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

摘要

Let F be a differentiable manifold endowed with an almost Kahler structure (J, omega), alpha a J-holomorphic action of a compact Lie group (K) over cap on F, and K a closed normal subgroup of (K) over cap which leaves omega invariant. The purpose of this article is to 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 Sigma. Our main results concern the special case of the triple (Hom ((C-r, C-r0), alpha(can), U(r)), where alpha(can) denotes the canonical action of (K) over cap = U(r) x 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 invariants explicitly 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), alpha(can), U(])). We find the following formula for the full Seiberg-Witten invariant of a ruled surface over a Riemann surface of genus g: SWX,(O1,H0)-sign[c,[f]](c) = 0, SWX,(O1,H0)sign[c, [F]](c) = sign[c, [F]] [Sigma(igreater than or equal tomax(0, g-wc/2))(g) Theta(c)(i)/i! boolean AND l, l(O1)], where [F] denotes the class of a fibre. The computation of the invariants in the general C, case r > 1 should lead to a generalized Vafa-Intriligator formula for "twisted" Gromov-Witten invariants associated with sections in Grassmann bundles. [References: 37]