Formalism for Relative Gromov-Witten Invariants

We develop a formalism for relative Gromov-Witten invariants of Li that is analogous to the Symplectic Field Theory of Eliashberg, Givental, and Hofer. This formalism allows us to express natural degeneration formulae in terms of generating functions and re-derive the formulae of Caporaso-Harris, Ran, and Vakil. In addition, our framework gives a homology theory analogous to SFT Homology.