Kobayashi-Hitchin correspondence for perturbed Seiberg-Witten equations
type: Item
instance of: scholarly article
title: Kobayashi-Hitchin correspondence for perturbed Seiberg-Witten equations
zbMATH Open document ID: 0922.32016
author: Ken-ichi Seimiya
published in: Tokyo Journal of Mathematics
publication date: 1999-09-30T00:00:00Z
review text: 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 t…
Mathematics Subject Classification ID: 32Q20
Mathematics Subject Classification ID: 58J99
Mathematics Subject Classification ID: 57R99
zbMATH DE Number: 1207084
zbMATH Keywords: Kähler surfaces
zbMATH Keywords: Seiberg-Witten equations
reviewed by: Steven B. Bradlow
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1270041986
OpenAlex ID: W2047925243
DOI: 10.3836/TJM/1270041986
description: scientific article; zbMATH DE number 1207084