This article was accepted into the corpus but its outbound wikilinks were never NER-processed — typical at the deepest BFS hop or when the run's entity cap was reached. No expansion funnel to show.
| Gromov–Witten/Donaldson–Thomas correspondence | |
|---|---|
| Name | Gromov–Witten/Donaldson–Thomas correspondence |
| Field | Algebraic geometry, Symplectic geometry, Mathematical physics |
| Discovered by | Mikhail Gromov; Simon Donaldson; Richard Thomas |
| Year | 1990s–2000s |
Gromov–Witten/Donaldson–Thomas correspondence
The Gromov–Witten/Donaldson–Thomas correspondence relates enumerative invariants arising from holomorphic curve counting and sheaf-theoretic ideal sheaf counting on Calabi–Yau threefolds, connecting techniques and conjectures from Mikhail Gromov, Simon Donaldson, and Richard Thomas. It synthesizes developments from the study of moduli spaces by Maxim Kontsevich, the physical insights of Edward Witten and Cumrun Vafa, and mathematical formulations by Davesh Maulik, Nikita Nekrasov, and Rahul Pandharipande. The correspondence has driven progress linking Mirror symmetry, Donaldson invariants, and enumerative geometry on varieties studied by Pierre Deligne, Jean-Pierre Serre, and David Mumford.
The origins trace to enumerative problems treated by Mikhail Gromov in symplectic topology and to gauge-theoretic moduli problems advanced by Simon Donaldson; subsequent work by Edward Witten and Cumrun Vafa framed curve-counting in the language of topological string theory, inspiring conjectures formalized by Maxim Kontsevich and Richard Thomas. Key mathematical milestones include the construction of virtual fundamental classes by Kai Behrend, virtual cycles used by Jun Li, and degeneration techniques developed by Bertram Fantechi and Barbara Fantechi. Seminal collaborative proofs and proposals emerged from researchers such as Davesh Maulik, Nikolaus Nekrasov (as Nekrasov in physics contexts), Rahul Pandharipande, Andrew Okounkov, and Helenius Thomas (note: Richard Thomas), consolidating perspectives across Princeton University and Harvard University groups. The correspondence has been influenced by insights from Maxim Kontsevich–Soibelman wall-crossing ideas and from enumerative developments at institutions such as Institute for Advanced Study.
Foundational structures include moduli spaces: the moduli of stable maps studied by Maxim Kontsevich and the moduli of ideal sheaves introduced in the work of Richard Thomas and collaborators. Virtual fundamental classes and obstruction theories were developed by Kai Behrend, Barbara Fantechi, and Jun Li to define invariants on spaces that lack expected smoothness. Key tools appear from Hodge-theoretic perspectives associated with Phillip Griffiths and from deformation theory influenced by Alexander Grothendieck and Pierre Deligne. Intersection-theoretic calculations draw on techniques by William Fulton and localization methods by Nikita Nekrasov and Andrei Okounkov. The formalism also uses ideas from Mirror symmetry proponents such as Strominger–Yau–Zaslow and enumerative frameworks shaped by Maxim Kontsevich's homological proposals.
Roughly, the correspondence asserts an equality between generating functions of Gromov invariants and Donaldson–Thomas invariants after a change of variables and a normalization dictated by multiple-cover contributions described by Edward Witten and Cumrun Vafa. In the case of toric Calabi–Yau threefolds the conjecture was formulated and verified by collaborations involving Davesh Maulik, Andrei Okounkov, Rahul Pandharipande, and Richard Thomas, building on localization computations pioneered by Nikita Nekrasov and partition function techniques familiar from Seiberg–Witten theory. The statement ties together curve classes in the Picard lattice associated to Calabi–Yau threefolds studied in works of Shing-Tung Yau and Philip Candelas with ideal sheaf counts expressed via moduli schemes used by Grothendieck.
Proof strategies exploit torus localization in the presence of a Toric variety action, degeneration formulae due to Jun Li, and the machinery of virtual localization developed by Kai Behrend and Andrei Okounkov. Other approaches use wall-crossing formulas and Hall algebra techniques inspired by Maxim Kontsevich and Yan Soibelman (Kontsevich–Soibelman), and use of stable pairs introduced by Richard Thomas and Pandharipande to interpolate between theories. Connections to integrable systems and representation-theoretic methods draw on work by Alexei Borodin and Andrei Okounkov, while algebraic combinatorics and topological vertex techniques trace to contributions from Mina Aganagic, Albrecht Klemm, and Marcos Mariño.
Explicit verifications occurred for toric Calabi–Yau threefolds such as the resolved conifold treated by Philip Candelas-era computations and for local surfaces like local P^2 studied in enumerative projects at Harvard University. Calculations employ the topological vertex formalism developed by Mina Aganagic, Albrecht Klemm, and Marcos Mariño and combinatorial evaluations using plane partitions linked to classical partitions studied by Srinivasa Ramanujan and G. H. Hardy in asymptotic contexts. Concrete equalities were checked by teams including Davesh Maulik, Andrei Okounkov, and Rahul Pandharipande using equivariant techniques with roots in Nikita Nekrasov's gauge theory partition functions.
Generalizations extend to the stable pairs theory of Pandharipande–Thomas, wall-crossing phenomena formalized by Maxim Kontsevich and Yan Soibelman, and relationships with Vafa–Witten invariants and Seiberg–Witten invariants in four-dimensional contexts studied by Edward Witten and Nathan Seiberg. Broader conjectures connect to homological mirror symmetry proposed by Maxim Kontsevich and to categorical enumerative invariants developed in research programs at Institute for Advanced Study and Mathematical Sciences Research Institute.
The correspondence has deep implications for Mirror symmetry predictions in string theory contexts framed by Edward Witten, for computations of black hole microstate counts in work of Cumrun Vafa and Andrew Strominger, and for understanding topological string partition functions explored by Marcos Mariño and Albrecht Klemm. In algebraic geometry it yields enumerative predictions for curve counting on Calabi–Yau threefolds studied by Shing-Tung Yau and guides developments in moduli theory championed by Alexander Grothendieck and David Mumford.
Category:Enumerative geometry