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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.
Extending our reduction construction in (S. Hu, Hamiltonian symmetries and reduction in generalized geometry, Houston J. Math., to appear, math.DG/0509060, 2005.) to the Hamiltonian action of a Poisson Lie group, we show that generalized Kähler reduction exists even when only one generalized complex structure in the pair is preserved by the group action. We show that the constructions in string theory of the (geometrical) T-duality with H-fluxes for principle bundles naturally arise as reductions of factorizable Poisson Lie group actions. In particular, the groups involved may be non-abelian.