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Stability Conditions and Algebraic Hearts for Acyclic Quivers

Takumi Otani, Dongjian Wu

TL;DR

The paper addresses stability conditions on the derived category $\mathcal{D}^b(Q)$ of a finite connected acyclic quiver $Q$ and shows that every stability condition $\sigma=(Z,\mathcal{P})$ has a heart $\mathcal{P}(\theta,\theta+1]$ that is algebraic after a phase rotation. The approach combines the support property, Kac's root-system correspondence, and cones built from the imaginary roots to locate a phase gap, proving the existence of an algebraic heart and thus connecting all stability conditions to algebraic hearts via rotation and simple tilts. Consequently, the authors prove the connectedness of $\mathrm{Stab}(\mathcal{D}^b(Q))$ and that every $\sigma$ admits a monochromatic full $\sigma$-exceptional collection, leveraging the algebraic exchange graph and DK-type results. The results unify and extend known cases (Dynkin, affine Dynkin, generalized Kronecker), providing a robust framework for understanding the topology of stability spaces and their links to root systems and exceptional collections. The work lays groundwork for potential Frobenius-structure interpretations and contractibility questions within stability spaces of quiver-derived categories.

Abstract

We study stability conditions on the derived category of a finite connected acyclic quiver. We prove that, for any stability condition on the derived category, its heart can be obtained from an algebraic heart by a rotation of phases. Consequently, we establish the connectedness of the space of stability conditions. Furthermore, we prove that every stability condition $σ$ admits a full $σ$-exceptional collection.

Stability Conditions and Algebraic Hearts for Acyclic Quivers

TL;DR

The paper addresses stability conditions on the derived category of a finite connected acyclic quiver and shows that every stability condition has a heart that is algebraic after a phase rotation. The approach combines the support property, Kac's root-system correspondence, and cones built from the imaginary roots to locate a phase gap, proving the existence of an algebraic heart and thus connecting all stability conditions to algebraic hearts via rotation and simple tilts. Consequently, the authors prove the connectedness of and that every admits a monochromatic full -exceptional collection, leveraging the algebraic exchange graph and DK-type results. The results unify and extend known cases (Dynkin, affine Dynkin, generalized Kronecker), providing a robust framework for understanding the topology of stability spaces and their links to root systems and exceptional collections. The work lays groundwork for potential Frobenius-structure interpretations and contractibility questions within stability spaces of quiver-derived categories.

Abstract

We study stability conditions on the derived category of a finite connected acyclic quiver. We prove that, for any stability condition on the derived category, its heart can be obtained from an algebraic heart by a rotation of phases. Consequently, we establish the connectedness of the space of stability conditions. Furthermore, we prove that every stability condition admits a full -exceptional collection.

Paper Structure

This paper contains 13 sections, 27 theorems, 45 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Stability condition
  3. Stability condition
  4. Global dimension function and totally semistable stability condition
  5. Algebraic Heart
  6. Root system and Kac's theorem
  7. Root systems associated with acyclic quivers
  8. Indecomposable objects and Kac's theorem
  9. Stability conditions for acyclic quiver
  10. Stability condition and algebraic heart
  11. Proof of Theorem \ref{['thm : main 1']}
  12. Connectedness of the space of stability conditions
  13. Full $\sigma$-exceptional collections

Key Result

Theorem 1.1

For any stability condition $\sigma = (Z, \mathcal{P})$ on $\mathcal{D}^b(Q)$, there exists a real number $\theta \in \mathbb{R}$ such that $\mathcal{P}(\theta, \theta + 1]$ is an algebraic heart.

Figures (1)

  • Figure 1: An example of the case $n = 3$ and $m = 2$. Left: The images of cones $A_i^\pm$. Right: The cones $C^\pm_j$ and phases $\theta$ and $\theta'$.

Theorems & Definitions (54)

  • Theorem 1.1: Theorem \ref{['thm : main 1']}
  • Theorem 1.2: Theorem \ref{['thm : simple tilts and rotation']}
  • Theorem 1.3: Theorem \ref{['thm : stability condition and full exceptional collections']}
  • Definition 2.1: Bri
  • Definition 2.2: KS
  • Definition 2.3: Bri
  • Proposition 2.4: Bri
  • Definition 2.5: IQ
  • Lemma 2.6
  • Proof
  • ...and 44 more