Relative Seiberg-Witten invariants and a sum formula

Abstract

We study relative Seiberg-Witten moduli spaces and define relative invariants for a pair $(X,\Sigma )$ consisting of a smooth, closed, oriented 4-manifold $X$ and a smooth, closed, oriented 2-dimensional submanifold $\Sigma \subset X$ with positive genus. These relative Seiberg-Witten invariants are meant to be the counterparts of relative Gromov-Witten invariants. We also obtain a sum formula (aka a product formula) that relates the SW invariants of a sum $X$ of two closed oriented 4-manifolds $X_1$ and $X_2$ along a common oriented surface $\Sigma$ with dual self-intersections to the relative SW invariants of $(X_1,\Sigma )$ and $(X_2,\Sigma )$. Our formula generalizes Morgan-Szabó-Taubes' product formula.

51 pages