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

Teleman, Andrei ; Okonek, Christian

HAL, hal-00881744 / Harvested from HAL

Résumé

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}),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: 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.

Détail

Publié le : 2002-07-05
Classification:  Gromov-Witten invariants,  Hamiltonian Gromov-Witten invariants,  Seiberg-Witten invariants,  rules surfaces,  53D45, 53C07, 32L05, 32L07, 32G15, 35Q40, 14D20,  [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG],  [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG],  [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG]