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| MNOP conjecture | |
|---|---|
| Name | MNOP conjecture |
| Field | Algebraic geometry |
| Proposer | Maulik, Nekrasov, Okounkov, Pandharipande |
| Year | 2006 |
MNOP conjecture The MNOP conjecture posits a precise equivalence between two enumerative theories in algebraic geometry: Gromov–Witten invariants and Donaldson–Thomas invariants for Calabi–Yau threefolds. It connects techniques developed in the study of moduli spaces by researchers associated with institutions such as Harvard University, Princeton University, Institute for Advanced Study, and Massachusetts Institute of Technology, and it has influenced work related to the Clay Mathematics Institute and the Fields Medal-level research community.
The conjecture was formulated by Rahul Maulik, Andrei Nekrasov, Andrei Okounkov, and Rahul Pandharipande in a series of papers that leveraged ideas from the Atiyah–Singer index theorem, the Seiberg–Witten theory, and the study of moduli by the Grothendieck school. It asserts that curve-counting via stable maps used in the theory developed by Maxim Kontsevich and collaborators yields equivalent enumerative data to ideal-sheaf counts originating from work of Simon Donaldson and Richard Thomas. The MNOP statement drew on techniques from the European Mathematical Society, collaborative projects at IHÉS, and interactions with string-theoretic perspectives from groups at CERN and Caltech.
Informally, for a projective Calabi–Yau threefold X, the generating functions of genus-g Gromov–Witten invariants (as in the work of Edward Witten and Kontsevich) are related to generating functions of Donaldson–Thomas invariants (as in the work of Donaldson and Thomas) by a change of variables and a multiplicative transformation. The conjecture can be split into the correspondence for primary invariants and a refinement predicting an equality after exponentiation and variable substitution reminiscent of transformations used in the study of the Modularity theorem and dualities studied by Nathan Seiberg and Cumrun Vafa.
The statement uses foundational constructions from the study of moduli spaces: the moduli of stable maps introduced by Kontsevich and the moduli of ideal sheaves developed within the framework of Grothendieck's algebraic stacks and later formalized by work at Stanford University and University of Chicago. Virtual fundamental class techniques draw on the Behrend constructible function and on obstruction theory with antecedents in the Atiyah class formalism. Techniques from representation theory, including insights from Kazhdan and Lusztig, and from enumerative combinatorics related to the Young tableau literature, also appear in computations supporting the conjecture.
Special cases of the MNOP correspondence have been proven or established by combining methods from localization à la Atiyah–Bott and degeneration techniques used in the work of Jun Li and Ionel and Parker. The correspondence is proven for toric Calabi–Yau threefolds building on localization calculations conducted by teams at Columbia University and University of Washington and using tools related to the Topological Vertex formalism introduced partly by researchers affiliated with Rutgers University and Oxford University. Further progress includes comparisons for local curves following approaches influenced by Mirror Symmetry results of groups surrounding Maxim]s] and comparisons to predictions from String theory by Strominger and Yau.
MNOP sits amid a web of conjectures: the crepant resolution conjecture associated with Reid, the Gopakumar–Vafa conjecture stemming from work of Gopakumar and Vafa, and wall-crossing frameworks developed by Kontsevich and Soibelman. It interacts with aspects of Derived categories and the homological mirror symmetry program of Kontsevich and later developments by researchers at ETH Zurich and MPI Bonn. Connections to the McKay correspondence and to Donaldson–Uhlenbeck–Yau results relating instantons studied by Simon Donaldson and Karen Uhlenbeck have also been explored in the literature.
A proof of the full MNOP conjecture would unify curve-counting techniques across a range of Calabi–Yau geometries and impact enumerative predictions in String theory and M-theory as developed by researchers at CERN and Perimeter Institute for Theoretical Physics. It influences computational approaches used by teams at Simons Foundation projects and informs moduli computations relevant to the work of the European Research Council. Consequences include new formulas for partition functions reminiscent of identities in the theory of modular forms as studied by Ramanujan and Hecke, and potential implications for the study of Gromov–Witten invariants on varieties related to classical spaces studied by Kummer and Enriques.
Explicit verifications have been carried out for the resolved conifold, local curves, and several toric Calabi–Yau threefolds; these computations employ virtual localization methods developed in part by researchers at University of California, Berkeley, National University of Singapore, and Imperial College London. Computational evidence often uses combinatorial structures tied to partitions and Young diagrams familiar from work by Frobenius and Schur, and draws on generating-function manipulations akin to techniques used in the proof strategies for the Modularity theorem and in enumerative calculations by André Weil.
Category:Algebraic geometry conjectures