# Stability conditions for contraction algebras

## TL;DR

This paper characterizes the space of Bridgeland stability conditions on contraction algebras from 3-fold flops, revealing a universal cover of hyperplane arrangements and deriving key algebraic and geometric corollaries.

## Contribution

It provides a comprehensive description of the stability manifold for contraction algebras, connecting it to hyperplane arrangements and ADE root systems, and proves several important corollaries.

## Key findings

- Stability manifold is the universal cover of a hyperplane arrangement.
- Proof of faithfulness of pure braid group actions without normal forms.
- Classification of tilting complexes in contraction algebra derived categories.

## Abstract

This paper gives a description of the full space of Bridgeland stability conditions on the bounded derived category of a contraction algebra associated to a 3-fold flop. The main result is that the stability manifold is the universal cover of a naturally associated hyperplane arrangement, which is known to be simplicial, and in special cases is an ADE root system. There are four main corollaries: (1) a short proof of faithfulness of pure braid group actions in both algebraic and geometric settings, the first that avoid normal forms, (2) a classification of tilting complexes in the derived category of a contraction algebra, (3) contractibility of a certain stability space associated to the flop, and (4) a new proof of the $K(\pi,1)$-theorem in various finite settings, which includes ADE braid groups.