Embedded curves and Gromov-Witten invariants of three-folds

Abstract

Associated with a prime homology class $\beta \in P_2(X,\mathbb{Z})$ (i.e. $\beta =p\alpha$ and $\alpha \in H_2(X,\mathbb{Z})$ imply $p=1$ or $p$ is an odd prime) on a symplectic three-manifold with vanishing first Chern class, we count the embedded perturbed pseudo-holomorphic curves in $X$ of a fixed genus $g$ to obtain certain integer valued invariants analogous to Gromov-Witten invariants of $X$.