Gauge theoretical equivariant Gromov-Witten invariants and the full Seiberg-Witten invariants of ruled surfaces

Abstract

Let $(F,J,\omega)$ be an almost K\"ahler manifold, $\alpha$ a $J$-holomorphic action of a compact Lie group $\hat K$ on $F$, and $K$ a closed normal subgroup of $\hat K$ which leaves $\omega$ invariant. We introduce gauge theoretical invariants for such triples $(F,\alpha,K)$. The invariants are associated with moduli spaces of solutions of a certain vortex type equation on a Riemann surface. We give explicite descriptions of the moduli spaces associated with the triple $(\Hom(\C^r,\C^{r_0}), \alpha_{\rm can},U(r))$, where $\alpha_{\rm can}$ denotes the canonical action of $\hat K=U(r)\times U(r_0)$ on $\Hom(\C^r,\C^{r_0})$. In the abelian case $r=1$, the new invariants can be computed expliciteley and identified with the full Seiberg-Witten invariants of ruled surfaces.