Bridgeland stability conditions on the acyclic triangular quiver

TL;DR

This paper proves that the space of Bridgeland stability conditions on the derived category of representations of the acyclic triangular quiver is contractible, enhancing understanding of its topological structure.

Contribution

It establishes the contractibility of the stability space for the acyclic triangular quiver, building on prior work on its connectivity.

Findings

The stability space is contractible.

The space is connected.

Topological structure is clarified.

Abstract

Using results in a previous paper "Non-semistable exceptional objects in hereditary categories", we focus here on studying the topology of the space of Bridgeland stability conditions on $D^b(Rep_k(Q ))$, where $Q$ is the acyclic triangular quiver (the underlying graph is the extended Dynkin diagram $\widetilde{\mathbb A}_2$). In particular, we prove that this space is contractible (in the previous paper it was shown that it is connected).