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

Okonek, C; Teleman, A (2002). Gauge theoretical equivariant Gromov-Witten invariants and the full Seiberg-Witten invariants of ruled surfaces. Communications in Mathematical Physics, 227(3):551-585.

## Abstract

Let F be a differentiable manifold endowed with an almost Kähler structure (J,ω), α a J-holomorphic action of a compact Lie group on F, and K a closed normal subgroup of 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
where αcan denotes the canonical action of on . 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< r 0),αcan, U(1)). We find the following formula for the full Seiberg–Witten invariant of a ruled surface over a Riemann surface of genus g:
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.

## Abstract

Let F be a differentiable manifold endowed with an almost Kähler structure (J,ω), α a J-holomorphic action of a compact Lie group on F, and K a closed normal subgroup of 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
where αcan denotes the canonical action of on . 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< r 0),αcan, U(1)). We find the following formula for the full Seiberg–Witten invariant of a ruled surface over a Riemann surface of genus g:
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.

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