---
tag: F0G92Q55DM
kind: concept
name: Stability condition
msc: [18G80, 14F08]
sources:
  - url: https://en.wikipedia.org/wiki/Bridgeland_stability_condition
same_as:
  wikidata: Q96373948
  wikipedia: Bridgeland stability condition
---

## Summary

A stability condition on a [[triangulated category|T2V7GJJ2EN]] is a pair made of a central charge and a slicing.
The central charge is a homomorphism from the Grothendieck group to $\C$.
The slicing assigns to each real number a class of semistable objects of that phase.
The notion was introduced by Tom Bridgeland to give a uniform meaning to stability of sheaves and of quiver representations.
The stability conditions on a fixed category form a space with a natural topology.
The study of this space links algebraic geometry with string theory and mirror symmetry.

## Explanations

### L1 human

A stability condition gives each object of a category a complex number, its central charge, which records a kind of mass and an angle.
The angle, divided by $\pi$, is called the phase.
The semistable objects are the basic building blocks, and every object splits in a unique way into layers of semistable pieces whose phases decrease.
For a smooth projective curve, the charge $Z(E) = -\deg E + i \operatorname{rk} E$ of a coherent sheaf $E$ recovers the classical slope stability of vector bundles.
Each stability condition has a heart, which is the heart of a bounded [[t-structure|2GD9V7SA0E]], so it refines the choice of a t-structure.