Gromov-Witten invariants

Gromov-Witten invariants
Name	Gromov-Witten invariants
Field	Algebraic geometry, Symplectic geometry
Introduced by	Mikhail Gromov, Andreas Floer, Simon Donaldson

Gromov-Witten invariants are a fundamental concept in algebraic geometry and symplectic geometry, introduced by Mikhail Gromov, Andreas Floer, and Simon Donaldson, and further developed by Edward Witten, Richard E. Gompf, and Ronald Fintushel. These invariants are used to study the properties of moduli spaces of pseudoholomorphic curves in symplectic manifolds, which are closely related to the work of William Thurston and John Milnor. The development of Gromov-Witten invariants has been influenced by the contributions of Shing-Tung Yau, Grigori Perelman, and Terence Tao.

Introduction to Gromov-Witten Invariants

Gromov-Witten invariants are a type of symplectic invariant that can be used to distinguish between different symplectic structures on a smooth manifold, as studied by Andreas Floer and Dusa McDuff. They are defined using the moduli space of pseudoholomorphic curves in a symplectic manifold, which is a fundamental object in symplectic geometry, and have been applied to the study of Calabi-Yau manifolds by Andrew Strominger and Cumrun Vafa. The invariants are closely related to the work of Simon Donaldson on instantons and Yang-Mills theory, and have been used by Nathan Seiberg and Edward Witten to study the properties of supersymmetric gauge theories. The development of Gromov-Witten invariants has also been influenced by the work of Richard Hamilton and Robert Geroch.

Definition and Mathematical Background

The definition of Gromov-Witten invariants involves the use of pseudoholomorphic curves, which are maps from a Riemann surface to a symplectic manifold that satisfy a certain partial differential equation, as introduced by Mikhail Gromov. The moduli space of these curves is a fundamental object in symplectic geometry, and the Gromov-Witten invariants are defined using the cohomology of this space, which is closely related to the work of Raoul Bott and Clifford Taubes. The invariants are defined using the intersection theory of the moduli space, which is a fundamental tool in algebraic geometry, developed by David Mumford and William Fulton. The mathematical background for Gromov-Witten invariants includes the work of Hermann Weyl and Elie Cartan on differential geometry and symplectic geometry, as well as the contributions of Lars Hörmander and Joseph Kohn to the study of partial differential equations.

Properties and Symmetries

Gromov-Witten invariants have several important properties and symmetries, which make them a powerful tool for studying symplectic manifolds and algebraic varieties, as demonstrated by the work of Claude Chevalley and Jean-Pierre Serre. They are homotopy invariant, meaning that they do not change under homotopy equivalences of the symplectic manifold, a property that has been used by Stephen Smale and Michael Atiyah. They also satisfy a certain gluing formula, which allows them to be computed using the invariants of smaller symplectic manifolds, as developed by Dennis Sullivan and William Thurston. The invariants also have a certain symmetry under orientation reversal, which is closely related to the work of René Thom and John Milnor on cobordism theory. The properties and symmetries of Gromov-Witten invariants have been studied by Ralph Cohen and Peter May, and have been used to study the properties of topological quantum field theories by Gregory Moore and Nathan Seiberg.

Calculations and Examples

Calculating Gromov-Witten invariants can be a challenging task, but there are several examples where they have been computed explicitly, such as the work of Simon Donaldson on K3 surfaces and Calabi-Yau manifolds. The invariants have been computed for several symplectic manifolds, including projective spaces and Grassmannians, by Andreas Floer and Dusa McDuff. They have also been used to study the properties of moduli spaces of algebraic curves, as demonstrated by the work of David Mumford and William Fulton. The calculations of Gromov-Witten invariants have been influenced by the contributions of George Lusztig and Michel Brion to the study of representation theory and algebraic geometry. The examples of Gromov-Witten invariants have been used to study the properties of topological strings by Andrew Strominger and Cumrun Vafa, and have been applied to the study of black holes by Leonard Susskind and Gerard 't Hooft.

Applications in Geometry and Physics

Gromov-Witten invariants have several important applications in geometry and physics, including the study of mirror symmetry and topological strings, as developed by Andrew Strominger and Cumrun Vafa. They are also closely related to the study of supersymmetric gauge theories and string theory, as demonstrated by the work of Nathan Seiberg and Edward Witten. The invariants have been used to study the properties of Calabi-Yau manifolds and K3 surfaces, which are important objects in string theory and M-theory, as studied by Shing-Tung Yau and Andrew Strominger. The applications of Gromov-Witten invariants have been influenced by the contributions of Richard Feynman and Julian Schwinger to the study of quantum field theory and particle physics. The invariants have also been used to study the properties of black holes and cosmology, as demonstrated by the work of Stephen Hawking and James Hartle.

Relationship to Other Invariants

Gromov-Witten invariants are closely related to other invariants in geometry and topology, such as the Donaldson invariants and the Seiberg-Witten invariants, as developed by Simon Donaldson and Nathan Seiberg. They are also related to the topological invariants of manifolds, such as the homotopy groups and the homology groups, as studied by René Thom and John Milnor. The invariants are also closely related to the invariants of algebraic varieties, such as the Chow groups and the K-groups, as developed by David Mumford and William Fulton. The relationship between Gromov-Witten invariants and other invariants has been studied by Ralph Cohen and Peter May, and has been used to study the properties of topological quantum field theories by Gregory Moore and Nathan Seiberg. The invariants have also been used to study the properties of moduli spaces of algebraic curves and Riemann surfaces, as demonstrated by the work of Lipman Bers and Lars Ahlfors.

Category:Symplectic geometry