Gromov–Witten invariants

Gromov–Witten invariants
Name	Gromov–Witten invariants
Caption	Moduli space schematic
Field	Symplectic geometry; Algebraic geometry; Mathematical physics
Introduced	1990s
Contributors	Mikhail Gromov; Edward Witten; Maxim Kontsevich; Aleksey Tikhomirov

Gromov–Witten invariants are intersection-theoretic invariants arising from counts of holomorphic curves in compact symplectic manifolds and projective varieties, developed in the 1990s during interactions between Mikhail Gromov, Edward Witten, Maxim Kontsevich and groups interested in enumerative geometry and quantum field theory. They formalize curve-counting problems connected to enumerative questions studied by Riemann, Enrico Bombieri, and techniques later refined by researchers at institutions such as Institute for Advanced Study, Princeton University, and IHÉS. These invariants link ideas from Symplectic geometry, Algebraic geometry, and String theory and play a central role in modern research programs influenced by work at Harvard University, Caltech, and collaborations around the Clay Mathematics Institute.

Introduction

Gromov–Witten theory grew out of foundational contributions by Mikhail Gromov on pseudo-holomorphic curves, by physicists around Edward Witten on topological sigma models, and by mathematicians such as Maxim Kontsevich who connected these ideas to moduli of maps and mirror symmetry conjectures pursued at IHÉS and University of Paris. The subject unifies techniques from the study of moduli problems in Deligne–Mumford style compactifications, intersection theory central to work of Alexander Grothendieck and Pierre Deligne, and enumerative predictions from Mirror symmetry proposals tied to computations linked to Candelas and collaborators. Early rigorous frameworks were built by researchers associated with Stony Brook, UC Berkeley, and projects involving American Mathematical Society conferences.

Definitions and formalism

Formally, one fixes a compact symplectic manifold or smooth projective variety X studied by teams at Princeton University and University of Cambridge, a genus g curve type related to classical questions from Riemann and Abel, marked points reminiscent of constructions by Mumford and Grothendieck, and a homology class beta inspired by cycles in the study of Lefschetz pencils and Hodge theory. A Gromov–Witten invariant is then defined by integrating cohomology classes pulled back along evaluation maps from a compactified moduli space of stable maps, an approach paralleling intersections in the work of Fulton and MacPherson and later formalized with virtual fundamental class techniques influenced by research at Stanford University and Massachusetts Institute of Technology. The construction uses obstruction theories similar to those appearing in deformation problems studied by Kodaira and Spencer and virtual cycle methods refined by groups at Rutgers University and University of Chicago.

Moduli spaces of stable maps

The moduli space of stable maps, central to the definition, is a Deligne–Mumford stack type that generalizes compactifications used by Deligne and Mumford in curves, and it parametrizes holomorphic maps from marked nodal curves to X studied in contexts at Yale University and Imperial College London. Compactness results rely on Gromov compactness theorems linked to work by Gromov and analytic foundations echo techniques used by researchers at Courant Institute and ETH Zurich. The virtual fundamental class on these moduli spaces employs obstruction theory ideas developed by teams including those at University of Michigan and University of Oxford, and it enables intersection numbers that remain invariant under deformation, as in stability analyses familiar from Nash and Morse style methods.

Properties and axioms

Gromov–Witten invariants satisfy axioms analogous to those in topological field theories studied by Atiyah, Segal, and Witten, including deformation invariance, splitting (gluing) axioms related to degeneration formulas used by Li and Ruan, and divisor axiom statements echoing intersection principles from Poincaré and Noether. These properties are organized into algebraic structures such as quantum cohomology rings first investigated in seminars at IHÉS and formal Frobenius manifold structures explored by researchers connected to Dubrovin and Saito. Further formal properties mirror structures in representation-theoretic programs at Institute for Advanced Study and categorical frameworks pursued by groups at University of Toronto.

Computational techniques and examples

Computations employ localization techniques inspired by the fixed-point formula of Atiyah–Bott and Berline–Vergne, degeneration and relative theories developed by Jun Li and others, and mirror symmetry predictions verified in computations like the quintic threefold counts studied by Candelas, Green, and Parkes. Toric geometry methods from work by Gelfand and Zelevinsky and virtual localization approaches linked to researchers at Harvard University and Stanford University allow explicit calculations for toric varieties and flag manifolds as in studies by Givental, Coates, and Cox. Examples include classical enumerative results going back to Schubert calculus generalizations and modern enumerations on Calabi–Yau manifolds examined in collaborations involving Maxim Kontsevich and Sergei Gukov.

Applications in geometry and physics

Applications span proofs and formulations of mirror symmetry conjectures tied to work by Candelas and Kontsevich, insights into symplectic topology problems influenced by research at Princeton and UCLA, and connections to topological string theory developed by groups around Edward Witten and Cumrun Vafa. In algebraic geometry, these invariants inform curve-counting problems and birational geometry investigations associated with programs by Mori and Yau, while in mathematical physics they relate to partition functions and dualities studied in conferences at Perimeter Institute and CERN.

Generalizations and related theories

Generalizations include relative Gromov–Witten theory developed by teams associated with Microsoft Research workshops and absolute-relative compatibilities proven by researchers affiliated with Columbia University and University of Illinois, orbifold Gromov–Witten theory tied to developments by Chen and Ruan, and Donaldson–Thomas invariants that form a parallel enumerative framework explored by Donaldson and Thomas. Further related theories include Fan–Jarvis–Ruan–Witten theory from collaborations linking Fan and Jarvis with others, and categorical enhancements such as the study of derived categories connected to work by Bondal and Orlov.

Category:Symplectic geometry Category:Algebraic geometry Category:Mathematical physics