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| Stability conditions | |
|---|---|
| Name | Stability conditions |
| Field | Algebraic geometry; Mathematical physics; Category theory |
| Introduced | 2002 |
| Introduced by | Tom Bridgeland |
| Notable applications | Mirror symmetry; Donaldson–Thomas theory; Moduli of sheaves |
Stability conditions
Stability conditions are structures on triangulated categories that organize objects by a notion of "phase" and "slope", providing a framework to construct and study moduli spaces of semistable objects. Originating in interactions between derived categories, complex geometry, and string theory, they connect ideas from Tom Bridgeland, Maxim Kontsevich, Edward Witten, Simon Donaldson, and Richard Thomas and serve as a bridge between algebraic geometry and mathematical physics through links to mirror symmetry, wall-crossing phenomena, and enumerative invariants.
A stability condition on a triangulated category assigns to each object a central charge in the complex plane and a slicing by phases so that semistable objects satisfy an analogue of the Harder–Narasimhan property. The formalism generalizes classical notions such as Gieseker stability for coherent sheaves and slope stability on curves, while interfacing with derived equivalences like those studied by Alexander Beilinson, Paul Seidel, and Denis Auroux. Stability conditions produce wall-and-chamber decompositions in spaces of central charges, interacting with structures studied by Maxim Kontsevich–Soibelman and shaping moduli controlled by results of Georgios Dimitrov and Tom Bridgeland.
Motivated by problems in enumerative geometry and mirror symmetry, the modern concept of stability conditions was introduced by Tom Bridgeland building on earlier work on moduli of sheaves by Simon Donaldson and Klaus Hulek and on stability of vector bundles by David Mumford and Takuro Mochizuki. Insights from Edward Witten and Cumrun Vafa in string theory, as well as predictions of Maxim Kontsevich in homological mirror symmetry, prompted the algebraic formalization. Subsequent developments linked the subject to wall-crossing formulas of Maxim Kontsevich and Yan Soibelman and to enumerative theories by Richard Thomas, Dmitri Joyce, and Bridgeland–Macri.
Formally, a stability condition σ = (Z, P) on a triangulated category D consists of a group homomorphism Z: K(D) → C called the central charge and a slicing P = {P(ϕ)}_{ϕ∈R} of full additive subcategories P(ϕ) of semistable objects of phase ϕ. The central charge must satisfy the support property relative to a fixed norm on K(D) ⊗ R, an axiom analogous to boundedness in classical moduli problems as in the work of Geometric Invariant Theory by David Mumford. The existence of Harder–Narasimhan filtrations in this setting ensures every object decomposes into semistables, paralleling results of Klaus Hulek and Takuro Mochizuki for sheaves.
Key examples include stability conditions on the derived category of coherent sheaves D^b(Coh(X)) for curves X like Elliptic curves and projective lines, reproducing slope stability of vector bundles studied by Atiyah and Mumford. For K3 surfaces, classification results intertwine with lattices and the work of Shigeru Mukai and Igor Dolgachev, while stability on quiver-derived categories connects to King’s notion of θ-stability for representations of quivers introduced by Alastair King. On the derived category of the projective plane, constructions involve exceptional collections related to Alexander Beilinson and braid group actions studied by Paul Seidel.
In algebraic geometry, stability conditions underpin constructions of moduli spaces of semistable objects, enabling definitions of Donaldson–Thomas invariants and proving wall-crossing formulas conjectured by Maxim Kontsevich and Yan Soibelman. They inform birational geometry via wall-crossing for moduli of sheaves as in the work of Arend Bayer and Emanuele Macrì, influencing the study of derived equivalences like those of Bridgeland–King–Reid. In physics, stability conditions formalize BPS state counting in supersymmetric theories described by Edward Witten and Cumrun Vafa, relate to D-brane categories predicted in homological mirror symmetry by Maxim Kontsevich, and enter the study of spectrum stability and wall-crossing in N=2 supersymmetric gauge theory contexts illuminated by Seiberg–Witten theory.
Important structural results include the local homeomorphism theorem: the space Stab(D) of stability conditions is a complex manifold locally modeled on Hom(K(D), C), proven by Tom Bridgeland under the support property. Wall-crossing behavior of invariants under variation of stability conditions is governed by wall-crossing formulas due to Kontsevich–Soibelman and refined by Dmitri Joyce and collaborators. Existence and uniqueness results for stability conditions in specific geometric settings, such as on K3 surfaces and quiver categories, rely on lattice-theoretic methods of Shigeru Mukai and tilting techniques connected to Happel–Reiten–Smalø theory.
Computational approaches to stability conditions use tilting of t-structures, Bridgeland slicing algorithms, and Harder–Narasimhan recursion to enumerate semistable objects in concrete categories; implementations draw on techniques from Alastair King for quivers and on wall-crossing computations inspired by Kontsevich–Soibelman formulae. Moduli spaces of semistable objects constructed via stability conditions admit structure theorems relating them to classical moduli of sheaves studied by Simon Donaldson and David Gieseker, and to birational models studied by Mukai and Arend Bayer. Wall-and-chamber decompositions in Stab(D) guide explicit enumerative calculations of Donaldson–Thomas invariants linked to predictions from homological mirror symmetry of Maxim Kontsevich and physical BPS counts from Edward Witten.