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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Introduction to Symplectic Field Theory

We describea formalism based on quantizationof quadratichamil- tonians and symplectic actions of loop groups which provides a convenient home for most of known general results and conjectures about Gromov-Witten invariants of compact symplectic manifolds and, more generally, Frobenius structures at higher genus. We state several results illustrating the formalism and its use. In particular, we establish Virasoro constraints for semisimple Frobenius structures and outline a proof of the Virasoro conjecture for Gro- mov - Witten invariants of complex projective spaces and other Fano toric manifolds. Details will be published elsewhere.

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Gromov - witten invariants and quantization of quadratic hamiltonians

Symplectic surgery and Gromov-Witten invariants of Calabi-Yau 3-folds

In this article, we introduce the notion of good map and use it to establish Gromov-Witten theory for orbifolds.

Orbifold Gromov-Witten Theory

This is the sequel to the paper [23]. In this paper, we construct the virtual moduli cycles of the degeneration of the moduli of stable morphisms constructed in [23]. We also construct the virtual moduli cycles of the moduli of relative stable morphisms of a pair of a smooth divisor in a smooth variety. Based on these, we prove a degeneration formula of the Gromov-Witten invariants.

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A Degeneration Formula of GW-Invariants

Since Gromov introduced his pseudo-holomorphic curve theory in the 80’s, pseudoholomorphic curve has soon become an eminent technique in symplectic topology. Many important theorems in this field have been proved by this technique, among them, the squeezing theorem [Gr], the rigidity theorem [E], the classification of rational and ruled symplectic 4-manifolds [M2], the proof of the existence of non-deformation equivalent symplectic structures [R2]. The pseudo-holomorphic curve also plays a critical role in a number of new subjects such as Floer homology theory,etc. In the meantime of this development, a great deal of efforts has been made to solidify the foundation of pseudo-holomorphic curve theory, for examples, McDuff’s transversality theorem for “cusp curves” [M1] and the various proofs of Gromov compactness theorem. In the early day of Gromov theory, Gromov compactness theorem was enough for its applications to symplectic topology. However, it was insufficient for its potential applications in algebraic geometry, where a good compactification is often very important. For example, it is particularly desirable to tie Gromov-compactness theorem to the DeligneMumford compactification of the moduli space of stable curves. Gromov’s original proof is geometric. Afterwards, many works were done to prove Gromov compactness theorem in the line of Uhlenbeck bubbling off. It was succeed by Parker-Wolfson [PW] and Ye [Ye]. One outcome of their work was a more delicate compactification of the moduli space of pseudo-holomorphic maps. But it didn’t attract much attention until several years later when Kontsevich and Manin [KM] rediscovered this new compactification in algebraic geometry and initiated an algebro-geometric approach to the same theory. Now this new compactification becomes known as the moduli space of stable maps. The moduli space of stable maps is one of the basic ingredients of this paper. During last several years, pseudo-holomorphic curve theory entered a period of rapid expansion. We has witnessed its intensive interactions with algebraic geometry, mathematical physics and recently with new Seiberg-Witten theory of 4-manifolds [T2]. One should mention that those recent activities in pseudo-holomorphic curve theory did not come from the internal drive of symplectic topology. It was influenced mostly by mathematical physics, particularly, Witten’s theory of topological sigma model. Around 1990, there were many activities in string theory about “quantum cohomology” and mirror symmetry. The core of quantum cohomology theory is so called “counting the numbers of

https://journals.tubitak.gov.tr/math/issues/mat-99-23-1/mat-23-1-10-98066.pdf

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. We also establish GW-invariants for a family of symplectic manifolds. As a consequence, we prove Arnold conjecture for nondegenerate Hamiltonian symplectomorphisms.

Yongbin Ruan

Papers

Virtual neighborhoods and pseudo-holomorphic curves

Virtual neighborhoods and pseudo-holomorphic curves