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| A-model | |
|---|---|
| Name | A-model |
| Discipline | Mathematical physics |
A-model
The A-model is a two-dimensional topological quantum field theory arising in string theory and symplectic geometry that encodes enumerative invariants of Calabi–Yau manifolds and general symplectic manifolds. It plays a central role in the interplay between Witten's topological field theories, Kontsevich's homological mirror symmetry, and enumerative problems studied by Gromov and others. The A-model informs computations in Gromov–Witten theory, influences developments in Fukaya category construction, and connects to physical frameworks like Type II string theory and topological string theory.
The A-model appears as the topological twist of the N=(2,2) supersymmetric nonlinear sigma model studied by Witten and is formulated on a Riemann surface mapped into a target Calabi–Yau manifold or more general symplectic manifold. Early work linking the model to enumerative geometry involved figures such as E. Witten, Kontsevich, Gromov, Bott, and Donaldson, and it sits alongside the complementary B-model studied in mirror symmetry by researchers including Candelas and Aspinwall.
Mathematically, the A-model is defined by a path integral over maps from a worldsheet Riemann surface Σ to a target symplectic manifold (X,ω) with action built from the pullback of ω and a topological term determined by the homology class of the map. The field content and BRST operator arise from the N=(2,2) supersymmetry algebra after topological twisting as described by Witten and formalized in works by Wehrheim and Salamon. Correlation functions reduce to integrals over moduli spaces of holomorphic maps, connecting to structures studied by Gromov and the foundations of Gromov–Witten theory developed by McDuff, Salamon, Ruan, and Tian.
Physically, the A-model captures topological sectors of Type IIA string theory compactified on Calabi–Yau manifolds and computes instanton corrections to mirror symmetry predictions used in string compactification analyses by groups including CERN researchers and theorists such as Witten, Vafa, and Strominger. Applications span counts of rational curves relevant to enumerative geometry problems tackled by Candelas and collaborators, computations of worldsheet instantons in Heterotic string theory settings, and relationships to categorical invariants central to Kontsevich proposals and constructions by Seidel and Fukaya.
The moduli spaces central to the A-model are spaces of stable maps from genus-g curves into X, studied via techniques introduced by Givental, Li, Graber, and Pandharipande. Virtual fundamental class constructions developed by Behrend, Fantechi, Ruan, and Tian allow rigorous definitions of Gromov–Witten invariants that the A-model computes. These invariants are essential in formulations by Kontsevich and appear in enumerative predictions confirmed in examples by Candelas and de la Ossa.
Mirror symmetry conjectures a duality between the A-model on a symplectic target X and the B-model on a complex mirror Y, a paradigm championed by Candelas, Kontsevich, and Witten. Kontsevich's homological mirror symmetry equates the A-model's Fukaya category to the derived category of coherent sheaves on the B-model, with major contributions from Seidel, Fukaya, Abouzaid, and Zorich. Computational techniques linking the two sides were advanced by Givental, Hosono, and Tsygan in studies of period integrals and Picard–Fuchs equations used to match A-model enumerative data to B-model complex-structure data.
Classic examples include the A-model on the quintic threefold studied by Candelas, de la Ossa, Green, and Parks which produced counts of rational curves matched to B-model calculations. Toric varieties and local Calabi–Yau geometries used by Aganagic, Klemm, Marino, and Vafa allow explicit computations via localization techniques developed by Witten, Givental, and Nekrasov. Low-genus invariants for surfaces studied by McDuff and Salamon and computations in symplectic mapping tori explored by Thomas and Donaldson illustrate concrete A-model outputs matched to algebraic predictions of Kontsevich.
The A-model emerged from Witten's 1988 topological field theory program and gained mathematical formulation through the work of Gromov, Witten, Kontsevich, McDuff, Ruan, Tian, Fukaya, Seidel, Givental, Candelas, and Vafa. Subsequent developments in virtual perturbation techniques and categorical approaches involved contributors such as Behrend, Fantechi, Abouzaid, Li, Pandharipande, and Graber. The collaborative interplay between physicists and mathematicians at institutions like IAS, Princeton University, Harvard University, University of Cambridge, and research programs at Simons Foundation fostered rapid growth in A-model theory.
Category:Topological field theory