Introduction to Symplectic Field Theory

@article{Eliashberg2000IntroductionTS,
  title={Introduction to Symplectic Field Theory},
  author={Yakov M. Eliashberg and Alexander Givental and Helmut Hofer},
  journal={arXiv: Symplectic Geometry},
  year={2000},
  pages={560-673}
}
We sketch in this article a new theory, which we call Symplectic Field Theory or SFT, which provides an approach to Gromov-Witten invariants of symplectic manifolds and their Lagrangian submanifolds in the spirit of topological field theory, and at the same time serves as a rich source of new invariants of contact manifolds and their Legendrian submanifolds. Moreover, we hope that the applications of SFT go far beyond this framework.1 
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References

SHOWING 1-10 OF 81 REFERENCES
First steps in symplectic topology
CONTENTSIntroduction § 1. Is there such a thing as symplectic topology? § 2. Generalizations of the geometric theorem of Poincare § 3. Hyperbolic Morse theory § 4. Intersections of Lagrangian
Symplectic geometry and topology
This is a review of the ideas underlying the application of symplectic geometry to Hamiltonian systems. The paper begins with symplectic manifolds and their Lagrangian submanifolds, covers contact
Gromov-Witten invariants of general symplectic manifolds
We present an approach to Gromov-Witten invariants that works on arbitrary (closed) symplectic manifolds. We avoid genericity arguments and take into account singular curves in the very formulation.
A symplectic fixed point theorem for toric manifolds
In this paper, by a toric manifold we mean a non-singular symplectic quotient M = ℂ n //T k of the standard symplectic space by a linear torus action. Such a toric manifold is in fact a complex
Virtual neighborhoods and pseudo-holomorphic curves
We use virtual neighborhood technique to establish GW-invariants, Quantum cohomology, equivariant GW-invariants, equivariant quantum cohomology and Floer cohomology for general symplectic manifold.
Properties of Pseudo-Holomorphic Curves in Symplectisations II: Embedding Controls and Algebraic Invariants
In the following we look for conditions on a finite energy plane ũ : = (a, u) : ℂ → ℝ × M, which allow us to conclude that the projection into the manifold M, u : ℂ → M, is an embedding. For this
Quantum cohomology of flag manifolds and Toda lattices
We discuss relations of Vafa's quantum cohomology with Floer's homology theory, introduce equivariant quantum cohomology, formulate some conjectures about its general properties and, on the basis of
Gromov-Witten classes, quantum cohomology, and enumerative geometry
The paper is devoted to the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry. In particular, it contains an axiomatic
Relative Gromov-Witten invariants
We define relative Gromov-Witten invariants of a symplectic manifold relative to a codimension-two symplectic submanifold. These invariants are the key ingredients in the symplectic sum formula of
Localization of virtual classes
We prove a localization formula for virtual fundamental classes in the context of torus equivariant perfect obstruction theories. As an application, the higher genus Gromov-Witten invariants of
...
...