Gromov–Witten

Gromov–Witten
Name	Gromov–Witten
Field	Symplectic geometry, Algebraic geometry, Mathematical physics
Introduced	1990s
Notable figures	Mikhail Gromov, Edward Witten, Maxim Kontsevich, Yakov Eliashberg

Gromov–Witten

Introduction

Gromov–Witten theory is a branch of symplectic geometry and algebraic geometry connecting Mikhail Gromov, Edward Witten, Maxim Kontsevich, Simon Donaldson, Andrei Okounkov through enumerative invariants, moduli spaces, and intersection theory. It arose from interactions among International Congress of Mathematicians, Harvard University, Princeton University, Institute for Advanced Study, Mathematical Sciences Research Institute programs and influenced work at Courant Institute, University of Cambridge, University of Oxford, University of Chicago. The theory provides counts of holomorphic curves in targets such as Calabi–Yau manifold, Kähler manifold, Deligne–Mumford stack, and has deep ties to Mirror Symmetry, String Theory, Topological Quantum Field Theory, Donaldson–Thomas theory.

Definitions and Formalism

Central objects include moduli spaces of stable maps to a target variety or symplectic manifold, with foundational techniques developed by researchers associated with Stanford University, École Normale Supérieure, Moscow State University, Brown University, Rutgers University. Formal definitions employ virtual fundamental classes from work at Max Planck Institute for Mathematics, Clay Mathematics Institute, Kavli Institute for Theoretical Physics, and use obstruction theory inspired by methods in papers by Kontsevich–Manin, Behrend–Fantechi, Ruan–Tian, Li–Tian. The construction relies on Gromov compactness theorems linked to Gromov, and on deformation theories related to Serre duality, Grothendieck, Alexander Grothendieck techniques, and intersections traced to Poincaré duality, Atiyah–Bott localization, Bott–Taubes methods.

Examples and Computations

Computations often occur for target spaces like Complex projective space, Grassmannian, Quintic threefold, K3 surface, Toric variety, Flag variety, and use mirror formulas from work at Institut des Hautes Études Scientifiques, California Institute of Technology, ETH Zurich. Effective calculations invoke techniques from Localization (mathematics), Virtual localisation, Mirror theorem (Givental), Givental, Aspinwall–Morrison conjectures and comparisons with predictions from Candelas, Strominger–Yau–Zaslow. Notable computed invariants include enumerative counts validated in collaborations among researchers at University of California, Berkeley, Columbia University, Yale University.

Mathematical Properties and Foundations

Foundational properties include invariance under symplectic deformation studied by groups at Northwestern University, University of Michigan, University of Illinois Urbana-Champaign, and relationships with cohomological field theories developed by Dijkgraaf–Witten inspired frameworks, Witten’s work on topological gravity, and axioms articulated in works tied to European Mathematical Society conferences. Technical foundations draw on compactification techniques related to Deligne–Mumford compactification, virtual perturbation schemes like Fukaya–Ono, Kuranishi structure approaches, and algebraic virtual cycle constructions by Li, Jun Li, Behrend, Fantechi.

Relations to Other Theories

Gromov–Witten invariants interface with Donaldson–Thomas theory, Pandharipande–Thomas theory, Seiberg–Witten theory, Floer homology, Symplectic Field Theory, Homological Mirror Symmetry, Kontsevich's homological mirror symmetry conjecture, and categorical frameworks in works at Perimeter Institute, Weizmann Institute of Science, Max Planck Institute for Physics. Cross-disciplinary influence links to Conformal Field Theory, Quantum Cohomology, Integrable systems, Painlevé equations, and conjectural correspondences inspired by N=2 supersymmetry research at CERN.

Applications and Impact

Applications span enumerative predictions for Calabi–Yau manifold geometry that influenced string compactification studies at Princeton Plasma Physics Laboratory and Los Alamos National Laboratory collaborations, algorithmic enumeration tools implemented in projects at Mathematical Sciences Research Institute, SageMath, and conceptual advances in mirror symmetry leveraged in courses at University of California, San Diego, University of Toronto. The theory shaped research agendas at funding bodies like National Science Foundation, European Research Council, Simons Foundation and inspired conferences at Banff Centre, ICM sessions and advanced graduate curricula at Massachusetts Institute of Technology.

Historical Development and Key Contributors

Origins trace to ideas by Mikhail Gromov on pseudoholomorphic curves, interactions with Edward Witten’s topological field theories, and formalizations by Maxim Kontsevich, Yongbin Ruan, Gang Tian, Kai Behrend, Barbara Fantechi, Kenji Fukaya, Kaoru Ono, Jun Li, Dusa McDuff, Dietmar Salamon, Yakov Eliashberg, Richard Thomas, Rahul Pandharipande, Claus Hertling, Alexander Givental, Lothar Göttsche, Clemens Voisin. Developments progressed through collaborative programs at Institute for Advanced Study, Mathematical Sciences Research Institute, Banff International Research Station, and major publications circulated via Journal of Differential Geometry, Inventiones Mathematicae, Communications in Mathematical Physics, accelerating the field across groups at Princeton University, Harvard University, Cambridge University.

Category:Symplectic geometry