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

Abstract

Let $(F,J,\omega )$ be an almost Kähler 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 $(\Hom(\C^r,\C^{r_0}), α_{\rm can},U(r))$, where ${\alpha }_{\mathrm{can}}$ 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.

arXiv comment41 pages, LaTeX, To appear in Commun. Math. Phys. Only one correction: $(pr_{\Pic^{d'}(Σ)})^*$ replaced by $(pr_{\Pic^{d'}(Σ)})_*$ on page 26

Cited by in corpus (2)

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