full Seiberg - Witten invariants of ruled surfaces

Let F be a differentiable manifold endowed with an almost Kähler structure (J, ω), α a J-holomorphic action of a compact Lie groupˆK on F , and K a closed normal subgroup ofˆK which leaves ω invariant. The purpose of this article is to introduce gauge theoretical invariants for such triples (F, α, K). The invariants are associated with moduli spaces of solutions of a certain vortex type equation on a Riemann surface Σ. Our main results concern the special case of the triple (Hom(C where α can denotes the canonical action ofˆK = U (r) × U (r 0) on Hom(C r , C r0). We give a complex geometric interpretation of the corresponding moduli spaces of solutions in terms of gauge theoretical quot spaces, and compute the invari-ants explicitely in the case r = 1. Proving a comparison theorem for virtual fundamental classes, we show that the full Seiberg-Witten invariants of ruled surfaces, as defined in [OT2], can be identified with certain gauge theoretical Gromov-Witten invariants of the triple (Hom(C, C r0), α can , U (1)). We find the following formula for the full Seiberg-Witten invariant of a ruled surface over a Riemann surface of genus g: 1 where [F ] denotes the class of a fibre. The computation of the invariants in the general case r > 1 should lead to a generalized Vafa-Intriligator formula for " twisted " Gromov-Witten invariants associated with sections in Grassmann bundles.