The Gromov Invariants of Ruan-Tian and Taubes

TL;DR

This paper establishes a connection between Taubes' Gromov invariants and Ruan-Tian invariants for symplectic four-manifolds, introduces a sequence of generalized invariants counting immersed curves, and confirms their relation to Seiberg-Witten invariants.

Contribution

It demonstrates the equivalence of Taubes' and Ruan-Tian invariants and generalizes Taubes' invariants to count immersed curves with double points.

Findings

Taubes' Gromov invariants equal certain Ruan-Tian invariants.

Introduction of a sequence of invariants $Gr_{ ext{delta}}$ for symplectic four-manifolds.

Some Ruan-Tian invariants match Seiberg-Witten invariants.

Abstract

Taubes has recently defined Gromov invariants for symplectic four-manifolds and related them to the Seiberg-Witten invariants. Independently, Ruan and Tian defined symplectic invariants based on ideas of Witten. In this note, we show that Taubes' Gromov invariants are equal to certain combinations of Ruan-Tian invariants. This link allows us to generalize Taubes' invariants. For each closed symplectic four-manifold, we define a sequence of symplectic invariants $Gr_{\delta}$, $\delta=0,1,2,...$. The first of these, $Gr_0$, generates Taubes' invariants, which count embedded J-holomorphic curves. The new invariants $Gr_{\delta}$ count immersed curves with $\delta$ double points. In particular, these results give an independent proof that Taubes' invariants are well-defined. They also show that some of the Ruan-Tian symplectic invariants agree with the Seiberg-Witten invariants.