Kontsevich–Soibelman

Kontsevich–Soibelman
Name	Kontsevich–Soibelman
Field	Mathematics
Subfield	Algebraic geometry, Symplectic geometry, Category theory, Mathematical physics
Notable works	Wall-crossing formula, Donaldson–Thomas theory, Stability conditions

Kontsevich–Soibelman

The Kontsevich–Soibelman theory encompasses a set of structures and results in modern Mathematics linking Maxim Kontsevich and Yan Soibelman via a program that connects Algebraic geometry, Symplectic geometry, Category theory, and Mathematical physics. Its central achievements include a wall-crossing formula for counting invariants, foundational contributions to Donaldson–Thomas theory, and formulations of stability conditions on triangulated categories, influencing research associated with Calabi–Yau varieties, Cluster algebras, and Mirror symmetry.

Definition and Overview

The Kontsevich–Soibelman framework provides a formalism for understanding how enumerative invariants change under variation of stability, synthesizing ideas from Maxim Kontsevich's work on Homological mirror symmetry and Yan Soibelman's work on Deformation theory. It codifies wall-crossing phenomena in terms of automorphisms of quantum torus algebras and group-like elements in pronilpotent Lie algebras, drawing on constructions related to Donaldson–Thomas invariants, Gromov–Witten invariants, and Seiberg–Witten theory. The formalism is situated within a network of results involving Bridgeland stability conditions, Fukaya category, and structures arising in String theory and Supersymmetric gauge theory.

Mathematical Background

Foundational prerequisites include Triangulated categories, Derived categories, and the theory of Stability conditions on triangulated categories developed by Tom Bridgeland. The algebraic underpinnings use Hall algebras and Quantum groups as seen in works by Ringel, Lusztig, and Drinfeld, while the geometric perspective leverages moduli of coherent sheaves on Calabi–Yau threefolds and moduli of objects in the Derived category of coherent sheaves. Analytic and categorical tools from Deformation quantization, A∞-categories, and the Fukaya category are essential, as are links to invariants studied by Donaldson, Thomas, Gromov, Witten, and constructions in Seiberg–Witten theory and Chern–Simons theory. The wall-crossing identities are formalized using pronilpotent completions of Lie algebras related to Poisson algebras and Symplectic geometry.

Kontsevich–Soibelman Wall-Crossing Formula

The Kontsevich–Soibelman wall-crossing formula describes how virtual counts like Donaldson–Thomas invariants transform when a stability condition crosses a wall in the space of Bridgeland stability conditions. The formula expresses continuity via factorization of automorphisms in a quantum torus algebra, paralleling products appearing in the theory of Cluster algebras and the work of Fomin–Zelevinsky. It refines earlier wall-crossing observations from Seiberg–Witten theory and Gaiotto–Moore–Neitzke proposals by encoding BPS state degeneracies in terms of algebraic identities connected to Kontsevich's formality theorem and Stokes phenomena studied by Deligne and Malgrange. The result has rigorous formulations using motivic refinements that reference Grothendieck ring of varieties and categories of Mixed Hodge structures as in research by Behrend and Joyce.

Applications and Consequences

Applications span enumerative geometry and mathematical physics. In Algebraic geometry the formalism informs computations of curve-counting invariants on Calabi–Yau threefolds and contributes to the proof strategies for relations between Gromov–Witten invariants and Donaldson–Thomas invariants as explored in works related to MNOP conjecture and research by Maulik–Nekrasov–Okounkov–Pandharipande. In Mathematical physics the wall-crossing identities describe BPS spectra in N=2 supersymmetric gauge theories and appear in analysis by Seiberg–Witten, Gaiotto, Moore, and Neitzke. The framework has informed developments in Cluster varieties, Scattering diagrams studied by Gross–Siebert, and constructions of stability conditions for Derived categories of coherent sheaves on K3 surfaces and Abelian varieties influenced by Bayer–Macrì and Arcara–Bertram. Consequences include new invariants with motivic and cohomological refinements and structural insights into Hall algebra representations and categorical Donaldson–Thomas theory advanced by Schiffmann and Bridgeland.

Examples and Computations

Concrete instances include computations of Donaldson–Thomas invariants for quivers with potential as developed by Derksen–Weyman–Zelevinsky and Ginzburg algebras, examples drawn from moduli of representations of Kronecker quiver and ADE quivers, and explicit scattering diagrams for cluster varieties in the program of Gross–Hacking–Keel–Kontsevich (noting Maxim Kontsevich's separate role). Calculations for compact Calabi–Yau threefolds build on techniques from Behrend–Fantechi virtual classes and use localization methods related to Graber–Pandharipande and vertex techniques developed by Okounkov–Pandharipande. In gauge-theoretic contexts, spectral network computations from Gaiotto–Moore–Neitzke illustrate wall-crossing in examples tied to Hitchin systems and moduli of flat connections studied by Hitchin and Simpson.

Related Concepts and Generalizations

The Kontsevich–Soibelman structures relate to a web of theories: Homological mirror symmetry by Maxim Kontsevich, Bridgeland stability by Tom Bridgeland, Cluster algebras by Fomin–Zelevinsky, and enumerative programs like the MNOP conjecture by Maulik, Nekrasov, Okounkov, and Pandharipande. Generalizations include motivic and cohomological refinements, categorified Hall algebras explored by Schiffmann and Toen, and connections to scattering diagrams and mirror constructions advanced by Gross, Siebert, Hacking, and Keel. The framework continues to interact with research in Representation theory, Integrable systems, and Quantum field theory through ongoing work by researchers including Joyce, Song, Kontsevich, Soibelman, Gaiotto, and Moore.

Category:Mathematical theories