The space of stability conditions for quivers with two vertices

TL;DR

This paper investigates the structure of the space of stability conditions for quivers with two vertices, revealing how the covering map behaves depending on the number of arrows, with implications for understanding derived categories.

Contribution

It characterizes the local homeomorphism from the stability space to ^2 for quivers with two vertices, showing it is a covering map after removing certain line arrangements, depending on the number of arrows.

Findings

For one or two arrows, the map is a covering map outside a line arrangement.

For more than two arrows, uncountably many lines must be removed for the map to be a covering.

The structure of the stability space varies significantly with the number of arrows.

Abstract

The purpose of this article is to study the space of stability conditions $\stab(P_n)$ on the bounded derived category $\D^b(P_n)$ of finite dimensional representations of a quiver $P_n$ with two vertices and $n$ parallel arrows. There is a local homeomorphism $\z:\stab(P_n)\rightarrow\C^2$. We show that, when the number of arrows is one or two, the map is a covering map if we restrict it to the complement of a line arrangement. When the number of arrows is greater than two we need to remove uncountably many lines to obtain a covering map.