On Symplectic Sum Formulas in Gromov-Witten Theory

TL;DR

This paper critically examines symplectic sum formulas in Gromov-Witten theory, clarifies misconceptions, and proposes a reformulation to address technical issues and limitations in existing approaches.

Contribution

It reformulates the symplectic sum formula, analyzes two analytic approaches, and clarifies the roles of rim tori and the S-matrix in Gromov-Witten theory.

Findings

Identifies errors in Ionel-Parker's gluing argument

Highlights vagueness in Li-Ruan's approach

Proposes a clearer reformulation of the symplectic sum formula

Abstract

This manuscript describes in detail the symplectic sum formulas in Gromov-Witten theory and related topological and analytic issues. In particular, we analyze and compare two analytic approaches to these formulas. The Ionel-Parker formula contains two unique features, rim tori refinements of relative invariants and the so-called S-matrix, which have been a mystery in GW-theory over the past decade. We explain why the latter, which appears due to imprecise reasoning, should not be present and how the former should be interpreted. While the gluing argument in the IP work attempts to address all of the issues relevant to certain "semi-positive" cases, it contains several highly technical, but crucial, mistakes, which invalidate it and thus the whole paper almost completely. The SFT type idea behind the Li-Ruan approach has the potential of avoiding many issues with the degeneration of the metric on the target occurring in the IP approach. Unfortunately, the LR paper is vague about the key notions and aspects of the setup, including the definition of relative stable maps, and does not contain even a description of the local structure of the relative moduli space or an attempt at a complete proof of any major statement. The only technical arguments in this paper concern fairly minor points and are either incorrect or unnecessary. Neither of the two papers even considers gluing stable maps with extra rubber structure, which is necessary for defining the relevant invariants outside of a relatively narrow collection of "semi-positive" cases. In this manuscript, we re-formulate the (numerical) symplectic sum formula, describe the issues arising in both approaches, and explain how the Li-Ruan SFT type idea can be used to address them.