Journal of Symplectic Geometry

An algebraic formulation of symplectic field theory

Eric Katz

Full-text: Open access

Abstract

We develop a formalism for relative Gromov–Witten invariants following Li J. Li, Stable morphisms to singular schemes and relative stable morphisms, J. Differential Geom. 57 (3) (2001), 509–578, J. Li, A degeneration formula of GW-invariants, J. Differential Geom. 60 (2) (2002), 199–293 that is analogous to the symplectic field theory (SFT) of Eliashberg, Givental and Hofer Y. Eliashberg, A. Givental and H. Hofer, Introduction to symplectic field theory, Geom. Funct. Anal. (Special Volume, Part II) (2000), 560–673 GAFA 2000 (Tel Aviv, 1999). This formalism allows us to express natural degeneration formulae in terms of generating functions and re-derive the formulae of Caporaso–Harris L. Caporaso and J. Harris, Counting plane curves of any genus, Invent. Math. 131 (2) (1998), 345–392, Ran Z. Ran, Enumerative geometry of singular plane curves, Invent. Math. 97 (3) (1989), 447–465, and Vakil R. Vakil, The enumerative geometry of rational and elliptic curves in projective space, J. Reine Angew. Math. 529 (2000), 101–153 for counting rational curves. In addition, our framework gives a homology theory analogous to SFT homology.

Article information

Source
J. Symplectic Geom., Volume 5, Number 4 (2007), 385-437.

Dates
First available in Project Euclid: 19 June 2008

Permanent link to this document
https://projecteuclid.org/euclid.jsg/1213883790

Mathematical Reviews number (MathSciNet)
MR2413309

Zentralblatt MATH identifier
1151.14033

Citation

Katz, Eric. An algebraic formulation of symplectic field theory. J. Symplectic Geom. 5 (2007), no. 4, 385--437. https://projecteuclid.org/euclid.jsg/1213883790


Export citation