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Let $(F,J,\omega)$ be an almost K\"ahler manifold, $\alpha$ a $J$-holomorphicaction of a compact Lie group $\hat K$ on $F$, and $K$ a closed normal subgroupof $\hat K$ which leaves $\omega$ invariant. We introduce gauge theoreticalinvariants for such triples $(F,\alpha,K)$. The invariants are associated withmoduli spaces of solutions of a certain vortex type equation on a Riemannsurface.  We give explicite descriptions of the moduli spaces associated with thetriple $(\Hom(\C^r,\C^{r_0}), \alpha_{\rm can},U(r))$, 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 becomputed expliciteley and identified with the full Seiberg-Witten invariants ofruled surfaces.Comment: 41 pages, LaTeX, To appear in Commun. Math. Phys. Only one  correction: $(pr_{\Pic^{d'}(\Sigma)})^*$ replaced by  $(pr_{\Pic^{d'}(\Sigma)})_*$ on page 2

Gauge Theoretical Equivariant Gromov-Witten Invariants and the Full Seiberg-Witten Invariants¶of Ruled Surfaces