Quiver braid group action for a 3-fold crepant resolution

TL;DR

This paper demonstrates that the derived category of a specific crepant resolution of a 3-fold cyclic quotient singularity admits a faithful action of a quiver braid group, revealing new symmetries.

Contribution

It introduces a quiver braid group action on the derived category of a 3-fold crepant resolution based on the intersection data of Hirzebruch surfaces.

Findings

Derived category D(X) admits a faithful quiver braid group action.

The quiver is a 3-cycle encoding intersection data.

The singularity is a 1/7(1,2,4) cyclic quotient.

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.