Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Stability scattering diagrams and quiver coverings

Qiyue Chen, Travis Mandel, Fan Qin

TL;DR

The paper establishes that, under a nice-grading assumption on quiver covers, stability scattering diagrams restrict to give equivalent diagrams for the quotient quivers with potential, thereby linking wall-crossing data across coverings. It applies this to quivers from marked surfaces, proving that the stability diagram matches the cluster diagram except for the once-punctured torus, and that the bracelets basis coincides with the theta basis for the stability diagram in triangulable surfaces. The work further shows that non-wrapping potentials permit a broader equivalence for cyclic and solvable coverings, providing a robust framework for transferring canonical bases (bracelets vs. theta) across surface-derived quivers and their Jacobian algebras. Collectively, these results highlight the stability scattering diagram as a natural, canonical object for constructing and comparing bases in surface-related cluster- and Hall-algebra contexts.

Abstract

Given a covering of a quiver (with potential), we show that the associated Bridgeland stability scattering diagrams are related by a restriction operation under the assumption of admitting a nice grading. We apply this to quivers with potential associated to marked surfaces. In combination with recent results of the second and third authors, our findings imply that the bracelets basis for a once-punctured closed surface coincides with the theta basis for the associated stability scattering diagram, and these stability scattering diagrams agree with the corresponding cluster scattering diagrams of Gross-Hacking-Keel-Kontsevich except in the case of the once-punctured torus.

Stability scattering diagrams and quiver coverings

TL;DR

The paper establishes that, under a nice-grading assumption on quiver covers, stability scattering diagrams restrict to give equivalent diagrams for the quotient quivers with potential, thereby linking wall-crossing data across coverings. It applies this to quivers from marked surfaces, proving that the stability diagram matches the cluster diagram except for the once-punctured torus, and that the bracelets basis coincides with the theta basis for the stability diagram in triangulable surfaces. The work further shows that non-wrapping potentials permit a broader equivalence for cyclic and solvable coverings, providing a robust framework for transferring canonical bases (bracelets vs. theta) across surface-derived quivers and their Jacobian algebras. Collectively, these results highlight the stability scattering diagram as a natural, canonical object for constructing and comparing bases in surface-related cluster- and Hall-algebra contexts.

Abstract

Given a covering of a quiver (with potential), we show that the associated Bridgeland stability scattering diagrams are related by a restriction operation under the assumption of admitting a nice grading. We apply this to quivers with potential associated to marked surfaces. In combination with recent results of the second and third authors, our findings imply that the bracelets basis for a once-punctured closed surface coincides with the theta basis for the associated stability scattering diagram, and these stability scattering diagrams agree with the corresponding cluster scattering diagrams of Gross-Hacking-Keel-Kontsevich except in the case of the once-punctured torus.
Paper Structure (8 sections, 31 theorems, 27 equations, 10 figures)

This paper contains 8 sections, 31 theorems, 27 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Scattering Diagrams
  3. Restriction of Scattering Diagrams
  4. Covering of Representations
  5. Applications to Surfaces
  6. Projection of Surface Jacobian Algebras
  7. Proofs of Theorem \ref{['thm:1p_surface_diagrams']} and Theorem \ref{['thm:theta-stability']}
  8. Cyclic coverings with non-wrapping potentials

Key Result

Theorem 1.3

Let $\pi:(Q,W)\rightarrow (\overline{Q},\overline{W})$ be a covering as above. If every indecomposable projective module of $J$ has a nice grading, then the restriction $\overline{\mathfrak{D}^\mathrm{st}(J)}$ is equivalent to $\mathfrak{D}^\mathrm{st}(\overline{J})$.

Figures (10)

  • Figure 4.1: A $2:1$ covering of a Kronecker quiver
  • Figure 5.2: An ideal triangulated torus with one puncture $p$
  • Figure 5.3: The adjacency quiver for the torus, shown in polygon presentation and compact form
  • Figure 5.4: Maps on arrows
  • Figure 5.5: The leftmost is a nonzero path in $J$, while the right two equal zero.
  • ...and 5 more figures

Theorems & Definitions (60)

  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6: Theorem \ref{['thm:non-wrap']}
  • Corollary 1.7
  • Theorem 2.1: gross2018canonical
  • Theorem 2.2: gross2018canonical
  • Lemma 2.3
  • proof
  • Remark 2.4
  • ...and 50 more