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**Introduction to Symplectic Field Theory.
Special volume of the journal Geometric and Functional Analysis.**
*(English)*
Zbl 0989.81114

Alon, N. (ed.) et al., GAFA 2000. Visions in mathematics–Towards 2000. Proceedings of a meeting, Tel Aviv, Israel, August 25-September 3, 1999. Part II. Basel: Birkhäuser, 560-673 (2000).

The authors describe in this article a new theory, which is called Symplectic Field Theory (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.

The first part of the paper contains the necessary background in symplectic-geometric and analytic information (Reeb vector fields, splitting of a symplectic manifold along a contact hypersurface, compatible almost complex structures, holomorphic curves in symplectic cobordisma, compactification and dimension of moduli spaces, coherent orientation of the moduli spaces of holomorphic curves), and a first attempt of algebrization (including a recollection of finite-dimensional Floer theory, Floer homology for the action functional, examples and relative contact homology and contact non-squeezing theorems).

The second part deals with the algebraic formalism; it starts with its own introduction – which includes a brief sketch of SFT, and develops further specific SFT topics: contact manifolds, symplectic cobordisms, the SFT-version of the chain homotopy statement in Floer homology theory, the composition formula for the SFT-invariants of symplectic cobordisms, invariants of contact manifolds, invariants of Legendrian submanifolds via SFT, examples and possible generalizations of SFT.

The article is part of a series of papers devoted to the foundations, applications, and further development of SFT. As the authors say, the presented theoretical issues and applications are just “the tip of the iceberg”; e.g., the wide range of applications of SFT – some of which are mentioned in the paper, includes: new invariants on contact manifolds and Legendrian knots and links, new methods for computing Gromov-Witten invariants, new restrictions on the topology of Lagrangian submanifolds and new non-squeezing type theorems in contact geometry.

For the entire collection see [Zbl 0960.00035].

The first part of the paper contains the necessary background in symplectic-geometric and analytic information (Reeb vector fields, splitting of a symplectic manifold along a contact hypersurface, compatible almost complex structures, holomorphic curves in symplectic cobordisma, compactification and dimension of moduli spaces, coherent orientation of the moduli spaces of holomorphic curves), and a first attempt of algebrization (including a recollection of finite-dimensional Floer theory, Floer homology for the action functional, examples and relative contact homology and contact non-squeezing theorems).

The second part deals with the algebraic formalism; it starts with its own introduction – which includes a brief sketch of SFT, and develops further specific SFT topics: contact manifolds, symplectic cobordisms, the SFT-version of the chain homotopy statement in Floer homology theory, the composition formula for the SFT-invariants of symplectic cobordisms, invariants of contact manifolds, invariants of Legendrian submanifolds via SFT, examples and possible generalizations of SFT.

The article is part of a series of papers devoted to the foundations, applications, and further development of SFT. As the authors say, the presented theoretical issues and applications are just “the tip of the iceberg”; e.g., the wide range of applications of SFT – some of which are mentioned in the paper, includes: new invariants on contact manifolds and Legendrian knots and links, new methods for computing Gromov-Witten invariants, new restrictions on the topology of Lagrangian submanifolds and new non-squeezing type theorems in contact geometry.

For the entire collection see [Zbl 0960.00035].

Reviewer: Vladimir Balan (Bucureşti)

### MSC:

53D42 | Symplectic field theory; contact homology |

53D35 | Global theory of symplectic and contact manifolds |

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |

53D45 | Gromov-Witten invariants, quantum cohomology, Frobenius manifolds |

53D40 | Symplectic aspects of Floer homology and cohomology |