Introduction to Symplectic Field Theory

By Y. Eliashberg, Alexander Givental, Helmut Hofer. 2000

798 citations recorded by the source corpus.

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

Subfields: Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/ffe30f7b0d548c3d9020e52bc1ed3b12cdc4b5a9

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Citations in the corpus

Integration theory on the zero sets of polyfold Fredholm sections (2007, background): “In order to make the algebraic formalism proposed in [ 3 ] rigorous one needs an integration theory for differential forms over the solution spaces.”
Conjectures on Bridgeland stability for Fukaya categories of Calabi-Yau manifolds, special Lagrangians, and Lagrangian mean curvature flow (2014, background): “[20]: given L with conical singularities at p1; : : : ; pk modelled on stable SL conesC1; : : : ; Ck C, we delete p1; : : : ; pk fromL;M , and treatM nfp1; : : : ; pkg as a noncompact symplectic manifold with concave cylindrical ends modelled on S 1 .”
Relative asymptotic behavior of pseudoholomorphic half‐cylinders (2007, background): “In symplectic field theory [3] (and related theories: symplectic Floer homology, contact homology, embedded contact homology, etc.”
Amoebas of Algebraic Varieties and Tropical Geometry (2004, methodology): “complex amoebas. Hopefully the technique developed in the Symplectic Field Theory, see [ 9 ] and [7] can help to verify this conjecture.”
Holomorphic disks, link invariants and the multi-variable Alexander polynomial (2008, background): “This kind of degeneration has been extensively studied in the literature, cf Ionel and Parker [8], Li and Ruan [14], Eliashberg, Givental and Hofer [3], Bourgeois, Eliashberg, Hofer, Wysocki and Zehnder [1], and of course [15].”
Punctured holomorphic curves and Lagrangian embeddings (2014, background): “The following result is due to Viterbo [58] for ”strongly Fano manifolds”, and to Eliashberg, Givental and Hofer [25] in the general case.”
Gluing pseudoholomorphic curves along branched covered cylinders II (2007, result): “The geometric setup here is essentially that of Hofer, Wysocki, and Zehnder [7], and the four-dimensional case of the setup used to define symplectic field theory [5].”
Symplectic Floer homology of area-preserving surface diffeomorphisms (2008, background): “5 Directions for further research Cylindrical contact homology Symplectic field theory and contact homology invariants (see Eliashberg, Givental and Hofer [10]) exist for mapping tori due to the existence of a Hamiltonian structure (see Bourgeois et al [3], Cieliebak and Mohnke [5] and Fabert [11]).”
A cylindrical reformulation of Heegaard Floer homology (2005, background): “There are strong general results about the existence of coherent orientations (see [6], [3] or [5]).”
Morse flow trees and Legendrian contact homology in 1-jet spaces (2005, neutral): “ar X iv :m at h/ 05 09 38 6v 1 [ m at h. SG ] 1 6 Se p 20”
Invariants of Legendrian Knots and Coherent Orientations (2001, methodology): “See [ 8 ] for more details, especially with regards to the cut and paste operation.”
Infinitely many exotic monotone Lagrangian tori in ℂℙ2 (2014, methodology): “4, we state, from [3] and [7], the main compactness theorem of pseudo-holomorphic curves for neck-stretching.”
A topological view of Gromov-Witten theory (2004, methodology): “P 3 [3,4] l −→ P 3 , S3, C3,4, (P 3 [2,3], S2) 2 −→ P 3 [ 2 ,3], S2,”
Knot and braid invariants from contact homology II (2003, background): “This interpretation allows us to extend the knot invariant to embedded graphs and higher-dimensional knots.”