Self-dual instantons and holomorphic curves
type: Item
instance of: scholarly article
title: Self-dual instantons and holomorphic curves
zbMATH Open document ID: 0812.58031
DOI: 10.2307/2118573
published in: Annals of Mathematics. Second Series
publication date: 1995-05-16T00:00:00Z
review text: Let \(P \to \Sigma\) be a nontrivial \(\text{SO}(3)\)-bundle over a compact oriented Riemann surface of genus at least 2 and let \(f : P \to P\) be an automorphism, \(h : \Sigma \to \Sigma\) the induced diffeomorphism. In this situation two Floer type homology groups can be constructed. First one considers the moduli space \({\mathcal M}(P)\) of flat connections on \(P\). This is a simply connected compact manifold with \(\pi_ 2({\mathcal M}(P)) \cong \mathbb{Z}\). It carries a natural symplectic structure and \(f\) induces a symplectomorphism \(\phi_ f : {\mathcal M}(P) \to {\mathcal M}(P)\). Extending a construction of \textit{A. Floer} [Commun. Math. Phys. 120, No. 4, 575-611 (1989; Zbl 0755.58022)] one can define a chain complex generated by the fixed points of \(\phi_ f\), provided these are nondegenerate; otherwise one passes to a perturbation of \(\phi_ f\). These fixed points are the critical points of the (perturbed) symplectic action functional. The boundary operator is determined by certain gradient flow lines of this functional (holomorphic curves). The homology groups of this chain complex are denoted by \(HF^{\text{symp}}_ *({\mathcal M}(P),\phi_ f)\). One can also consider the induced bundle \(P_ f \to \Sigma_ h\) where \(P_ f\) and \(\Sigma_ h\) are the mapping cylinders of \(f\) and \(h\), respectively. This is a principal \(\text{SO}(3)\)-bundle over a compact oriented 3-manifold. The (perturbed) Chern-Simons functional can be used to define a chain complex generated by the critical points (flat connections modulo gauge equivalence). The boundary operator is again determined by certain gradient flow lines (self-dual Yang-Mills instantons on the 4-manifold \(\Sigma_ h \times \mathbb{R}\)); cf. \textit{A. Floer} [Commun. Math. Phys. 118, No. 2, 215-240 (1988; Zbl 0684.53027)]. The homology groups of this chain complex are denoted by \(H F^{\text{inst}}_ *(\Sigma_ h, P_ f)\). Main Theorem. There is a natural isomorphism of Floer homologies \[ HF^{\tex…
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 58E05
zbMATH DE Number: 640754
zbMATH Keywords: Floer homology
zbMATH Keywords: instanton homology
zbMATH Keywords: holomorphic curves
zbMATH Keywords: instantons
author: Stamatis Dostoglou
author: Dietmar A. Salamon
reviewed by: Thomas J. Bartsch
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/2118573
OpenAlex ID: W2077239907
Wikidata QID: Q126027949
description: scientific article; zbMATH DE number 640754