T}\mathcal{S}(Q)$ to $({\mathbb{R}}_{>0}{)}^{Q_0}$ is linearly equivalent to $\mathbb{R}\times {\mathbb{R}}_{>0}^{Q_1}$.

11 pages. v2: totally rewritten in terms of Bridgeland stability conditions instead of classical slope function