Quiver potential and King theta stability data

Work over $\mathbb{C}$. A finite quiver is a tuple $Q=(Q_0,Q_1,s,t)$ of finite vertex and arrow sets with source and target maps $s,t:Q_1\to Q_0$. It is 2-acyclic when it has no loops and no oriented cycles of length two. Its path algebra $\mathbb{C}Q$ has basis the finite directed paths, including one length-zero idempotent at each vertex, and multiplication by concatenation when endpoints match and by zero otherwise.

A finite potential $W$ is a finite complex linear combination of directed cycles, considered modulo cyclic permutation of the factors. For an arrow $a$, the cyclic derivative of a cycle $a_1\cdots a_{\mathrm{\ell }}$ is ${\mathrm{\partial }}_a(a_1\cdots a_{\mathrm{\ell }})={\sum }_{j:a_j=a}{a}_{j+1}\cdots a_{\mathrm{\ell }}{a}_1\cdots a_{j-1}$, extended linearly. The Jacobian algebra is $J(Q,W)=\mathbb{C}Q\mathrm{/}⟨{\mathrm{\partial }}_{a}W:a\in Q_1⟩$, where the denominator is the two-sided ideal generated by these derivatives. No finite-dimensionality of $J(Q,W)$ is assumed. Write $\mathcal{A}={\mathrm{mod}}_{\mathrm{f}\mathrm{d}}J(Q,W)$ for its abelian category of finite-dimensional unital left modules, equivalently finite-dimensional quiver representations satisfying the Jacobian relations.

For $E\in \mathcal{A}$, its dimension vector is $d(E)\in {\mathbb{N}}^{Q_0}$, with coordinate ${\dim }_{\mathbb{C}}(e_{i}E)$ at vertex $i$; extend this additively to ${\mathbb{Z}}^{Q_0}$. A weight is $\theta \in {\mathrm{Hom}}_{\mathbb{Z}}({\mathbb{Z}}^{Q_0},\mathbb{R})$, and $\theta (E)$ abbreviates $\theta (d(E))$. A nonzero representation $E$ is King $\theta$-semistable when $\theta (E)=0$ and every subobject $0\mathrm{\ne }A\subseteq E$ satisfies $\theta (A)\le 0$. It is $\theta$-stable when $\theta (E)=0$ and $\theta (A)<0$ for every proper nonzero subobject $A⊊E$. This abelian-category notion is distinct from a slicing on a triangulated category.