Contents Online

# Journal of Symplectic Geometry

## Volume 5 (2007)

### Number 4

### An algebraic formulation of symplectic field theory

Pages: 385 – 437

DOI: http://dx.doi.org/10.4310/JSG.2007.v5.n4.a2

#### Author

#### 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.