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| Stability conditions on triangulated categories | |
|---|---|
| Name | Stability conditions on triangulated categories |
| Introduced | 2002 |
| Introduced by | Tom Bridgeland |
| Area | Algebraic geometry; Mathematical physics; Representation theory |
Stability conditions on triangulated categories are a framework introduced to study objects in triangulated categories using notions analogous to slope stability for coherent sheaves on algebraic varieties. They connect influences from Tom Bridgeland, Maxim Kontsevich, Alexander Beilinson, Paul Deligne, Pierre Deligne, David Mumford, and Shinichi Mochizuki to bring together techniques from Alexander Grothendieck, Jean-Pierre Serre, Sergei Gelfand, and Vladimir Drinfeld. This theory interacts with constructions and conjectures originating in Michael Atiyah, Isadore Singer, Simon Donaldson, Edward Witten, and Cumrun Vafa and has become central in research linking Yau–Tian–Donaldson conjecture, Homological mirror symmetry, Geometric invariant theory, and moduli problems studied by institutions such as Institute for Advanced Study and Clay Mathematics Institute.
A stability condition on a triangulated category formalizes a notion of semistability that mirrors classical definitions by David Mumford and Shigefumi Mori for sheaves on projective varieties such as Projective space, Calabi–Yau manifold, and K3 surface. The original construction by Tom Bridgeland builds on categorical foundations laid by Alexandre Grothendieck, Jean-Louis Verdier, and Pierre Deligne and connects to conjectures in Maxim Kontsevich's program for Homological mirror symmetry and proposals by Edward Witten in String theory and Topological quantum field theory.
A stability condition comprises a group homomorphism Z: K(T) → C called the central charge and a slicing P of the triangulated category T into semistable phases. This draws on K-theory work of Michael Atiyah and Friedrich Hirzebruch and cohomological formalisms of Alexander Beilinson and Joseph Bernstein. Key properties include the Harder–Narasimhan property inspired by results of David Mumford and Georges Harder, and support conditions reflecting ideas from Shinichi Mochizuki and Christophe Soulé. The space of stability conditions Stab(T) is a complex manifold under suitable hypotheses, a phenomenon connected to deformation theories developed by Klaus Hulek and Maxim Kontsevich. Group actions on Stab(T) by autoequivalences relate to the work of Paul Seidel and Richard Thomas on derived categories and braid group actions studied by Vladimir Drinfeld and Jean-Pierre Serre.
Constructions start from hearts of bounded t-structures by methods of Alexei Bondal, Dmitri Orlov, and Raphaël Rouquier and use tilting procedures reminiscent of tilting modules in representation theory of Bernhard Keller and Idun Reiten. Classical examples arise from derived categories D^b(Coh(X)) for varieties X studied by Kunihiko Kodaira, Shing-Tung Yau, Max Noether, and Francesco Severi; notable cases include K3 surface, Abelian variety, and Fano variety. Bridgeland stability on curves recovers slope stability originating with David Mumford and Enrico Arbarello, while quiver examples connect to work by Pierre Gabriel, Henning Haahr Andersen, and William Crawley-Boevey. Construction techniques also use tilting bundles from Helene Esnault-type approaches and Fourier–Mukai transforms introduced by Shigeru Mukai and expanded by Daniel Huybrechts.
Stability conditions parameterize moduli spaces of semistable objects akin to moduli of sheaves studied by David Gieseker, Simon Donaldson, Kurt Mumford, and Frances Kirwan. Wall-crossing phenomena follow wall-and-chamber structures that echo results of Maxim Kontsevich and Yuri Manin in enumerative geometry and generate identities formulated by Andrei Okounkov and Richard Thomas. Wall-crossing formulas of Kontsevich–Soibelman relate to studies by Jan Manschot, Greg Moore, and Atish Dabholkar in string-theoretic BPS counts, and to Donaldson–Thomas invariants developed by Tom Bridgeland and Kai Behrend. Techniques employ tools from symplectic geometry used by Mikhail Gromov and Yakov Eliashberg and birational geometry techniques associated with Vladimir Voevodsky and Shigefumi Mori.
Hearts of t-structures, introduced by Jean-Pierre Verdier and popularized by Alexander Beilinson, give abelian categories whose simple objects admit stability analysis. Tilting theory connects to Hugh Thomas's work and to derived Morita theory developed by Murray Gerstenhaber and Bernard Keller. Mutations of hearts mirror cluster transformations studied by Fomin Zelevinsky and relate to cluster algebras arising in studies by Vladimir Fock and Andrei Zelevinsky. These operations interact with exceptional collections examined by Daniel Orlov and geometric operations on varieties such as flops studied by Miles Reid.
In algebraic geometry, stability conditions inform birational geometry of moduli spaces of sheaves on surfaces and threefolds studied by Dmitri Orlov, Michael Thaddeus, and Brendan Hassett. In mathematical physics, connections appear in Seiberg–Witten theory, Topological string theory, and BPS state counting linked to Edward Witten, Cumrun Vafa, and Greg Moore. The interplay with homological mirror symmetry of Maxim Kontsevich and symplectic geometry contributions of Paul Seidel yields predictions tested on Calabi–Yau threefolds and K3 surfaces explored by Shing-Tung Yau and Claire Voisin.
Recent developments include studies of stability conditions on categories of singularities by Michael Ballard and David Favero, existence results in noncompact settings investigated by Tom Bridgeland and Yukinobu Toda, and connections to Donaldson–Thomas theory advanced by Kontsevich–Soibelman and Yukinobu Toda. Active research explores spaces of stability conditions with actions of mapping class groups studied by John Hempel and autoequivalence groups described by Alexei Bondal and Christopher Brav. Contemporary directions involve interactions with derived algebraic geometry of Jacques Lurie, categorification programs of Mikhail Khovanov, and arithmetic aspects related to conjectures of Pierre Deligne and Kazuya Kato.