Kobayashi-Hitchin correspondence for perturbed Seiberg-Witten equations
On a four-manifold with the structure of a closed Kähler surface, the Seiberg-Witten equations acquire a holomorphic formulation and reduce to the so-called abelian vortex equations. The latter can be interpreted as equations for a Hermitian metric on a holomorphic line bundle, and a solution can be sought by conformal variation of a background metric. The equations then reduce to a non-linear partial differential equation which was first solved by Kazdan and Warner. In order to guarantee a smooth moduli space of solutions, it is necessary to perturb the original Seiberg-Witten equations. One method for doing this, introduced by \textit{E. Witten} [Math. Res. Lett. 1, No. 6, 769-796 (1994; Zbl 0867.57029)], is by the addition of a non-zero holomorphic 2-form. The perturbed equations can still be viewed as defining a Hermitian metric on a holomorphic line bundle, and can still be formulated in terms of a conformal variation of a background metric. The resulting PDE is a modified version of the Kazdan-Warner equation. The main result of this paper provides the details on how to adapt the Kazdan-Warner proof in order to solve this modified version. A similar, though less detailed, treatment can be found in the review article on `Seiberg-Witten invariants and vortex equations' by \textit{O. Garcia-Prada} [in `Quantum Symmetries', proceedings of Les Houches Summer School, Session LXIV, 1995, A. Connes, K. Gawedzki, and J. Zinn-Justin (eds.)]. The Kobayashi-Hitchin correspondence referred to in the title is a correspondence between an algebraic notion of stability (e.g. for a holomorphic bundle) and the existence of solutions to a gauge theoretic equation (e.g. the Hermitian-Einstein equations in the case of holomorphic bundles). Such a correspondence is known to apply to the vortex equations. However, when applied to a line bundle (i.e. a rank one bundle) it is not so interesting because in that case stability is automatic. The author alludes briefly to such aspects of the perturbed vortex equations, but does not add anything new on this topic.
scientific article; zbMATH DE number 1207084