Kontsevich–Soibelman wall-crossing

Kontsevich–Soibelman wall-crossing
Name	Kontsevich–Soibelman wall-crossing
Field	Algebraic geometry, Mathematical physics, Symplectic geometry
Introduced	2008
Authors	Maxim Kontsevich, Yan Soibelman

Kontsevich–Soibelman wall-crossing is a structural principle in algebraic geometry and mathematical physics describing discontinuities of invariants under variation of stability conditions, connecting work of leading figures such as Maxim Kontsevich, Yan Soibelman, Donaldson, Thomas, and Tom Bridgeland. It formalizes how enumerative invariants change across walls in parameter spaces studied by researchers at institutions like IHÉS, Institute for Advanced Study, and Perimeter Institute. The theory interacts with developments by Alexander Beilinson, Paul Seidel, Richard Thomas, Ed Witten, and Cumrun Vafa.

Introduction

The origin traces to interactions among ideas from Maxim Kontsevich and Yan Soibelman after insights by Simon Donaldson and Richard Thomas on stable objects; later developments involved contributions from Tom Bridgeland, Denis Auroux, Dmitry Orlov, and Mikhail Khovanov. The concept links moduli problems studied at Princeton University, Harvard University, and Cambridge University with wall phenomena investigated by Alexander Polyakov, Edward Witten, and Nathan Seiberg. It plays a role alongside frameworks by Andrei Okounkov, Rahul Pandharipande, Sheldon Katz, and Kentaro Hori.

Mathematical Framework and Definitions

The setting uses moduli of objects in triangulated categories studied by Paul Hacking, stability conditions pioneered by Tom Bridgeland, and deformation theory influenced by Maxim Kontsevich. One considers central charges as in work by Bridgeland and filtrations reminiscent of constructions by Alexander Beilinson and Joseph Bernstein. The algebraic structures employ Hall algebras developed by Philip Hall and motivic refinements related to ideas from Pierre Deligne, Alexander Grothendieck, and Jean-Pierre Serre. Key definitions reference Calabi–Yau categories treated by Paul Aspinwall, Michel Van den Bergh, and Bernhard Keller, while homological mirror symmetry contexts recall contributions by Kontsevich and Paul Seidel.

Wall-Crossing Formula and Invariants

The central formula attributes discontinuities of invariants to product decompositions first articulated by Kontsevich and Soibelman and uses techniques familiar from work by Nicholas Katz, Davesh Maulik, Rahul Pandharipande, and Yinbang Lin. The algebraic identity parallels identities in quantum groups explored by Vladimir Drinfeld and George Lusztig and reflects cluster transformations studied by Sergey Fomin and Andrei Zelevinsky. Invariants resemble Donaldson–Thomas counts introduced by Simon Donaldson and Richard Thomas, motivic enhancements advocated by J. Denef and F. Loeser, and categorifications inspired by Mikhail Khovanov and Jacob Lurie. Wall-crossing statements were influenced by physical indices from Edward Witten, Cumrun Vafa, and G. Moore.

Examples and Applications

Examples include moduli of sheaves on Calabi–Yau threefolds investigated by D. Joyce, constructions on K3 surfaces studied by Shigeru Mukai and Daniel Huybrechts, and quiver representations analyzed by William Crawley-Boevey and Henning Krause. Applications reach enumerative predictions tested by Andrei Okounkov, counts in local Calabi–Yau geometries of interest to Hironori Iritani, and cluster algebra computations connected to work by Fomin and Zelevinsky. Other domains include symplectic geometry applications by Paul Seidel, categorical dynamics considered by David Ben-Zvi, and moduli problems addressed by Yukinobu Toda.

Relation to Donaldson–Thomas and Cluster Algebras

The relation to Donaldson–Thomas theory follows developments by Donaldson and Thomas, with motivic refinements studied by Kontsevich and Soibelman and computational tools developed by Davesh Maulik and Rahul Pandharipande. Connections to cluster algebras arise from transformations of stability data analogous to combinatorial patterns found by Fomin and Zelevinsky, with structural input from Bernhard Keller and Philipp Gross. Cross-fertilization occurs with mirror symmetry programs led by Kontsevich, homological work by Tom Bridgeland, and wall-crossing phenomena explored by Andrew Neitzke.

Physical Interpretations and BPS States

Physically, the wall-crossing framework models jumps of BPS spectra studied by G. Moore, Andrew Strominger, Cumrun Vafa, and Edward Witten in supersymmetric theories formulated by Seiberg–Witten. The algebraic identities mirror structures in gauge theory research at Princeton University and black hole microstate counting pursued by Strominger and Ashoke Sen. Interactions with string dualities connect to insights by Juan Maldacena, Cumrun Vafa, Edward Witten, and Michael Douglas, while scattering diagrams and tropical techniques recall methods from Grigory Mikhalkin and Mark Gross.

Proofs, Methods, and Extensions

Proof approaches integrate motivic integration techniques from Pierre Deligne, categorification strategies influenced by Mikhail Khovanov and Jacob Lurie, and analytic methods reminiscent of Edward Witten and Nathan Seiberg. Extensions include quantized wall-crossing proposed by Kontsevich and Soibelman, categorical enhancements pursued by Tom Bridgeland and Bernhard Keller, and applications in cluster categories developed by Andrei Zelevinsky and Fomin. Ongoing research involves collaborations at IHÉS, Institute for Advanced Study, Perimeter Institute, and universities where figures like Maxim Kontsevich, Yan Soibelman, Tom Bridgeland, Bernhard Keller, and Davesh Maulik continue to advance the subject.

Category:Algebraic geometry