Quiver braid group action for a 3-fold crepant resolution

TL;DR

The paper constructs a faithful action of a quiver braid group on the derived category of a 3-fold crepant resolution X of the cyclic quotient singularity $\mathbb{C}^3/\mu_7$. By focusing on a concrete X with three exceptional surfaces $S_k\cong \mathbb{F}_2$ arranged in a cycle, the authors define a quiver with potential $(Q,W)$ whose associated algebraic twist group $G=\mathrm{AT}(Q,W)$ acts via spherical twists $T_i$ on $D(X)$. They verify the essential braid-type relations and an additional relation using explicit Hom-space calculations for the exceptional divisors, plus orthogonality arguments among twists, and then prove faithfulness by embedding into a known faithful Br$_4$ action and identifying $G \cong \mathrm{Br}_4$. The work extends 2D braid-action paradigms to a 3-fold setting, combining toric geometry, Calabi–Yau Serre duality, and derived-category techniques to provide an explicit, geometric realization of a quiver braid group action on $D(X)$. This contributes to understanding how quiver-based symmetries manifest in higher-dimensional crepant resolutions and their derived categories, with potential implications for mirror symmetry and noncommutative refinements of 3-fold birational geometry.

Abstract

The 3-fold cyclic quotient singularity denoted $\tfrac{1}{7}(1,2,4)$ admits a crepant resolution X with three exceptional Hirzebruch surfaces intersecting pairwise along curves. We show that the derived category D(X) carries a faithful action of a quiver braid group, where the relevant quiver is a 3-cycle encoding the intersection data.

Figures (9)

• Figure 1.1: The three exceptional surfaces in $X$
• Figure 1.2: Quiver $(Q,W=cba)$
• Figure 1.3: Junior simplex of $X$
• Figure 3.1: The fan of $\mathbb{F}_e$
• Figure 4.1: The fan $\Sigma$ of $X$

Theorems & Definitions (37)

• Theorem 1.1: Theorem \ref{['thm.main']}
• Definition 2.1
• Definition 2.2
• Lemma 2.3
• proof
• Proposition 2.4
• proof
• Proposition 2.5
• proof
• Lemma 2.6