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.

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.

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## Additional indexing

Item Type: | Journal Article, refereed, original work |
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Communities & Collections: | 07 Faculty of Science > Institute of Mathematics |

Dewey Decimal Classification: | 510 Mathematics |

Uncontrolled Keywords: | Gromov-Witten invariants; Seiberg-Witten type invariants |

Language: | English |

Date: | 2002 |

Deposited On: | 29 Nov 2010 16:27 |

Last Modified: | 05 Apr 2016 13:25 |

Publisher: | Springer |

ISSN: | 0010-3616 |

Additional Information: | The original publication is available at www.springerlink.com |

Publisher DOI: | https://doi.org/10.1007/s002200200637 |

Related URLs: | http://www.ams.org/mathscinet-getitem?mr=1910831 http://www.zentralblatt-math.org/zbmath/search/?q=an%3A1037.57025 |

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