Formalism for Relative Gromov-Witten Invariants

TL;DR

This paper introduces a new formalism for relative Gromov-Witten invariants that parallels Symplectic Field Theory, enabling degeneration formulas and homology theories, thus advancing the mathematical understanding of symplectic and algebraic geometry.

Contribution

It develops a novel formalism for relative Gromov-Witten invariants, allowing for natural degeneration formulas and a homology theory similar to SFT Homology.

Findings

Derived degeneration formulas using the new formalism

Re-derived classical formulas of Caporaso-Harris, Ran, and Vakil

Established a homology theory analogous to SFT Homology

Abstract

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.