An Introduction to Spaces of Bridgeland Stability Conditions with Examples from Representation Theory

Core Concepts

This article provides an introduction to Bridgeland stability conditions, focusing on their role in encoding homological properties of triangulated categories and reviewing their application in representation theory, particularly in the context of quiver categories from marked surfaces.

Abstract

Barbieri, A. (2024). Spaces of Bridgeland stability conditions in representation theory. arXiv.

This article aims to introduce the concept of Bridgeland stability conditions, particularly for those interested in module categories, and illustrate their application in representation theory using the example of quiver categories from marked surfaces.

Key Insights Distilled From

Spaces of Bridgeland stability conditions in representation theory

by Anna Barbier... at arxiv.org 11-04-2024

https://arxiv.org/pdf/2409.15131.pdf

Deeper Inquiries

How can the understanding of Bridgeland stability conditions be leveraged to solve open problems in algebraic geometry or mathematical physics?

Bridgeland stability conditions provide a powerful tool for studying triangulated categories, which arise naturally in algebraic geometry and mathematical physics. Here are some ways they can be leveraged to address open problems:
Algebraic Geometry:

Construction of Moduli Spaces: One of the main motivations for introducing stability conditions was to construct well-behaved moduli spaces of objects in triangulated categories. While classical GIT stability has been successful for abelian categories, it doesn't directly translate to the derived setting. Bridgeland stability offers a way to define stability for complexes, potentially leading to new moduli spaces of sheaves, complexes on varieties, or even derived categories themselves.
Understanding Derived Categories: Bridgeland stability conditions encode rich information about the structure of a triangulated category. Studying the geometry of the stability manifold, its wall-crossing phenomena, and the associated moduli spaces can shed light on the intricate structure of derived categories of algebraic varieties. This can lead to progress on problems like characterizing derived equivalences, understanding autoequivalence groups, and classifying varieties with equivalent derived categories.
Mirror Symmetry: Bridgeland stability conditions play a crucial role in mirror symmetry, a profound duality between symplectic and algebraic geometry. The stability manifold is conjectured to be closely related to the base of the SYZ fibration on the mirror side. Understanding this connection can lead to new insights into mirror symmetry and its implications for enumerative geometry and Gromov-Witten theory.
Mathematical Physics:

String Theory and D-branes: Bridgeland stability conditions were inspired by Douglas's work on Π-stability for D-branes in string theory. The stability manifold can be interpreted as a space of allowed vacua in string theory, and its geometry is expected to have deep connections with the physics of these vacua.
Topological String Theory: The study of Donaldson-Thomas invariants, which count stable objects in certain derived categories, has close ties to topological string theory. Bridgeland stability conditions provide a natural framework for defining and studying these invariants, potentially leading to new connections between algebraic geometry and string theory.
Gauge Theory and Integrable Systems: The wall-crossing phenomena observed in the stability manifold have intriguing connections with wall-crossing formulas in gauge theory and the theory of integrable systems. Exploring these connections can lead to a deeper understanding of both areas and potentially uncover new dualities.
Overall, Bridgeland stability conditions provide a rich and versatile framework for addressing open problems in algebraic geometry and mathematical physics. Their deep connections to moduli spaces, derived categories, mirror symmetry, string theory, and gauge theory make them a powerful tool for advancing our understanding of these fundamental areas of mathematics and physics.

Could there be alternative mathematical frameworks that capture similar properties of triangulated categories without relying on the concept of stability conditions?

While Bridgeland stability conditions have proven incredibly fruitful, it's natural to wonder if alternative frameworks could capture similar properties of triangulated categories. Here are some possibilities:

t-structures and their mutations: t-structures are fundamental to the definition of stability conditions, providing the abelian categories where stability functions are defined. One could explore alternative notions of "stability" directly within the framework of t-structures, perhaps by considering different types of filtrations or decompositions within the heart of a t-structure. Mutations of t-structures, which generalize tilting, could also offer a way to move between different "stability conditions" without explicitly using central charges.
Slicings and their deformations: Slicings, as generalizations of t-structures, offer another potential avenue. One could investigate different ways to deform slicings, potentially leading to a notion of "stability" based on the behavior of objects under these deformations. This approach might be particularly relevant for categories where defining suitable central charges is challenging.
Non-commutative geometry: Triangulated categories often arise in non-commutative geometry, where they play the role of derived categories of non-commutative spaces. Techniques from this field, such as cyclic homology and K-theory, could potentially be used to define alternative notions of "stability" that capture the non-commutative nature of these categories.
Higher categorical structures: Triangulated categories are inherently "1-categorical," but there's growing interest in higher categorical structures, such as stable ∞-categories. These frameworks might offer a more natural setting for studying "stability," potentially leading to new insights that are not visible in the classical triangulated setting.
It's important to note that these are just potential directions, and it remains to be seen if they can lead to frameworks as powerful and versatile as Bridgeland stability conditions. Nevertheless, exploring alternative approaches is crucial for deepening our understanding of triangulated categories and their applications.

How does the concept of stability, as explored in this mathematical context, relate to the notion of stability in other scientific disciplines, such as physics or systems biology?

The concept of "stability" in the context of Bridgeland stability conditions, while deeply rooted in abstract mathematics, shares intriguing parallels with the notion of stability in other scientific disciplines. Here's a comparative exploration:
Mathematics (Bridgeland Stability):

Definition: An object is considered "stable" if it cannot be decomposed into "smaller" objects in a specific way, determined by a central charge and a slicing of the triangulated category.
Focus: The focus is on the intrinsic structure of objects and their behavior under morphisms within a categorical framework.
Goal: To classify and understand objects based on their stability properties, often leading to the construction of moduli spaces.
Physics (e.g., Stability of Systems):

Definition: A physical system is considered "stable" if it returns to its equilibrium state after being subjected to small perturbations.
Focus: The focus is on the system's dynamics and its response to external influences.
Goal: To predict the long-term behavior of the system and determine its resistance to change.
Systems Biology (e.g., Stability of Biological Networks):

Definition: A biological network (e.g., gene regulatory network) is considered "stable" if it maintains its functionality despite fluctuations in its components or environment.
Focus: The focus is on the robustness and adaptability of biological systems.
Goal: To understand how biological systems maintain stability and adapt to changing conditions.
Common Themes:

Resistance to Decomposition/Perturbation: In all cases, "stability" implies a certain resistance to change. In mathematics, it's resistance to decomposition; in physics, it's resistance to perturbations; and in biology, it's resistance to fluctuations.
Classification and Prediction: The concept of "stability" allows for classification and prediction. In mathematics, it classifies objects; in physics, it predicts system behavior; and in biology, it helps understand system robustness.
Underlying Structure and Dynamics: "Stability" often reflects deeper underlying structures and dynamics. In mathematics, it reveals categorical structures; in physics, it reflects the system's governing equations; and in biology, it reveals network architecture and interactions.
Differences:

Abstraction Level: The level of abstraction varies significantly. Bridgeland stability is highly abstract, while stability in physics and biology is more concrete and often empirically observable.
Formalism: The mathematical formalism used to define and study "stability" differs across disciplines. Bridgeland stability relies on category theory, while physics and biology use differential equations, network theory, and statistical mechanics.
Despite the differences, the shared themes highlight a fundamental connection between these seemingly disparate notions of "stability." They all reflect a desire to understand how systems, whether mathematical objects, physical entities, or biological networks, maintain their integrity and resist change. This suggests that the mathematical tools developed for studying Bridgeland stability might inspire new approaches to understanding stability in other scientific disciplines, and vice versa.